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1 A GENERALIZED METHOD OF MOMENTS APPROACH TO SPATIAL DISCRETE CHOICE MODELS INVOLVING MICRO LEVEL DATA By LEDIA GUCI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF TH E REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011
2 2011 Ledia Guci
3 To my f amily and the memory of my uncle Terry A. Terezi
4 ACKNOWLEDGMENTS Th e completion of this dissertation was made possible wit h valuable contribution by many i ndividuals so I would like to take this opportunity to thank them all First and foremost I wish to extend my deepest gratitude to my dissertation committee chair Dr. Alfonso Flores Lagunes ; h i s guidance and encouragement went well beyond the mere supervision of a research project His h igh standards made me l earn to a ppreciate the rigor of research and find the motivation needed to accomplish this task I am especially indebted to Dr. Flores Lagunes for assisting in shapi ng my career path by providing me with o utstanding mentorship and nume rous professional opportunities I am also utterly grateful to my other supervisory committee members Dr. Charles Moss, Dr. Carmen Carrion Flores, and Dr. Dawn Jourdan for their constr uctive criticism regarding various aspects of the project w hich encourage d me to broaden my research perspective. In addition, I am very thankful to Dr. Ray Huffaker for his involvement and recommendations on earlier stages of this work Dr. David Brasingt on of the University of Cincinnati for compiling and supplying the data set used in one of the empirical studies and James Colee, a consultant with the IFAS Statistics Department, for his suggestions regarding the generation of the data for some of the Mo nte Carlo experiments. I feel fortunate to have collaborated with and learned from such dedicated scholars. However, i t goes without saying that I am solely responsible for any errors found in this manuscript. I would also like to express my appreciation to the Food and Resource Economics Department for the financial support that made my graduate studies possible Special thanks go to Mrs. Jessica Herman for her help with the graduate program logistics and
5 my professors and fellow graduate students for mak ing this pursuit of knowledge a memorable experience. Finally, and most importantly, I wish to extend my heartfelt thanks to my family and friends for providing the much needed moral support as they patiently waited for the completion of this work Specia l appreciation is offered to my grandmother, my parents, my sisters, and my fianc for sharing the burden as well as the excitement of this experience I can only hope that I have made you proud.
6 TABLE OF CONTENTS page A CKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBR EVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 2 SPATIAL LIN EAR REGRESSION MODELS AND DISCRETE CHOICE MODELS: A LITERATURE REVIEW ................................ ................................ ...... 18 Spatial Linear Regression Models ................................ ................................ .......... 18 Spatial Dependence and Spatial Heterogeneity ................................ ............... 18 Spatial Weights ................................ ................................ ................................ 21 Spatial Lag Dependence and Spatial Error Dependence ................................ 26 Diagnostic Tests for Spatial Dependence ................................ ......................... 32 Estimation of Spatial Linear Regression Models ................................ .............. 35 Maximum likelihood estimation ................................ ................................ .. 35 Instrumental variables/generalized method of moments estimation ........... 41 Economic Models of Discr ete Choice ................................ ................................ ..... 47 Random Utility Maximization Framework ................................ ......................... 49 Unordered choice alternatives ................................ ................................ ... 49 Ordered choice alternatives ................................ ................................ ....... 55 Estimation of Discrete Choice Models ................................ .............................. 57 Logit Based Models ................................ ................................ .......................... 60 Conditional, multinomial, and mixed logit models ................................ ...... 60 Independence of irrelevant alternatives (IIA) ................................ .............. 64 Ordered logit model ................................ ................................ ................... 68 3 AN ESTIMATION APPROACH TO DISCRETE CHOICE MODELS WITH SPATIAL LAG DEPENDENCE INVOLVING LARGE SAMPLES ............................ 70 Spatial Dependence in Discrete Choice Models ................................ ..................... 70 Spatial Logit Estimators for Large Samples ................................ ............................ 77
7 4 SPATI AL ANALYSIS OF LAND USE CONVERSION AT THE RURAL URBAN FRINGE ................................ ................................ ................................ .................. 88 Background ................................ ................................ ................................ ............. 88 Impacts of Urban Growth ................................ ................................ ........................ 92 Economic Model of Land Use Conversion ................................ .............................. 96 Econometric Model ................................ ................................ ............................... 100 Monte Carlo Analysis ................................ ................................ ............................ 102 Determinants of Land Use Choice ................................ ................................ ........ 107 Final Remarks ................................ ................................ ................................ ....... 117 5 NEIGHBO RHOOD INFLUENCE AND PUBLIC POLICY: ADOPTION OF OPEN ENROLLMENT POLICIES BY SCHOOL DISTRICTS IN OHIO ........................... 134 Background ................................ ................................ ................................ ........... 13 4 Interdist rict Open Enrollment in Ohio ................................ ................................ .... 141 Estimation Framework ................................ ................................ .......................... 147 Data ................................ ................................ ................................ ...................... 150 Estimation Results ................................ ................................ ................................ 154 Final Remarks ................................ ................................ ................................ ....... 161 6 CONCLUSIONS AND FUTURE WORK ................................ ............................... 171 APPENDIX A LOGIT CHOICE PROBABILITIES AND MARGINAL EFFECTS ........................... 175 Derivation of Logit Probabilities ................................ ................................ ............ 175 Marginal Effe cts and Elasticities ................................ ................................ ........... 178 B DERIVATION OF GRADIENTS FOR SPATIAL LOGIT MODELS ........................ 181 Spatial Mixed Logit (SMXL) Model ................................ ................................ ........ 181 Spatial Multinomial Logit (SMNL) Model ................................ ............................... 188 Spatial Conditional Logit (SCL) Model ................................ ................................ .. 190 Spatial Ordered Logit (SOL) Model ................................ ................................ ....... 192 C DESCRIPTION OF MONTE CARLO EXPERIMENTS ................................ .......... 198 Scenario 1: SMNL Model with Equal C hoice Probabilities ................................ .... 198 Scenario 2: SMNL Model with Different Choice Probabilities ................................ 199 LIST OF REFERENCES ................................ ................................ ............................. 203 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 222
8 LIST OF TABLES Table page 4 1 Simulation results for the SMNL model with e qual choice probabilities (Sample size: 1,000 observations) ................................ ................................ ... 123 4 2 Simulation results for the SMNL model with equal choice probabilities (Sample size: 5,000 observations) ................................ ................................ ... 124 4 3 Simulation results for the SMNL model with different choice probabilities (Sample size: 1,000 observations) ................................ ................................ ... 125 4 4 Simulation resu lts for the SMNL model with different choice probabilities (Sample size: 5,000 observations) ................................ ................................ ... 126 4 5 Land use conversion model: variable description ................................ ............. 127 4 6 Land use conversion model: variable descriptive statistics .............................. 127 4 7 Proportion of parcels in each land use category ................................ ............... 127 4 8 Estimated coefficients of the land use change model ................................ ....... 128 4 9 Marginal effects of the estimated coefficients of the land use change model ... 130 5 1 School districts in each enrollment category ................................ .................... 163 5 2 Policy adoption model: variable description ................................ ...................... 163 5 3 Policy adoption model: variable descriptive statistics ................................ ....... 164 5 4 Estimated coefficients of the policy adoption model ................................ ......... 165 5 5 Marginal effects for the estimated coefficients of the policy adoption model .... 167
9 LIST OF FIGURES Figure page 4 1 Medina County, Ohio ................................ ................................ ........................ 122 4 2 Land use changes in Medina County, Ohio (1970 2000) ................................ 122 5 1 Interdistrict open enrollment policies in Ohio ................................ .................... 162 5 2 ................................ ................................ ... 162
10 LIST OF ABBREVIATION S CDF Cumulative distribution function ( of a random variable) CL Conditional logit (model) CLR Classical linear regression ( model ) FG LS Feasible generalized least squares (estimation) GLS Generalized least squares (estimation) GMM Generalized method of moments (estimation) IIA Independence of irrelevant alternatives (property) IID Independent and identically distributed (random variable) IV Instrumental variable (estimation) LM Lagrange multiplier (test) LR Likelihood ratio (test) ML Maximum likelihood (estimation) MNL Multinomial logit (model ) MXL Mixed (multinomial) logit (model) OL Ordered logit (model) OLS Ordinary least squares (estimation) PDF Probability density function (of a random variable) RMSE Root mean squared error (statistic) RUM Random utility maximization (framework) SAE Spatia l autoregressive error (model) SA L Spatial autoregressive lag (model) SAL SAE Spatial autoregressive model with spatial autoregressive errors SCL Spatial conditional logit (model) SMNL Spatial multinomial logit (model)
11 SMXL Spatial mixed logit (model) SOL Spatial ordered logit (model) TSLS Two stage least squares (estimation)
12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philos ophy A GENERALIZED METHOD OF MOMENTS APPROACH TO SPATIAL DISCRETE CHOICE MODELS INVOLVING MICRO LEVEL DATA By Ledia Guci August 2011 Chair: Alfonso Flores Lagunes Major: Food and Resource Economics Many economic problems that require micro level analys is within a discrete choice framework are fundamentally spatial processes. Under these circumstances, the estimation of standard discrete choice models results in in consistent parameter estimates While several estimation methodologies have been developed for spatial discrete choice models, their implementation become s i nfeasible in empirical applications involving micro level data This dissertation extends an esti mation methodology for discrete choice models with spatial lag dependence that is computatio nally feasible with large samples The methodology consists of linearizing the original spatial model around a convenient point of parameter values and estimating the linearized model using a generalized method of moments (GMM) approach The model lineariz ation avoids the repeated matrix inversion involved in GMM estimation which hinders its use with large samples and breaks up the estimation procedure into two steps: a standard discrete choice model followed by two stage leas t squares estimation Spati al estimators based on this methodology are derived for various polychotomous logit models T he performance of the proposed methodology in finite samples is asse ssed using Monte Carlo methods.
13 Simulation result s indicate that the linearized model provides a good approximation to the original spatial model for a reasonable range of induced spatial dep endence in the simulated data. The proposed spatial estimation approach is employed in t wo empirical studies The first study uses parcel level data to estimat e a spatially explicit model of land use conversion occurring at the rural urban f ringe. The results from this study corroborate previous research findings regarding the tendencies of new urban development. Moreover the results provide new evidence of sig nificant spatial dependence in land use conversion decisions T he presence of spatial spillover effects suggests that policies designed at a small scale could lead to sub optimal land use pattern s The s econd study examines adoption decis ion s of interdistrict open enrollment policies using a spatially explicit model of policy adoption to accommodate for potential influence from neighboring districts The empirical results s ubstantiate extant descriptive evidence regarding the determinants of adoption of open enrollment policies. I mportantly the results show strong neighborhood influence in policy adoption decisions calling attention to the critical role that r eference groups play in shaping behavior toward the adoption of new policies
14 CHAPTER 1 INTRODUCTION The economic behavior of decision makers faced with discrete choice decisions that involve various alternatives is often influenced by the behavior of decision makers in reference groups n eighbors, peers, or other individuals that face similar decisions because it is costly to evaluate all alternatives T he economic s literature has documented numerous settings that have prompted such interdependence in decision making Examples include technology adoption decision s (e.g. Katz and Shapiro, 1986; Case, 1992), financial investment decisions (e.g. Scharfstein and Stein, 1990), firm use conversion decisions (e.g. Irwin and Bockstael, 2004; Zhou and K ockelman, 2008), firm location decisions (Klier and McMillen, 2008), etc. Thus, modeling individual discrete choice decisions often requires accommodating for interdependence in decision making which involves specifying and estimating spatial discrete c hoice model s A s patial discrete choice model formalizes the spatial relationships that exist between the decision makers by defining a spatial structure in the form of a spatial weight s matrix that expl icitly links each of the observations in the sample and assigns spatial weights based on the strength of the potential interaction between observations at different locations. In the presence of such spatial interactions, t he estimation of standard discrete choice models i s problematic due to a number of ec onometric challenges such as spatial heteroskedasticity and autocorrelation, spatial heterogeneity, and selection bias. Thus, t raditional modeling methods generally lead to inconsistent estimates and are inappropriate for hypothesis testing and prediction (McMillen, 1992).
15 The estimation of spatial discrete choice models is challenging because of nonlinear optimization procedures involved in the estimation of discrete choice models which are further complicated by spatially dependent observations. In addit ion, t he sample size plays a critical role T he maximum likelihood procedure, typically employed in discrete choice model estimation, becomes increasingly cumbersome with larger samples because the likelihood function i nvolves numerous integrals under non spherical disturbances G eneralized method of moments estimation become s infeasible as well in large data sets because it involves i terative numerical optimization process es which require repeated inversion o f large matrices Several spatial discrete choi ce estimators in the literature provide consistent estimates under spatially dependent data T heir application however, has been limited to the binary choice case and small sample sizes These estimators attempt to preserve the estimation structure implie d by maximum likelihood by either making simplifying assumptions about the spatial weight s matrix or directly simulating the choice probabilities under a set of assumptions for the error terms (e.g. Case, 1992; LeSage, 2000; McMillen, 1992, Beron and Vijve rberg, 2004). More recent attempts to incorporate spatial dependence in a multinomial setting follow a Bayesian framework (e.g. Kakamu and Wago 200 5; Wang and Kockelman 2009 ; Chakir and Parent, 200 9 ) T hese approaches are computationally intensive as w el l in moderate to large sample sizes This dissertation develops a n estimation methodology for cross sectional polychotomous choice models with spatial lag dependence that is computationally feasible in large samples The methodology an extension of the methodology developed by Klier and McMillen (2008) for the estimation of a spatial binary logit model
16 consists of linearizing the original spatial model around a convenient point of initial parameter values and estimating the linearized version of the sp atial model The extension of methodology is possible because t he key result that enables the linear approximation of the spatial model in a binary choice case holds in a multinomial setting. The model linearization avoids the r epeated matrix inversion involved in GMM estimation and breaks up the estimation procedure into two simple sequential steps: a standard polychotomous choice model with no spatial dependence followed by a two stage least squares estimation of the linearized spatial model which accounts for the spatial dependence. Spatial estimators based on this linearization methodology are derived for various polychotomous models, a ll of which have a wide applicability in analyzing economic decisions. Because the se spatia l estimators are based on linearized version s of the original spatial model s how well the y perform depends on how well the former model s approximate the latter Thus, t he performance of the spatial estimator s in finite samples is asse ssed in a controlled setting using Monte Carlo methods. Simulation results indicate that the linearized spatial model s provide a good approximation to the original spatial model s and produce fairly accurate estimates for a reasonable range of induced spatial dep endence in the simulated data. Judging from these results, the linearization approach appears successful. However, t he ultimate assessment of the use and performance of the proposed estimation method rests in the empirical application s T he proposed spatial estimation a pproach is used in two empirical studies. The first study involves a spatial analysis of land use conversion decisions. The main objective of this study is to understand what drives land use conversion and
17 identify th ose factors that play a sig nificant role in the conversion of land to urban use. It uses parcel level data from Medina County, Ohio, to estimate a spatially explicit model of land use conversion occurring at the rural urban f ringe. The focus of the second study is neighborhood influ ence in the adoption of public policies For this study school district data are used to examine the neighborhood influence in the adoption of interdistrict open enrollment policies by school districts in Ohio. Both studies find significant spatial depende nce in the discrete choice decisions under consideration Knowledge of the se spatial effects i s of value to inform p olicy The outline of this dissertation is as follows. Chapter 2 provides an overview of the literature on spatial linear regression models and discrete choice models t o lay the foundation necessary for the development of the estimation methodology proposed in the next chapter. Chapter 3 reviews the literature on spatial nonlinear models and discusses the challenges that arise when estimating discrete choice models in the presence of spatial effects. It then proposes an estimation approach for a class of spatial multinomial logit model s that is computationally simple In C hapter 4 the proposed methodology is first assessed using Monte Carlo ex periments and then used to empirically estimate a spatially explicit model of land use conversion Chapte r 5 examine s the neighborhood influence on adoption decision s of interdistrict open enrollment policies using a spatially explicit mo del of policy adoption The last chapter provides final remarks and outlines venues of future work
18 CHAPTER 2 SPATIAL LINEAR REGRE SSION MODELS AND DIS CRETE CHOICE MODELS: A LITERATURE REVIEW Spatial Linear Regression Models Spatial D ependence and Spatia l H eterogeneity Spatial effects are present in various empirical applications in which observations in the data sample are arranged in space 1 Prominent e xamples include empirical applications that utilize data on population, employment, or other economic activities recorded over administrative units such as states, counties, districts, etc. Based on spatial features of the data the spatial econometric literature distinguishes between t wo types of spatial effects: spatial dependence and spatial heterogenei ty 2 Spatial dependence is defined as the existence of a functional relationship bet ween what happens at one point in space and what happens elsewhere (Anselin, 1988 a p a g e 11 ) Anselin and Bera (1998) define spatial dependence as the coincidence of va lue similarity with location similarity T he values of a random variable exhibit a pattern over space as values associated with two locations that are cl oser tend to be more correlated either positively or negatively than the ones associated with loca tions that are further apart. S imilar values of a random variable th a t tend to cluster in space are an indication of positive spatial dependence. In contrast, ne gative spatial dependence is present when a random variable takes on dissimilar values for 1 The concept of space is not confined to geographic space. It can be generalized to include policy space, economic space, technological space, social networks, etc. Refer to Isard ( 1969) for further discussion. 2 In the spatial econometrics literature, the terms spatial dependence and spatial autocorrelation are often used synonymously. Strictly speaking, spatial dependence implies a multidimensional relationship of the observations in the sample which is e xpressed by a joint density function, whereas spatial autocorrelation is a process characterized by moments of this joint distribution. The latter, a weaker expression of the former, is the focus of most applications. Spatial dependence and spatial autocor relation are used interchangeably throughout this manuscript.
19 obse rvations in neighboring locations resulting in dispersed patterns P ositive spatial dependence has a more intuitive interpretation and it is the kind of spatial dependence most commonly encountered in empirical applications. Other formal definitions of sp atial dependence are also available in the literature (e.g. Griffith, 1992; Hepp le 2000). Consider the classical linear regression (CLR) model : (2.1) where is the dependent variable ; denotes the ex planatory variable s ; is a vector of stochastic errors ; and represents the model parameter s In this model, s patial dependence between two observations correspond ing to locations and can be formally expressed as 3 : (2.2) Note that the o bservations and in (2.2) do not obey the structure of the data generating process of a cross sectional data sample of independent observations implied by the linear regression model in (2.1) Thus the estima tion of the model by means of ordinary least squares (OLS) generally leads to biased and i nconsistent estimates and invalid statistical inference. These undesirable properties of the OLS estimator have led to the development of specialized estimation techn iques that explicitly account for the structure of spatial dependence. A discussion of t hese estimation methods follows. S patial heterogeneity on the other hand, is defined as the lack of structural st ability (or uniformity) of spatial relationships over space (Anselin, 1988 a ). In this case, the spatial process differs across spatial units resulting in spatial effects that vary over 3 In terms of the moment condition, spatial autocorrelation is formally defined as:
20 space As a result, the estimated spatial effects over all locations m ay not adequately describe the spatial process at a pa rticular location. A general case of spatial heterogeneity in a CLR model can be expressed as : (2.3) w here the model parameters imply the existence of a different linear relationship between the dependent variable and the explanatory variables for every observation in the sample 4 In general problems caused by spatial heterogeneity (e.g. spatial heteroskedasticity) can be addressed with standard econometric techniques (Anselin, 1988a). Thus, th is review on spatia l econometric models as well as the research presented in this dissertation focu s exclusively on issues of spatial dependence. In cross sectional data, both types of spatial effects may appear alike For instance, an observed spatial cluster of outliers in OLS residuals could be a case of spatial heteroskedasticity or it could be due to a spatial process that generated the residual cluster, thus a case of spatial autocorrelation. The problem of discerning between these effects (Anselin and Bera, 1998) Moreover, s patial dependence and spatial heterogeneity may co exist I n this case distinguishing between the two types of spatial effects is especially complex. The spatial econometric s literature deal s with the se spatial effects by incorporating the m explicitly in the model and testing for the specification of the spatial model 5 4 In this case, s between t he dependent variable and the explanatory variables for all th e observations in the data sample. It may also involve a different functional form of this relationship for different observation s. 5 This model driven approach is what distinguishes the two closely related fields of spatial econometrics and spatial stati stics. The latter adopts a data driven approach in which randomness of spatial effects is assumed and the spatial patterns and interactions are derived from the data. In contrast, the model driven
21 Spatial Weights A nalysis of spatial dependence r equires an explicit expression of the spatial pattern which quantif ies t he extent and the strength of spatial interactions b etween spatial unit s (observations) A cross section al data sample with observations contains insufficient information for the estimation of the parameters corresponding to all spatial interactions 6 Thus, it is necessary to impose a structure on these interactions and express them as a function of a small er number of param eters that can be estimated 7 (Anselin, 1988 a ) T h e structure of spatial dependence is formally expressed by defining a spatial weight s matrix that links all the observation s in the samp l e This is done by assuming that spatial dependence arise s w ithin a of data points surrounding the spatial unit of interest Thus, t he spatial weight s matrix contains non negative constant elements for those observations that are defined as neighbors based on some criterion that properly describe s t he nature of the interaction process. In addition, the spatial weights matrix is symmetric and, by convention, the diagonal elements are set to zero ( indicating that an observatio n is not a neighbor to itself. Typically the spat ial weights are assumed to be exogenous and determined a priori. approach starts with a theory and is eventually confronted with the data, but focuses on model estimation, specification testing, and statistical inference in the presence of spatial effects (Anselin, 1988a). 6 Spatial dependence implies the existence of a covariance (correlations) matrix; hence, it requir es estimation of up to parameters with observations. 7 The structure on the spatial interactions can be imposed based on t wo approaches The geostatistical or ons as a continuous function of the distance between their locations. The lattice the correlations indirectly from a particular spatial stochastic process specified to describe the interaction process. The lattic e approach is more appropriate for economic analyses where economic agents are associated with discrete locations in space (Anselin, 2002). For this reason, the lattice approach is reviewed in this section and adopted in the empirical studies presented in the subsequent chapters.
22 The origin al form of the spatial weights matrix ( by Moran 1948 ; Geary 1954) is based on the concept of binary co ntiguity between two spatial units; is set equal to one if two spa tial units share a common border ( i.e. ar e contiguous ) and zero otherwise. This definition of contiguity assumes the existence of a map like layout with distinguishable boundaries and works well for areal spatial units with irregular boundaries. When spati al units are organized in a regular lattice a common border may be defined as a common edge, a common vertex, or a combination of both, resulting in different definitions of contiguity These definitions of contiguity are known as the rook, the bishop, an d the queen respectively. Illustrations of the various definitions of contiguity are provided in Anselin (1988 a ). Spatial units may also consist of points (rather than areas) that are arranged in space. In this case, areal spatial units are generated by u sing various tessellations which partition the space into polygons that correspond to the location of the point s 8 In addition to simple contiguity, n eighbors can be defined based on h igher orders of contiguity ; th order contiguity results from spatial units that are first order contiguous to a th order contiguous spatial unit but not contiguous to spatial units of lower order contiguity than In a regular lattice, neighbors based on higher orders of c ontiguity can be visualized as a series of concentric bands around the spatial unit of interest (Anselin, 1988a). A nother criterion commonly used to specify the spatial weights is based on the distance between two spatial units ; is set equal to one if location and are contained within a specified distance threshold and zero otherwise. To measure the 8 Thiessen polygons (also known as Dirichlet polygons or Voronoi polygons) are the most common types of spatial tessellations. See Ripley (1981), Amrhein et al (1983), and Upton and Fingleton (1985) for an overview.
23 strength of the potential interaction between spatial units, s patial dependence is assumed to decrease with distance and the spatial weights ar e assigned according to an inverse function of distance ; where is the distance between observations in locations and Measures of di stance can be defined based o n geographical information ( e.g. Euclidean distanc e t ravel time distance ) as well as based on socio economic indices. For instance, Case et al. (1993) defined between counties based on differences in per capita income and proportion of black population Doreian (1980) specifie d spatial weight s based on in dividuals social network s More general spatial weights matrices combine distance and contiguity relationships. For instance, Cliff and Ord (1973, 1981) combine d a distance measure with the length of the border shared between sp atial units t o capture the physical characteristics of the spatial unit. Their criterion results in an asymmetric spatial weight s matrix with elements where is the distance between location and i s the share of the boundary of location contiguous to and and are parameters ( set a priori ) S imilar weighing schemes can be found in Dacey ( 1968 ), B odson and Peeters ( 1975 ) etc Despite how the spatial weights are specified the resulting sp atial process must satisfy some regularity conditions in order to obtain desirable properties of estimators and test statistics 9 (Anselin and Bera 19 9 8 ). The spatial weights matrix is often s tandardized such that the sum of row elements equals one. R ow s tandardization is mainly done for statistical reasons so as to constrain the spatial autoregressive parameters to ensure that the likelihood function of 9 These are known as the mixing conditions; the spatial weights must be nonnegative, finite, and generated by a proper metric. In addition, they have to be exogenous (Anselin, 1988a).
24 the spatial model is well behaved over a non continuous parameter space 10 In addition, it ensures that all the spatial weights are between zero and one which facilitates the interpretation of the and makes the parameters of different spatial stochastic processes comparable between models (Anselin and Be ra, 1998) Row standardizing, however, may change the structure of the assumed interdependence because it generally rescales each row of the matrix differently As a result, the relative dependence among neighbors does not change, but the total impact of neighbors across observations changes (Bell and Bockstael, 2000). In this case, the economic interpretation of the total spatial impacts sh ould be made with caution, especially for distance decay type matrices because the total impact of neighbors varies over observations depending on the density and the spatial pattern that surround s the spatial unit (observation) of interest 11 To avoid the interpretation problems from row standardization of distance decay spatial matrices, a series of higher order contiguity matrices of a more flexible form have been developed (Plantiga and Irwin, 2006). Row standardization may also complicate the model est imation since the spatial weights matrix may become asymmetric after standardization In practice, a number of issues may arise when generating spatial weights based on any of the criteria described above. The contiguity criterion may result in different q uantitative representations of the spatial relationships for the same spatial pattern. The distance criterion based on geographic information cre ates problems when there is a 10 The likelihood function is well defined over the parameter space when the spatial autoregressive parameters are constrained within the interval and where and are the smallest and the largest eigenvalues of respectively (Anselin and Bera, 1998). 11 A discussion on the sensiti vity of model results to both specification and row standardization of is provided in Plantiga and Irwin (2006) and Bell and Bockstael (2000)
25 high degree of heterogeneity in the distribution of spatial unit s or the size o f spatial regions because it results in a highly disproportionate number of neighbors for smaller spatial units compared to larger spatial units. A lternative measures of distance such as may produce weights that are not meaningful For i nstance, a metric based on inverse distance produces weights that are undefined when two individuals have identical socio economic indicators In addition, the spatial weights are unlikely to be exogenous when the v ariables used to generate the weights are included in the model An selin (2002) provides a thorough discussion and provides examples for each of these issues. It becomes evident from this discussion that t he specification of the structure of the spatial dependence is criti cal to the model estima tion First, the specification of the spatial weights matrix is made a priori and it is not a testable Once specified, the model estimation is carried out under the assumption that the spatial weights matrix represents the true spatial structure of the ne ighborhood T hus t he estimates are valid as long as the spatial structure is correctly reflected in the spatial weights. Second, d ifferent specifications of the spatial dependence yield d ifferent covariance structures (Anselin, 200 2 ) As a result m odel e stimates may be sensitive to the s pecification of the spatial weights matrix. For this reason, it has become customary for r esearchers to conduct sensitivity analys e s by specifying several spatial weights matrices based on different metrics and neighborhoo d sizes For any particular empirical application, t here is little formal guidance for the proper c hoice of the spatial weights matrix. T h is choice depends on the nature of the problem being modeled and any information on the problem that might be availab le.
26 Anselin (1988 a ) argues that the choice of spatial weights should have a direct link with t he theoretical conceptualization of the structure of dependence, rather than describe the spatial pattern ad hoc P roper specification of the spatial weights matr ix however, is complex and remains one of the most controversial methodological issues in the analysis of spatial data Spatial Lag Dependence and Spatial Error Dependence In applied work, s patial dependence is generally an outcome of two conditions : m ea surement error or the existence of socio economic phenomena that involve spatial interaction a nd result in spatial externalities and spatial spillover effects (Anselin, 1988a) Spatial dependence as a byproduct of measurement error arises when s patial data boundaries do not coincide with the spatial scale of the phenomenon under study when a ggregating spatial data, or when combining data samples defined at different spatial scales. As a result, errors from neighboring locations are correlated resulting in spatial autocorrelation. I n this case spatial dependence is considered a nuisance and the main objective of the spatial analysis is to obtain proper statistical inference When spatial interaction characterizes the socio economic p henomenon under study t he purpose of modeling spatial dependence is to understand the nature of the spatial interaction and the underlying economic and social process es that generated th e interaction In this case, spatial dependence is in itself of interest to explain economic behavior, hence considered a substantive S patial dependence is typically modeled by specifying a relationship between the dependent variable or the error term a nd the corresponding spatial lag s, and The resulting model specifications are known as the spatial autoregressive lag ( SAL ) and spatial autoregressive error (SAE) models, respectively.
27 Spatial dependence in the dependent variable known as spatial lag dependence is analogous to the inclusion of a serially autoregressive term of the dependent variable in a time series model (Anselin and Bera, 1998). In this model, the value of the dependent variable corres pondi ng to each cross sectional unit is assumed to depend in part on the weighted average of the values of the dependent variable corresponding to neighboring cross sectional units. The SA L model takes the form: (2.4) w here is the dependent variable ; is a spatially lagged dependent variable ; is a matrix of explanatory variables ; is a vector of independent and identically distributed (IID) ho moskedatic error terms ; and is the spatial autoregressive parameter. Anselin (1993) refers to this model as a spatial autoregressive model with substantive spatial dependence. The presence of the spatial lag as a n explanatory variable in the model induces correlation with the error terms similar to an endogenous variable. In fact, the spatial lag for a particular observation is correlated not only with the respective error term but also with the error terms of all other observations Estimating a non spatial model thus, ignoring the presence of the spati ally lagged dependent variable results in a mi sspecification error akin to an omitted variable (Anselin and Bera, 1998). Consequently the OLS paramete r estimates are biased and inconsistent So me exceptions are discussed in Lee (2002) 12 The model in the reduced form becomes: (2.5) 12 The results in Lee (2002) are generated by a very special structure of the spatial weights matrix.
28 with the following covariance matrix : T he covariance matrix indicates a global dependence structure; ev ery observation is correlated with every other observation in the sample but the strength of the interaction decays with distance ( or order of contiguity ) following Tobler s (1970 ) first law 13 This relationship is made more explicit by expanding the spatial multiplier as follows 14 : (2.6) This is the reason that despite the similarity of the SA L mod el with a time series model the properties of the OLS estimat or in the former model do not extend to the latter In a time series m odel the lagged dependent variable is uncorrelated with the error term in the absence of serial co rrelation in the errors and consistent estimates can be obtained by OLS 15 (Anselin, 1988 a ) This is not the case for the SA L model regardless of the properties of the error term Anselin (1988 a ) attributes th is lack of analogy in the properties of the OLS e stimat or to the two dimensional and multi directional nature of spatial dependence Due to the endogeneity induced by the spatial lag variable this model can be consistently estimated in a maximum likelihood framework or by means of instrument al variables. The interpretation of the spatial autoregressive parameter in the SAL model depends on th e source of spatial dependence. When spatial interaction is a feature of the economic behavior under study a statistically significant estimate for the spatial autoregressive parameter indicates true contagion or substantive spatial 13 Toble things" (Tobler, 1970). 14 is known as a spatial multiplier because, as (2.6) shows, a change occurring in a particular location aff ects first order neighbors, second order neighbors, third order neighbors, and so on. 15 In this case the covariance matrix is triangular rather than a full matrix (Anselin and Bera, 1998).
29 dependence (Anselin and Bera, 1998) In other words, it measures the extent of spatial externalities or spatial spillovers. Examples in the applied economics literature include among others, farmer s hnology adoption behavior in Case (1992), state expenditures and tax setting behavior in Case et al. (1993), and strategic interaction among cities in the choice of growth controls in California in Brueckner (199 8 ). If the spatial scale of the data does no t match the spatial scale of the phenomenon under study, the SA L model can be used to filter out t he spatial autocorrelation due to the scale mismatch 16 In this case, the spatial autoregressive parameter is a nuisance parameter, but allows for proper inter pretation of the significance of the other model parameters. The SA L model in this context has been used for instance, to model urban housing and mortgage markets since they operate at different scales (e.g. Can, 1992; Can and Megbolugbe, 1997; Anselin an d Can, 1996). Spatial dependence in the error term known as spatial error dependence, is typically modeled under the assumption that the error term follows a spatial autoregressive process 17 The SAE model can be written as follows: (2.7) where is the dependent variable ; is a matrix of explanatory variables ; is the spatial error lag ; is a vector of uncorrelated and homoskedastic error terms ; and is the spatial autoregressive parameter This model is appropriate when there is no theoretical anticipation of spatial interactions among economic agents and it is useful for 16 The SAL model can be expressed as: wh ere is a spatially filtered dependent variable (Anselin and Bera, 1998). For alternative approaches to spatial filtering see Getis (1995). 17 Other specifications of the spatial autoregressive error process can be found in Cliff and Ord (1 981), Haining (1988, 1990), Kelejian and Robinson (1993, 1995). These specifications, however, have seen limited applications.
30 correcting or filtering out the spatial autocorrelation present in spatial da ta 18 The spatial autocorrelation may be a result of measurement error or unobserved variables that may spillover across spatial units but are otherwise not essential to the model (Anselin and Bera, 1998). The model in the reduced form becomes: (2.8) which results in the following covariance matrix : This covariance matrix highly resembles the one in the SA L model indicating a global dependence structure that decays wit h distance or order of contiguity Th e similarity between the two covariance matrices complicates specification test ing to properly discriminate betwee n spatial error dependence and spatial lag dependence. We turn to this issue shortly. T he estimated coeff icients using OLS are in general unbiased but inefficient due to the non spherical disturbance s These results are more in line with the properties of the OLS estimat or in time series models. More efficient estimates can be obtained by using robust methods that explicitly take into account the structure of spatial error dependence In this model, the spatial autoregressive parameter is interpreted as a nuisance parameter ( e.g. Benirschka and Binkley, 1994). Anselin (1981) shows that OLS estimation cannot produce a consistent estimate of the spatial autoregressive parameter ; thus, t he SAE mode l is also estimated with alternative estimation frameworks such as maximum l ikelihood T he consequences of ignoring spatial error dependence are generally argued to be less severe than i gnoring spatial 18 The SAE model can be expressed as: where and are the spatially filtered dependent variable and spatially filtered independent variables, respectively, and is the uncorrelated error term (Anselin and Bera, 1998).
31 lag dependence because while t he former invo lves stat istical considerations, the latter involves theoretical considerations (LeSage, 1997 a ). A more general spatial process combines both types of spatial dependence This model is known as a spatial autoregressive model with autoregressive disturbances ( SA L S AE ) The model can be written as follows 19 : (2.9) where is the dependent variable ; is the matrix of explanatory variables ; and are the spatial lag s ; is a vector of uncorrelate d and homoskedastic error terms ; and and are the spatial autoregressive parameter s. The spatial weights matrices and are generally assumed to be different to allow for two different spatial processes. If complications arise in the identification of the spatial autoregressive parameters, and Anselin (1980 ) shows that, in this case, and are identified only when a set of nonlinear constraints is strictly enforced. The red uced form model can be written as follows: (2.10) which generates a more complex covariance structure: Empirical a pplications of th e SAL SAE model are found in Case (1987,1991, 1992), Case et al. (1993), Besley and Case (1995), etc. Anselin and Bera (1998) poi nt out that the need for higher order processes can be considered to be more a result of a poorly specified spatial weights matri x than due to a more complex spatial data generating process For instance, if the spatial weights matrix for a spatial lag process does not 19 A more general but highly complex spatial process is the spatial autoregressive moving aver age (SARMA). Refer to Huang (1984) for a treatment of the SARMA process. Applications of this process can be found in Brandsma and Ketellapper (1979) and Blommestein (1983, 1985).
32 capture all the dependence in the data, it will result in spatial error autocorrela tion. Florax and Rey (1995) investigate the effects of misspecified weights in spatial linear regression models. The SA L and SAE models can be derived as special cases of the more general SA L SAE specification b y adding some restrictions to the parameters These models make the standard taxonomy of spatial linear regression models. A richer taxonomy of spatial linear models is developed in Anselin (2003) based on t he distinction between global and local dependence and the way in which it translate s into th e incorporation in a regression specification of a spati ally lagged dependent variable spatiall y lagged explanatory variables an d spatially lagged error terms T he relevance of a ny taxonomy however, depends o n its empirical applications Diagnostic Tes ts for Spatial Dependence While a theoretical argument should suggest the nature of spatial dependence to be specified in a model a number of diagnostic tests are a lso available in the literature A general test for spati al error dependence is Moran I Th is test, an extension of the test statistic in Moran (19 50 ), was presented b y Cliff and Ord (1972) The Moran I test statistic takes the form: (2.11) where is a vector of OLS residuals ; is the spatial weights mat rix ; and is a standardization factor ( 20 The test was initially suggested by Moran as a simple test for correlation between nearest neighbors, which explains the similarity of 20 For a row standardized spatial weights matrix, thus the Moran I test statistic simplifies to:
33 this test statistic with the Durbin Watson statistic (Anselin and Bera, 1998). A standardized standard normal distribution, thus the Moran I test is implemented based on an asymptotically normal standard z value. The test i s easy to i mplement because it only depends on OLS residuals But, it has low power because it is a general test ; it does not test against a specific alternative. A similar general test can be found in Kelejian and Robinson (1992). Kelejian and Prucha (2001) derive and examine the large samp le distribution of Moran I type test statistics for the SA L SAE model and a variety of limited dependent variable models. An alternative to the general tests discussed above is one which states explicitly an alternative hypothesis based on the data genera ting process. For instance, given the spatial process: we can test for versus Any of the testing principles Likelihood Ratio (LR), Wald, and Lagrange Mul tiplier (LM) can be used. These testing principles result in tests that are asymptotically equivalent. In practi ce, the LM test is more appealing because its implementation requires only OLS estimation under the null hypothesis. The Wald and LR tests require maximum likelihood (ML) estimation under the alternative hypothesis and the functional forms of these tests a re more complex. Interestingly, Monte Carlo evidence (e.g. G odfrey 1981; Bera and McKenzie 1986 ) suggest that the LM is not inferior to the LR or Wald test with respect to power although it does not use the information in the alternative hypothesis. An LM test for spatial error autocorrelation was suggested by Burridge (1980). The test takes the form: (2.12)
34 where represents the trace operator ; is a vector of OLS residuals ; is the spatial weights matrix ; and is an estimate for the error variance ( Under the null hypothesis of no spatial dependence this test statistic follows a distribution. This test is designed t o test a single specification, namely spatial error dependence, assuming this is the correct specification of the model. In the presence of spatial lag dependence ( ) this test is not valid even asymptotically (Anselin and Bera, 1998). A similar LM te st for testing spatial lag dependence was suggested in Anselin (1988 a ). Given the model specification: the test is formulated as : versus In this case several tests based on the ML principle are available as well, but the LM test is more attractive because it only requires OLS residuals. The test statistic takes a more complex form : (2.13) where denotes the trace operator ; is the spatial lag ; is a vector of model parameters ; and This test statistic also follows a distribution under the null hypothesis of no spatial depe ndence. Similar to the test for spatial error dependence this test is valid only if the assumed specification of the model is the correct specification. Anselin (1993) suggests that the type of spatial dependence can be diagnosed by comparing the LM test s tatistics in (2.12) and (2.13) The test with the most significant value tends to point to the correc t type of spatial dependence. We can also test for concurrent spatial lag and spatial error dep endence. Anselin(1988b) suggested two approaches. The first approach involves estimating an SA L SAE model and t est ing jointly for An LM test for this approach based on OLS residuals is developed in Anselin (1988b). A shortcoming of this test is that if the
35 null hypothesis is rejected, it is not possible t o infer whether the model misspecification is a result of spatial lag dependence or spatial error dependence. Another approach is to test for spatial error dependence in the presence of spatial lag dependence when to determine whether the SA L model fully accounts for the spatial dependence, or alternatively test for spatial lag dependence in the presence of spatial error dependence when to determine if the SAE model accounts for all spatial dependence. The LM tests proposed in Anselin (1988b) for t he second approach require nonlinear optimization or numerical techniques As an alternative, Anselin et al. (1996) develop ed computationally simple r diagnostic tests that utilize OLS residuals based on the modified L M test developed by Bera and Yoon (1993). Estimation of Spatial Linear Regression M odels The review of the estimation methods for spatial linear regression model in this section focuses on the classical estimation framework. For Bayesian a pproaches to spatial m odels see, for example, Hepple (1995a, 1995b), LeSage (1997 b ) etc. LeSage and Pace (2009) provides a more recent review. Maximum likelihood estimation In maximum likelihood estimation t he probability of the joint distribution of all observations ( i.e. t he likelihood) is maximized with respect to the model parameters. T he ML approach for models of spatial interaction received its first comprehensive treatment b y Ord (1975) 21 Under regularity conditions, the ML estimator for these 21 ML estimation for spatial models originates in the work of Whittle (1954) and Mead (1967). In addition to Ord (1975), ML approaches for spatial autoregressive models and spatial moving average models have been developed by Cliff and Ord (1973), Hepple (1976), Hordijk and Paelinck (1976), Haining (1978), Anselin (1980), Doreian (1982), Cook and Pocock (1983), Blommestein (1985),etc.
36 models achieves the desir able properties of con sistency, asymptotic efficiency, and asymptotic normality 22 The regularity conditions a re generally satisfied when the st ructure of spatial interaction, expressed b y the spatial autoregressive parameter and the spatial weights matrix, is non explosive (Anselin 1988a). C onsider the model specification for the general spatial process in (2.10) F o llowing the notation in Anselin (1988 a ), let and T he log likelihood function for a vector of observations based on a joint standard normal distribution for the error t erm s is given by: (2.14) w here is the error covariance matrix and is the sum of squares of the transformed error terms 23 It becomes apparent from (2.1 4 ) that m a ximizing the log likelihood function of this model is not equiv alent to minimizing the sum of squared error s The determinants of the spatial terms of the Jacobian namely and prevent the OLS estimates from being the ML estimates 24 In fact, t he difference in the estimates given by t he two estimators depends on the magnitude of these determinants and becomes greater for larger values of the spatial autoregressive parameters ( and ). 22 The regularity conditions consist of a non degenerate and continuous ly differentiable log likelihood, bounded partial derivatives, positive definite and non singular covariance matrices, and finite quadratic forms. In addition, the number of model parameters should be fixed and independent of the number of observations to avoid an incidental parameters problem (Anselin,1988a). 23 Since then where is the vector of uncorrelated and homoskedastic error terms in (2.10). 24 T he Jacobian is given by : To avoi d an explosive spatial structure, the Jacobian needs to satisfy which indirectly constrains the range of the spatial autoregressive parameters to guarantee a well behaved log likelihood function over the parameter space (see also footno te 10).
37 T he first order conditions for maximizing the log likelihood function in (2.14 ) involve taking the partial derivatives of t h is function with respect to the model parameters 25 : (2.15) The ML estimates are obtained by setting the first order conditions equal to zero and solving for the parameter values. The system of the first order conditions is nonlinear in parameters, thus its solution is obtained by numerical means For the SA L and SAE model specifications, solving this system of equations is simpler because conditional up on the respective spatial autoregressive parameter the first order conditions for the non spatial parameters have a n analytical solution, which are then used to concentrate the log likelihood function. The co ncentrated log likelihood function is nonlinear only in the spatial parameter and it is maximized by numerical optimization. We now turn the attention to these models. For the SA L model the log likelihood function can be derived by s etting in (2.14 ). Hence : (2.16) The ML estimates of and are obtained by the first order conditions as: (2.17) 25 F or derivation details see Appendix 6.A in Anselin (1988a).
38 For known th ese ML estimates are equivalent to the OLS estimates from the regression of a spatially filtered dependent variable on the explanatory variables (Anselin and Bera, 1998). A concentrated log likelihood function is obtained by s ubstituting (2.1 7 ) in to the log likelihood function (2.1 6 ) which is then maximized numerically to generate an ML estimate for 26 The formal conditions that lead to the desirable asympt otic properties of the ML estimator for the SA L model have been established by Bates and White (1985) and Heijmans and Magnus (1986a, 1986b, 1986c). More recently, Lee (2004) investigate d the asymptotic distributions for the ML estimator and quasi ML estim ator for the SA L model and show ed that the se estimators have the usual asymptotic properties of ML. The likelihood function for the SAE model can be written by s etting in (2.14 ) as follows: (2.18) T h e spatial error autocorrelation in the SAE model can be considered a special case of the general parameterized non spherical error terms and the model can be seen as a generalized least squares ( GLS ) type model T hus the ML estimation of the SAE model can be carried out as a special case of the general framework of the ML estimation of GLS models with unknown parameters in the covariance matrix developed by Magnus (1978) 27 The log likelihood function can be rewritten as: 26 Refer to Anselin and Bera (1998) for expressions of the concentrated log likelihood functions for both the SAL and SAE models and Anselin (1980) for more technical details. 27 The est imation of this model by feasible generalized least squares (FGLS) was developed more recently by Kelejian and Prucha (1999)
39 (2.19) The fir st order conditions for maximizing this log likelihood function result in the GLS estimate s of : (2.20) C onditional on the ML estimate s of are equivalent to the OLS estimate s obtained by regressing a s patially filtered dependent variable on spatially filtered independent va riable s (Anselin and Bera, 1998). A similar solution of the first order conditions for yields a ML estimate for To obtain a consistent estimate of the solutions of the first order conditions for and are substit uted into the log likelihood function to concentrate the log likelihood function A n a lternative approach that involves an iterative solution of the first order conditions is provided by Magnus (1978) The desirable asymptotic properties of the ML estimato r in the presence of unknown parameters of the error variance are shown by Magnus (1978), Rothenberg (1984), and Andrews (1986). In addition to the numerical procedures required for nonlinear optimization, ML estimation is complicated further by the deter minants in the Jacobian that n eed to be repeatedly evaluated in searching for the ML estimates of the spatial autoregressive parameters. The determinants of the spatial terms in the Jac obian are determinants of matrices, thus require numerically int ensive procedures. While in some cases it may be feasible to evaluate these determinants directly (e.g. Pace and Barry, 1997), a pproximations to simplify the ir computation have also been proposed ( e.g. Ord, 1975; Griffith 1990, 1992).
40 Ord (1975) derived a simplification for these determinants in terms of the eigenvalues of the respective spatial weights matrices as follows: (2.21) where are the eigenvalues of 28 T he convenience of this simplification largely depends on the size of the sample, because the precision of the numerical procedure required to evaluate the eigenvalues deteriorat es with the size of the spatial weights matrix. For instance, Kelejian and Prucha (1999) found using ISML routines that the eigenvalues of a row standardized matrix could not be reliably calculated for dimensions of greater than 400. With sparse matrix routines in MatLab, the eigenvalues for larger matrices have been computed but with unverified accuracy 29 Griffith ( 1 992) explored several simplification s of the Jacobian in SA L models for irregular lattices. In particular, a generalized Jacobian approxima tion seems to simplif y substantially the computation of the Jacobian ; however, its performance needs further examination 30 In general, ML estimation of spatial autoregressive models is computationally intensive and requires specialized statistical softwar e and advanced programming techniques. This is even more the case with empirical applications that involve micro level spatial data sets with large sample sizes. 28 The evaluation of the determinants using eigenvalues is simpler for symmetric matrices because the eigenvalues are real. However it becomes challenging when the spatial weights matrix is not symmetric and when the sample size is large. 29 See Bell and Bockstael (2000) for a discussion. 30 The functional form of the generalized Jacobian approximation by Griffith (1992) is:
41 Instrumental variable s / g eneralized method of moments estimation T he spatial lag i ncluded as a regressor in the SA L model ( 2.4) suggests that the endogeneity ( or simultaneity ) that it induces in the model can be addressed u sing instrumental variables. The instrumental variable (IV) approach is based on the existence of a n instrumental v ariable(s) that is strongly correlated with the endogenous variable but asymptotically uncorrelated with the error term These conditions are formally expressed as 31 : (2.22) where represents the set of instruments ( including the exogenous variables ) and is the matrix of explanatory variables C ons istent estimate s for the model parameters are obtained by : (2.23) provided that satisfies the rank condition, hence it is invertible. When there are more instruments available than endogenous variables in the model the dimension of is greater than the dimension of thus the model is overidentified In this case, the model can be estimated by using a generalized method of moment s (GMM) framework 32 The GMM framew ork provides a different interpretation for t he IV estimator as a solution to the system of moment conditions: (2.24) 31 Equivalently, these conditions can be expressed as: and where is a finite and uniformly positive definite matrix. 32 For a rigorous treatment of the GMM approach refer to Sargan (1958), Gallant and Jorgenson (1979), Hansen (1982), Bowden and Turkington (1984), White (1984), etc.
42 These moment conditions arise from a general optimization problem with the following objective function: (2.25) where is a positive definite matrix. When the estimator of achieves minimum variance, thus it is efficient (Hansen, 1982) The solution to this optimization problem is the IV estimator: (2.26) where is an idempotent projection matrix. Under exact identification, the IV GMM estimator in (2. 26 ) reduces to the IV estimator in (2.2 3 ). In addition, the former is computationall y equivalent to the two stage least squares (TSLS) estimator for simultaneous equations. The IV estimator has the desirable asymptotic properties of consistency and asymptotic normality, but it is in general, not the most efficient estimator. The efficien cy of this estimator largely depends on the choice of instruments (Anselin, 1988a) The implementation of this estimation approach in the context of an SA L model seems straightforward, but a major implementation challenge is the proper choice of instrumen ts for A sensible in strument choice is This expectation, h owever is a function of the unknown spatial parameter : Kelejian and Prucha (1998) showed that for a row sta ndardized matrix and , with Hence, the linearly independent columns in can serve as instruments for These instruments,
43 henceforth referred to as the KP instruments have seen extensive application In a similar vein, Lee (2003) arg ued an optimal instrument to be the expectation: Some earlier choices of instruments based on spat ially lagged predicted values, s patial lags of exogenous variables, or combinations of the exogenous variables have also been argued to give satisfactory results (e.g. Anselin, 1980, 1984) 33 Having specified the set of instruments we can obtain consist ent estimates for this model by applying TSLS. In the first stage is regressed on to get the fitted values In the second stage, the estimates of and are obtained by the regression of on where is replaced by As a side remark, although this discussion focuses on the presence of the spatial lag variable as the source of endogeneity, the IV approach can be easily applied to situations that involve additional endogenous explanatory variabl es The implementation of the IV /GMM approach for the SAE and SA L SAE model specifications requires consistent estimate s of the parameter ( s ) of the error covariance matrix D espite the similarities between these models and the counterpart specifications i n tim e series analysis, the IV estimation procedures developed for time series models have limited applications to spatial models 34 Anselin (1988a) outline d an iterative IV approach an d a nonlinear optimization approach for the SA L SAE model but ap p licatio ns of these approaches have been almost non existent an d the properties of 33 For instance, Anselin (1980) proposed the use of a spatially lagged predicted value of the depe ndent variable with 34 The IV estimator was first advocated for time series models (Anselin, 1988a). Recall that in time series models a consistent estimate of the autoregressive parameter can be obtained by OLS as long as the err ors are not serially correlated. This is not the case in the counterpart spatial specification.
44 these estimation procedures are largely unknown The GMM approach to these spatial autoregressive models gained momentum by the work of Kelejian and Prucha in the late 1990s Kelej ian and Prucha (1999) proposed a GMM estimator for the spatial autoregressive parameter in the SAE model and showed that the spatial parameter can be consistently estimated using FGLS based on t he moments of the error terms. Recall the SAE model in (2.7). The model has two stochastic error terms and that are related in the followin g way: (2.27) From the relationship of the error terms in (2. 27 ) the three moments of interest are 35 : (2.28) The moment conditions in (2 28 ) imply the following three equation system: (2.29) where is a matrix containing functions of the error terms ; and are the parameters to be estimated; and is a vector that contains the remaining right hand side expressions o f the moment conditions A sample analog of (2. 29 ) in terms of the predicted values of error terms can be written as : (2.30) 35 The last equality in (2.28) holds because the diagonal elements of the spatial weights matrix are zero by construction.
45 where contains functions of pre dictors of and can be thought of as a vector of residuals. Using OLS residuals as predictors of and imposing the restriction between and the generalized moments estimator for and is the nonlinear least squares estimator corresponding to the fo llowing objective function : (2.31) Once a consistent estimate of is obtained the estimates for are obtained by FGLS as follows: (2.32) This procedure is a straightforward application of nonlinear least squares and it does not involve computi ng the Jacobian determinants nor the eigenvalues of Thus, unlike ML estimation it is feasible even for large samples Bell and Bockstael (2000) provide d the first application of this estimator to household level data T hey empirically assess ed the performance of this approa ch relative to the conventional ML approach. Both estimators performed qualitatively similarly for sample sizes for which the ML estimation was feasible, leading the authors to conclude that th is GMM approach is a good alternative to the ML procedures A general ized spatial two stage least squares ( G STSLS) pro cedure for the SA L SAE model specification was developed by Kelejian and Prucha (1998) The proposed three stage procedure c ombin es the GM M procedures for the SA L and SAE models and involves a Cochran e Orcutt type transformation for the spatial model to correct for spatial autocorrelation T o outline this procedure, consider the SA L SAE model in (2.9). The model can be written more compactly as:
46 (2.33) where and To estimate this model, the first stage of the procedure consists of a TSLS estimation of the first equation of the model in (2.3 3 ) to get an estimate of and obtain the model residuals TSLS is employed because the model cannot be consistently estimated by OLS s ince The TSLS estimator is given by: (2.34) w here , and is the set of KP instruments 36 This TSLS estimator is consistent, but it does not utilize information relating to the spatial correlation of the error term. In the second s tage, the model residuals from the first stage, are used to set up the error moment conditions in (2.28 ) and a consistent estimate of is obtained by the GMM procedure for the SAE models developed in Kelejian and Prucha (1999) 37 In the third stage, the model is re estimated using T SLS after applying a Cochrane Orcutt type transformation to account for the spatial autocorrelation The transformed model takes the form: (2.35) where and The TSLS estimator of this model is then give n by: 36 The set of KP instruments in this case is extended to include subset s of the linear ly independent columns of See Kelejian and Prucha (1998) for more details. 37 This estimation procedure was initially proposed in a working paper version of Kelejian and Prucha (1999) in 1995.
47 (2.36) where , and is the consistent estimate of obtained from the second stage. Kele jian and Prucha (1998) also formally establish ed th e l arge sample properties of consistency and asymptotic normality of th is estimator Both the IV / GMM and ML estimator s of spatial autoregressive models have similar asymptotic properties The main advanta ge of GMM estimation is that it is easy to implement irrespective of the sa mple size. The computation al simplicity, in turn, allows for a larger set of problems to be analyzed spatially as well as it accommodates for more general and realistic spatial rela tionships (Bockstael and Bell, 2000) Moreover, the GMM estimator does not re ly on normality of errors, thus it is consistent for departures from this assumption. When the errors are indeed normally distributed, the GMM estimator is less efficient than the corresponding ML estimator Another potential disadvantage of the GMM framework is that it does not constrain the values of the spatial autoregressive parameters, as is the case in ML estimation, thus esti mates of these parameters may lie outside the interval. As a final remark, when dealing with nonlinear spatial models such as spatial discrete choice models, even the computationally simple r GMM estimation procedures become quite arduous A discussion of the estimation challenges of discrete c hoice models in the presence of spatial dependence is provided in the next chapter but first ly a review of these models follows Economic M odels of Discrete Choice A d iscrete choice model depicts choice behavior of economic agents who face a s et of discr ete economic alternatives. T h is probabilistic framework was introduced into
48 the economic literature by Marschak (1960) from the seminal work of Thurstone on psychophysical discrimination 38 Marschak showed that the choice probabilities were consistent with utility maximization for utility functions with random components and introduced the Random Utility Maximization (RUM) model (McFadden, 2001) The RUM framework provides the theoretical justification for discrete choice models as models of economic choice behavior (Quandt, 1956; McFadden, 1974). Applications of these models in the economic literature include studies of occupational choice, choice of transportation mode, firm and household location decision s recreation demand, demand for differentiated prod ucts etc. Discrete choice models belong to the larger class of qualitative response models 39 A defining characteristic of these models is that the dependent variable is categorical taking on a finite number of discrete outcomes Based on the number of o utcomes, categorical variables are classified as dichotomous ( binary ) and polychotomous ( multinomial ) Categorical variables with multiple outcomes can be further classified as unordered and ordered 40 In u nordered categorical variable s the values assigned to each outcome can be arranged in any order since the order is not meaningful An example of an unordered categorical variable is empl oymen t status employed unemployed and out of the labor force. In contrast, in ordered categorical 38 T hurstone (1927) modeled the response of human subjects to alternative levels of true stimuli as a series of pairwise comparisons based on perceived stimuli (perceived with some error). The interpretation of the perceived stimuli as satisfaction (or utility ) led to a model of economic choice (McFadden, 2001). 39 Qualitative response models also include count data models. A common aspect of discrete choice models and count data models is that the dependent variable in both cases assumes discrete values. As a point of departure, the dependent variable in count data models although discrete is not categorical. As a result, count data models and discrete choice modes require different estimation methods. 40 Sequential is often listed as a third category. For our purpose, a sequential categorical variable can be thought of as an ordered variable with a particular ordering scheme; f or example, education status (no high school/high school but no college/ college but no professional degree/ professional degree).
49 variables the outco mes are inherently ordered; hence, the values assigned to these outcomes are no longer arbitrary An example of an ordered categorical variable is health status poor good, and excellent. In this case, the quantitative values assigned to the qualitative outcomes need to convey that an excellent health status is better than good which in turn, is better than poor. E xcellent treatments of discrete choice models i n the literature can be found in Maddala ( 1 983 ), Wooldri d ge ( 2002 ), Cameron and Trivedi ( 2005 ) and Train ( 2007 ) among other references This review attends exclusively to multinomial choice models and, in particular to logit based models Random Utility Maximization Framework Unordered choice alternatives Discrete choice models are generally d erived within a RUM framework assuming utility maximizing behavior of economic agents 41 An economic agent face s a decision that involves making a choice from a choice set that contains a finite number of mutually exclusive and exhaustive alternatives Give n the choice set a rational a gent with well defined preferences is expected to choose the alternative that maximizes his utility subject to a ( budget ) constraint or equivalently, the alternative that provides him t he highest indirect utility 42 The resear cher Hence, it is further assumed that there are factors that jointly utility thus his choice some of which are unobserv able s viewpoint, 41 Th is discussion closely follows Train (2007) and Cameron and Trivedi (2005). D iscrete choice models are also consisten t with other types of behavior (Train, 2007). 42 e ranked in order of preference) and transitive (if X is preferred to Y, and Y is preferred to Z, then X is preferred to Z), which combined with a mutually exclusive and exhaustive choice set allows an individual to consistently rank all alternatives in te rms of his preferences and make an unambiguous choice.
50 a random function. In this setting, t he goal of the researcher is to understand the decision making process that led to a particular choice, having observed the choice and a number of choice and individual related factors (Train, 2007) To set up the model, c onsider an individual who makes a choice among ( unordered ) alternatives available in the choice set where D efine latent variable s to denote the indire ct utilities associated with different choice alternative s Th ese utilities are known to the decision maker but are unknown to the researcher D efine an outcome variable to take value if the th alternative is chosen by the decision maker Th e ch oice is observed by the researcher. T he decision maker chooses the alternative that provides the greatest indirect utility (assuming no ties); hence he chooses a particular alternative from the choice set say if and only if : (2.37) Equivalently the researcher observes : (2.38) Although t y is not directly observable the researcher observes some attributes of the choice alternatives as well as some a ttributes of the decision maker and can specify a function that relates these observed factors to as follows 43 : (2.39) 43 A clarification on notation: throughout this manuscript, refers generally to the alternatives in the choice set whereas (and used later) refers to a particular alternative in the choice set ; denotes alternative varying variables; denotes alternative invariant variables; denotes alternative invariant parameters; denotes alternative varying parameters; denotes all observed factors, and denotes all model parameters associated with the observed factors,
51 where denotes all observed attributes ], and denotes all the parameters associated with the observed factors that enter in the specification of is called a representative utility and differs from because of the unobserved factors The se unobserved factors capture any unobserved variations in tastes, unobserved attributes of alternatives or individuals and/or errors in perception and optimization by the decision maker s (Maddala, 1983). Denot ing the unobserved factors by th e indirect utility for any of the alternatives in the choice set can be expressed as : (2.40) Th e decomposition of the utility into a systematic component and a stocha stic component allows the researcher to analyze the choice of the decision maker probabilistic ally 44 Note that a higher value of the representative utility in (2.4 0 ) for a particular alternative does not guarantee the choice of that alternative The unobserved factors could be such that a different alternative may be sufficiently better and the latter may prevail The unobserved factors a re treated as a random vector with joint density ). Having spec ified this density, t he probability that decision maker chooses a particular alternative is given by: Thus, the prob ability that decision maker chooses alternative is the expected value of the following indicator function over all possible values of unobserved factors 45 : 44 choice can be predicted exactly. 45 The indicator functio n takes on value one for a combined value of differences in and that induces the decision maker to choose alternative (i.e. if the expression in parentheses is true) and zero otherwise. Also note that ties would have zero probability.
52 (2.41) where The functional form for (or more generally ) should be specified in a manner that ensures that the choice probabilities are statistically valid; and Diffe rent specifications of are obtained from different assumptions about the joint distribution of the error terms and result in d ifferent specifications of discrete choice models The two most common specifications are the logit model and the probit model The logit model is derived under the assumption that the error terms are distributed IID t ype I extreme value This distributional assumption facilitate s the estimation of the logit model because the choice probabilities in (2.4 1 ) take a closed form solu tion. A critical aspect of this assumption is the independence of the error terms for different alternatives, which impl ies that the unobserved factors are uncorrelated a cross alternatives. The latter gives rise to an important property of the logit model known as the independence of irrelevant alternatives (IIA) which i mplies the same degree of substitutability among alternatives irrespective of the composition of the choice set. This property of logit models will be discussed later in greater detail. Th e p robit model assume s that the unobserved factors are distributed jointly norma l. This model avoids the independence of errors assumption and accommodates for different patterns of correlations ; h owever the choice probabilities do not have a closed form solution T hus they have to be evaluated numerically th r ough simulation. A major advantage of the probit model is the flexibility in handling correlation across alternatives (and over time) A drawback is the more complex estimation procedures and its rel iance on the normal distribution which in some situations may not be
53 appropriate Given the similarity between the normal and the logistic distribution, i n practice, both the logit and the probit models produce (after proper scaling) very similar results T he choice of a proper distribution is difficult to justify on theoretical grounds 46 (Greene, 2000). In addition to the choice of multinomial models with unordered choice alternatives also differ from the type of explanatory variables included in the model depending on whether or not the se variables vary across alternatives Alternative varying regressors take on different values by alternative (and possibly by individual ) For example, the cost of transportation in a transportation choice model largel y depends on the mode of transportation In this case in (2.4 1 ) for any of the choice alternatives takes the form: where denotes alternative specific attributes and is a vector of alternative invariant parameters gy, this is known as a conditional model and it is on observed attributes of each alternative. In contrast regressors may be individual specific but alternative invari ant F or instance, in the transportation choice model, soci o economic characteristics of a decision maker are the same regardless of the transportation modes available In this case, in (2.4 1 ) takes the form: where denotes individual specific attributes and is a vector of alternative varying parameters This model, simply referred to as a multinomial m odel is appropriate in 46 The lo gistic distribution has heavier tails compared to the normal distribution. It closely resembles a distribution with seven degrees of freedom (Greene, 2000).
54 cases where attributes of alternatives are either unimportant or data on these attributes is not available For instance, in a model of occupation choice, the earnings of an individual in all occupations are not known. In this case, characteristics such as education and past experience are used to explain his occupation choice. Other applications such as the transportation choice example above, may involve both types of variables. In this case, a richer model th at combines both types of regressors is employed. This model is known as a mixed model. This distinction between the types of explanatory variables and the resultant discrete choice models is important not only for their applications, but also for practi cal reasons. As it will become more evident when discussing logit based models it is possible to change the format of the variables from alternative invariant to alternative varying; hence, the mixed and multinomial models have a conditional model formula tion In this regard, the conditional model is more general. Finally, equation (2.4 1 ) brings up two important points for the identification and estimation of discrete s choice is based on the difference in indirect utilities. This has implications in terms of the explanatory variables that can be included in the model and the interpretation of model parameters because they capture differences across alternatives. Second, the indirect utility is only determined up to scale, so the scale of utility needs to be normalized. This is typically done by normalizing the variance of the error terms 47 47 As Train (2007) points out, t he scale of utility and the variance of the error terms are link ed by definition. Hence, normaliz ing the scale of utility is equivalent to normalizing the variance of the error terms. For instance w hen changes by : See Train (2007) for a detailed discussion on identification of discrete choice models.
55 Ordered choice alternatives Ordered choice models are better conceptualized by considering a rather different decision making scenario The decision now involves the decision maker evaluating an experience (event) based on the satisfaction ( utility ) acquired f rom the experience. The choice set contains alternatives that correspond to different le vels of the decision utility but the alternatives are ordered such that a higher ranked alternate is associated with a higher level of utility. In this scenario, t he decision maker is expected to choose an alternative from the choice set that be st represents his overall satisfaction from the experience. To set up a RUM model c onsider an individual who makes a choice among ( ordered ) alternatives in the choice set Define a latent variable to denote the utility a ssociated with th e experience has multiple levels associated with different choice alternatives in the choic e set A s before, the researcher does not observe th is ut ility index, but observes the choice made. Define an outcome variable to take value if t he decision maker chooses the th alternative to represent his overall utility level associated with the experience. The researcher then observes : (2.42) where and are utility levels that define a utility interval s uch that if the utility perceived by the decision maker falls within this interval, it result s in the choice of alternative The u tility can be decomposed in to a systematic component that is expressed as a function of observed factors and a stochastic component due to unobserved factors as follows:
56 (2.43) Then, the probability that individual chooses any alternative in the choice set is : Hence, (2.44) where is the cumulative distribution function ( CDF ) of Different distributional a ssumptions about the error term result in different specifications of ordered models. An ordered logit specification resu lts from assuming that the error term is distributed l ogistic Similarly, an ordered probit model is obtained by assuming a standard normal distribution for the error term. Either distributional assumptions result in choice probabilities that are statistic ally valid 48 T he threshold values and of the utility are subjectively determined by the decision maker thus are not known to the researcher. The y are estimate d along with the rest of the model parameters that enter the systematic component of utility. These parame ters are a result o f the ordered feature of the model. The choice probabilities in (2.4 4 ) indicate that o rdered choice models face similar identification issues with unordered models ; the absolute level of utility is unimportant and the scale of utility ne eds to be normalized. However, t he e stimation of ordered models is simpler because the choice probabilities in (2. 44 ) do not involve multidimensional integrals. T he next section provides a general discussion on the estimation of discrete choice models 48 In addition to the logistic and normal distributions, other distributions used for ordered models are, for example, the log log, negative log log, and complem entary log log for skewed data (Angresti, 1996).
57 Es tima tion of Discrete Choice Models A common estimation method for d iscrete choice models in applications that involve cross sectional data is maximum likelihood. Consider the choice probabilities : wh ere is the CDF of the unobserved factors ; represents the observed factors ; and represents a vector of model parameters to be estimated. As emphasized previously, specification of should e nsure that the choice probabilities are not only statisti cally valid, but also consistent with maximization of a random utility function 49 Normal and logistic distributions for the error terms produce c hoice probabilities that satisfy both conditions and result in different m odel specifications namely multinomi al logit/probit ordered logit/probit, ne sted logit/probit etc. To set up the likelihood function, r edefine the dependent variable in a binary form by introducing binary variables as follows : F or each observation only one of the terms will be non zero T hus t he likelihood of observing is given by : (2.45) For a sample of independent observations, the likelihood function of the entire sample is the product of the individual likelihoods: (2.46) 49 The RUM consistency conditions are established by Williams (1977), Daly and Zachary (1979), and McFadden (1981).
58 where is the vector of parameters that enter the systematic component of the utility. The multiplicative form of this likelihood function makes it more convenient to work with the log likelihood function which is given by: (2.47) The log likelihood function is maximized at the point where: (2.48) Hence, th e ML estimates of are obtained as a solution to the first order conditions in (2.48) These equations are nonlinear in because is a nonlinear function of This estimator will be consistent if the choice probab ilities are correctly speci fied such that in which case the expectation of (2. 48 ) equals zero (Cameron and Trivedi, 2005) For logit models, the ML estimates are obtained by an iterative procedure that uses any of the familiar numerical optimization algorith ms such as Newton Raphson, method of scoring, etc. Probit models in turn, are computationally costly even for a small choice set because the probit choice probabilities involve multidimensional integrals (Maddala, 1983; Amemiya, 1985). ML estimates for p r obit models are typically obtained by s imulation methods 50 For ordered models, the ML estimates are easily obtained for both the logit and probit specifications because the choice probabilities do not involve multidimensional integrals. Th e ML estimator fo r discrete choice models has the usual asymptotic properties of consistency and asymptotically normal ity In the presence of endoge ne ity or correlation across the error t erms for different observations, the discrete choice model s can be estimated using a GMM framework. 50 Refer to Hajivassiliou and Ruud (1994) and Keane (1993) for reviews of simulation estima tion.
59 Provided that the choice probabilities are correctly specified, the GMM estimator will be the solution to the following moment conditions: (2.49) where is vector of instruments t hat does not depend on A choice for can be, for example, which makes the moment conditions in (2. 49 ) similar to the fi rst order c onditions in (2. 48 ). For correctly specified choice probabilities the expectation of (2. 49 ) equals zero. Hence, t he GMM estimator is consistent, but its efficiency depends on the choice of The moment conditions in (2. 49 ) are nonlinear in so obtaini ng GMM estimates involve s n on linear estimation procedures as well O nce the model is estimated, the interpretation of the model coefficients requires further computation to obtain marginal effects or elasticitites The model coefficients do not have a dir ect marginal effect interpretation because of the nonlinearity of the model. F or some model specifications, there is not even a direct correspondence between the coefficient sign and the sign of the respective marginal effect. Marginal effects measure the change in the choice probabilities for a ceteris paribus change in a regressor They a re typically computed for each individual as: (2.50) An average marginal effect is obtained by averaging the individual marginal effects E lasticities provide an interpretation of the model coefficients in terms of percentage changes as they measure the percentage change in the choice probabilities for a percentage change in the value of a regressor. They are obtained as follows: (2.51)
60 The effect of a n attribute change on choice probabilities can also be computed by taking first differences in predicted choice probabilities obtained before and after the attribute change Logit Based Model s Cond itional, m ultinomial and mixed logit models L ogit based model s with unordered choice alternatives are derived by assuming that the error terms in ( 2.41 ) are distributed IID type I extreme value (or Gumbel) with density : and CDF : (2.52) Th e type I extreme value distribution is a symmetrical (right skewed) with mean 0.58 and variance ( Cameron and Trivedi, 2005) This distributional assumption implicitly normalizes the scale of utility ( T rain, 2007). For any two type I extreme value random variables, and their difference is a random variable distributed l ogistic with mean zero and variance (Johnson and Kotz, 1 970) Hence for a logit model, in (2.41 ) is specified as the logistic CDF : (2.53) Th e logistic distribution al assumption about the error differences results in the following cl osed form formula for t he choice probabilities in (2.41 ) 51 : (2.54) 51 S ee Appendix A for a complete derivation of the logit choice probability formula from a RUM model set up. The logit probability formula can also be easily derived from a pure statistical perspective.
61 Th e functional form of the choice probabilities in ( 2.54 ) ensures well behaved probabilities because is inevitably between zero and on e and the sum of choice probabilities for all alternatives equals one. In addition, these choice probabilities have also been shown to be consistent with RUM maximization 52 This functional form of choice probabilities simplifies substantially the estimatio n of logit models because the log likelihood function is globally concave in parameters (McFadden, 1974). The simplest specification for the systematic component of indirect utility is linear in parameters as follows : (2.55) where represent s the attributes of the th alter native for individual ; represent s the attributes of the th individual ; and and are vector s of parameters relating alternative attributes and individual attributes to the decision maker Given the specificatio n of the representative utility in (2.55), the probability that individual chooses alternative is given by : (2.56) Th e model with choice probabilities (2.56) is known as a generalized or mixed logit (MXL) mod el because it incorporates both alternative specific and individual specific attributes 52 The logit formula was first derived by Luce (1959) from the IIA assumption. Marschak (1960) showed that given the IIA assumption, the model was consistent with utility maximization. Marley, cited in Luce and Suppes (1965), showed that an extreme value distribution for the unobserved portion of the u tility leads to the logit formula. McFadden (1974) completed the analysis by showing the opposite; that the logit formula necessarily implies that unobserved utility is distributed extreme value. This chronology is obtained from Train (2007).
62 If only the subset of the alternative varying explanatory variables in (2. 55 ) is included in the model such that in (2.56), the resulting model is referred to as a conditional logit (CL) model. In the CL model, t he choice probabilities are specified as : (2.57) T he parameter vector does not vary by choice S i nce an equivalent model can be specified by defining i n terms of deviations of regressor s from the values of one of the alternatives and setting the corresponding equal to zero (Cameron and Trivedi, 2005). In contrast, if only alternative invariant regressors are included in the model, such that in (2. 55 ), the ch oice probabilities take the form : (2.58) This model is know n as a multinomial logit (MNL) mo del 53 Since there are only probabilities that can be determined independently. Thus, a normalization is needed for the identification of the model parameters A common normalization set s one of the p a ra meters equal to ze ro. The interpretation of the model results is then made relative to the normalized alternative. Th e distinction between the types of explanatory variables and the resultant logit model specifications is important in empirical applications. For instance, land use research uses primarily location specific data whereas marketing research relies on category specific information. As a result, a MNL model is more appropriate for 53 Strictly s peaking, all the models reviewed in this section are multinomial logit models. This terminology is used in the literature with the understanding that this model accommodates only for individual specific regressors.
63 use choices whereas a CL model is more appropriate for a An interesting feature of these model s i s that the M N L and MXL model s can be viewed as special cases of the CL model and can be expressed in a CL formulation T o see this, l et be a vector of individua l attributes and corresponding parameter vector T his vector can be expressed as a vector of alternative varyin g (individual) characteristics by defining a vector ) with all zero elements except the elements corresponding to the th block, which are set equal to In addition, we can define a common parameter vector of dimension with elements More explicitly, (2.59) Then, H ence the choice probabilitie s can be equivalently expressed as: (2.60) The transformation of individual specific variables to alternative specific variables essentially involves expressing the se variables as interactions with a series of alternative specific dummy variables Identification of the parameters in the MNL model requi res that the parameters for one of the alternatives be normalized to zero This normalization is handled in the CL formulation b y omitting the variable interactions with the dummy variables corresponding to the a lternative chosen for normalization The exp ression of the MNL and MXL models as CL models is of p ractical importance because a c omputer program written for alternative varying regressors can be used to
64 estimate a ll three model specifications since it is possible to change the format of the variable s from alternative invariant to alternative varying 54 Independence of irrelevant alternatives (IIA) A key aspect of the distributional assumption that leads to the logit choice probabilities is the independence of the error terms across alternatives. The independence of the errors assumption implies that the unobserved portion of the utility for one alternative is unrelated to the unobserved portion of the utility for another alternative and results i n the IIA property 55 This assumption is likely to be vio lated in situations where two alternatives share common unobserved attributes or when unobserved individual characteristics influence how the observed individual and alternative attributes affec t choice. The IIA property is unattractive because it implies the same degree of substitutability between choice alternatives regardless of the composition of the choice set ; hence, it results in a restrictive s ubstituti on pattern The IIA property can be f ormally expressed using the odds ratio between two alternati ves in the c hoice set ( and ) as follows : (2.61) where and de note two different choice sets that contain alternatives and ; and denote the probability of the th individual choosing alternative and from the choice set (and similarly for choice set ) This od ds ratio depends only 54 This is the programming route taken in this dissertation. Refer to Cameron and Trivedi (2005) for more details. 55 All three models, CL, MNL, and MXL exhibit the IIA property.
65 on alternatives and thus the odds of choosing over are the same irrespective of the rest of the alternatives in the choice set s and and their attributes. A classical example in the literature is the red bus blue bus problem due to McFadden (1974). In the famous example, we would expect that adding a different color bus (red) to the choice set consisting of a car and a (blue) bus would not change the choice probabilities of choosing between a bus and a car (as suming commuters are indifferent about the color). Instead, the logit model predicts a decrease in choice probabilities for both the (blue) bus and the car once the new (red) bus is introduced in the choice set. T hus the model overstates the overall proba bility of choosing a bus Alternatively, consider improving an attribute of alternative We would expect the decision maker to substitute away from the other alternatives in favor of the alternative with the improved attribute. The effect of this chang e in the probabilities of other alternatives for example alternative is given by the following c ross elasticity 56 : (2.62) This cross elastici ty depen ds only on alternative This means that while the improvement in the attribute of alternative increases the choice probability of alternative it reduces the choice probabilities for alternative and all other alternatives in the choice set by the exact same proportion. Thus the reason that the ratio of the probabilities in (2. 61 ) is constant is because the change of an a ttribute of a third alternative affects both choice alternatives and in the same proportion This pattern of substituti on is called proportionate shifting (Train, 2007) 56 See Appendix A for derivations of elasticites and marginal effects.
66 Despite the restrictive model assumptions t he logit model accommodates for variations in taste as well as for dynamics of repeated choice as long as these variations are related to the observed factors. Any variation in unobserved factors enters the unobserved component of utility and violates the IID assumption of the error terms (Train, 2007). Interestingly, t he restrictive IIA property is also considered to be a n advantage of these models ( Domencich an d McFadden, 1996). Because the relative probabilities of alternatives within a subset are unaffected by the alternatives omitted, the IIA property allows the model to be estimated consistently with a subset of the choice alternatives or from data on binomi al choices (Hausman and McFadden, 1984). This is particularly helpful when the choice set is relatively large or when the researcher is only interested in examining a subset of alternatives. Various testing procedures are available in the literature to te st w hether or not the IIA assumption holds. O ne way i s to re estimate the model on a subset of alternatives. If IIA holds, parameter estimates for the subset should not be significantly different from the ones with the full set of choice alternatives A te st statistic for this type of test is found in Hausman and McFadden (1984). Another test involves estimating the model with cross alternative variables a nd testing for the significance of the effect of these variables Significant effects indicate a violat ion of the IIA assumption. A procedure along these lines is developed in McFadden (1987). In more flexible specification s (e.g. general extreme value (GEV) models) IIA can be tested by testing model restrictions provided the more general model is a prope r specification (Hausman and McFadden, 1984; McFadden, 1987; Train el al., 1989). However, when IIA fails, these tests do not provide much guidance as to the correct model specification McFadden (197 4 )
67 advocate s that the CL and MNL models be used only in cases when the alternatives can be viewed as distinct and weighted independently by the decision maker. S everal models that relax the IIA assumption have also been suggested 57 A multinomial probit model is theoretically attractive because it accommodates for relatively unrestricted correlation patterns but it has practical limitations d ue to the complexity of estimation 58 A nother approach to introduce some correlation into the model is to estimate a nested logit model 59 In a nested logit model, the decisi on making process assumes a tree like structure with several decision layers. Similar alternatives are grouped in to nests and the unobserved factors are permitted to be correlated across nests, but uncorrelated within nests. While the IIA property does not hold for alternatives in different nests, it holds for alternatives within nests A limitation of the nested logit model is that the decision making process m ight not fit an obvious nesting structure A third possibility involv e s specifying a more general model, known as a random parameters logit model, which decomposes the unobserved factors into a part that contains all the correlation and another part that is distributed IID extreme value 60 This model specification is quite general and can approximate a ny discrete choice model. 57 For overviews see Ben Akiva et al. (1997) and Horowitz et al. (1994). 58 Refer to McCulloch and Rossi (1994) and McCulloch, Polson, and Rossi (2000) for a general discussion. 59 Nested logit models belong to the class of generalized extreme value (GEV) models. They are derived by assuming that the un observed portion of utility for all alternatives is jointly distributed generalized extreme value. See McFadden (1984) for a detailed treatment of these models. 60 See McFadden and Train (2000) for a general discussion on random parameters logit (or probit ) models and their properties in approximating general choice patterns. This model is also known in the literature as a mixed logit model. This terminology is avoided here to prevent confusion with the mixed logit model that combines individual specific an d alternative specific variables.
68 Ordered l ogit model An ordered logit (OL) model is obtained by assuming that the error term in (2. 4 3 ) follows a logistic distribution with CDF: (2.63) Hence, t he choice probabilities in ( 2.4 4 ) take the form: (2.64) where denotes the explanat ory variables ; denotes the parameter s that enter the systematic component of utility ; and and are the utility threshold parameters. Despite the similarity in the specification of the systematic component of utility with a MNL model, an OL model fits the same slopes across all outcomes ( does not vary by alternative ). This is k nown as the parallel slopes assumption. It implies that the observed factors equally affect the likelihood of a person being in any of the different ordered cate gories. If t he parallel slopes assumption is not expected to hold the model should be estimated by MNL despite the ordered nature of the outcomes (Boroorah, 2002 ) One way to test for the validity of the parallel slope s assumption is to estimate the mode l with MNL and OL and perform a LR type test using the likelihood values from both models 61 (Boroorah, 2002 ) If the parallel slopes assumption does not hold, t reating unordered outcomes as ordered is likely to bias the estimates. In contrast if ordered ou tcomes are treated a s unordered, failing to impose the ordered structure may 61 This test is only suggestive because it is not a LR test; the restricted model (OL model) is not nested within the unrestricted model (MNL model).
69 lead to loss of efficiency. In the view of this trade off an ordered model should be used when there is an unequivocal ordering of outcomes. The discussion o n the models of discr ete choice in this chapter assume s that the decision makers make their choice s independently. However, e conomic decisions of individuals are at times influenced by choices made by neighbors, peers, or other individuals at different locations that face simi lar decisions. Hence, modeling individual discrete choice decisions may require accommodating for interdependence in decision making. S p atial dependence in discrete choice models is the focus of the next chapter.
70 C HAPTER 3 AN ESTIMATION APPROA CH TO DISC RETE CHOICE MODELS WITH S PATIAL LAG DEPENDENCE INVOLVING LARGE SAMPLES Spatial D ependence in D iscrete C hoice M odels Spatial dependence in discrete choice models has been essentially handled in the following ways. Standard probit or logit models have been e mployed to estimate a model with spatial data (e.g. Wallace, 1988; McMillen and McDonald, 1991; McMillen et al. 1992). Spatial dependence has been dealt with indirectly by using s ubsampling techniques to purge the data of spatial dependence (e.g. Carrion Flores and Irwin, 2004; Nelson et al. 2001; Newburn et al., 2006 ) or by including s patially derived variables (e.g. Staal et al., 2002) and spatially lagged characteristics to capture some of the spatial effects (e.g. Nelson and Hellerstein, 1997). Lastly spatial dependence has also been modeled directly (e.g. Case 1992; McMillen, 1992; LeSage, 2000; Beron and Vijverberg, 2003, 2004). T h e estimation of spatial discrete choice models however, remains challenging A spatial specification of a discrete cho ice model differs from a standard (non spatial) specification because the resultant spatially correlated covariance structure prohibits the expression of the multivariate distribution associated with the likelihood function as the product of univariate dis tributions; thus, the estimation of spatial discrete choice models results in a multidimensional integration problem (Fleming, 2004). The complexity of the functional form makes the direct maximization of the discrete choice likelihood fun ction quite diffi cult in practice. For instance, Beron and Vijverberg (2000) rep orted that the estimation of a spatially autoregressive lag (SAL) specification of a probit model lasted several hours fo r a dataset with 49 observations. With large r datasets the ML estimatio n of spatial discrete choice models becomes
71 computationally infeasible. Moreover the covariance structure implies that the model error terms are both autocorrelated and heteroskedastic, thus estimating a standard model leads to inconsistent parameter esti mates 1 (McMillen, 1992) Several spatial discrete choice estimators in the literature provide consistent estimates under spatially dependent data 2 A pplications of these estimators however, are limited to empirical problems that involve binary choice s and relatively small samples. To simplify the estimation of the spatial model, some of the proposed spatial estimation procedures focus on spatial heteroskedasticity and ignor e spatial autocorrelation Consequently they produce e stimates that are con sistent but not efficient Case (1992) developed an estimation methodology for a heteroskedastic probit model with a spati ally lagged dependent variable The estimation methodology involves transforming the model to produce homoskedastic errors and estimate the v ariance normalized version of the model using maximum likelihood. Th is estimation procedure is made possible by the block diagonal structure assumed for the spatial dependence 3 This approach can only be used in those empirical applications that can justif y the special form of the spatial dependence structure. For instance, Rincke (2006) used this estimation methodology to study policy innovation in local jurisdictions assuming that 1 The inconsistency of standard (probit or logit) models is a result of spatial heteroskedasticity. Spatial autocorrelation affects efficiency when full spatial information is not utilized. McMillen (1992) points out that the heteroskedasticity induced in the model is artificial because it arises from the location of spatial units relative to the boundary (not from differences in spatial units). Heteroskedasticity in spatial models is also examined by Haining (1988). 2 A more rigorous discussion of these estimation techniques is provided in Fleming (2004). 3 Case (1992) assu mes that spatial dependence in her technology adoption application arises between farmers that reside within a district (but not across districts). She specifies a block diagonal contiguity weights matrix where the only non zero elements correspond to a bl ock of households that reside in a district. The block diagonal structure simplifies the estimation of the model because it produces an algebraic expression for
72 the dependence in the policy adoption decision s of school districts arises within school districts in the same local jurisdiction Pinkse and Slade (1998) proposed a GMM estimator based on the moment conditions derived from a heteroskedastic maximum likelihood function o f a probit model with a spatial autoregressive error (SAE) structure. In the model, the heteroskedastic error variances are expressed as a function of the unknown spatial autoregressive parameter. As a result, the GMM estimation involves an iterative procedure that requires repeated evaluation of the covariance ma trix for different values of the spatial parameter This in turn, involves inversion of matrices with each iteration which makes the implementation of the procedure computational ly intensive To avoid the matrix inversion Klier and McMillen (2008 ) proposed a linearization of th is spatial estimator in the context of a SAL l ogit model An other application of Pinkse and is found in Flores Lagunes and Schiner (20 1 0 ) which is used to obtain first step estimates for sample selection m odels with spatial error dependence McMillen (1992) also considered a spatial binary probit model with heteroskedatic errors. T he form of heteroskedasticity in the model however, is not derived directly from the spatial dependence structure. A functiona l form for the heteroskedasticity is assumed based on a spatial expansion method and the model is estimated by weighted nonlinear least squares. The implementation of this estimator is fairly easy e ven in large samples and produces consistent estimates pro vided the form of heteroskedasticity is correctly specified. Since it is unclear how heteroskedasticity consistent estimators perform in small to moderate sample sizes as they may lead lo large variances McMillen (1995a)
73 conducted a Monte Carlo invest igation of the performance of a standard probit and a heteroskedastic probit when the dataset exhibits both heteroskedasticity and autocorrelation. To discern between the spatial effects, autocorrelation was specified as a function of the distance between two locations to ensure that it did not induce heteroskedasticity and heteroskedasticity was introduced separately in the model The si mulation results indicate that the standard probit is preferred to the heteroskedastic probit for small sample sizes and low degrees of heteroskedasticity. For larger sample sizes and higher degrees of heteroskedasticity the heteroskedastic probit outperformed the standard probit. Autocorrelation was found to have little to no effect in the results. Other estimation procedu res account for spatial heteroskedasticity by taking into account the full covariance structure of the model. These full information spatial estimators rely on the expectation maximization (EM) algorithm, simulation methods, and Bayesian techniques to solv e the multidimensional integration problem (e.g. McMillen, 1992; LeSage, 2000; Beron and Vijveberg, 2003, 2004). They attempt to preserve the estimation structure implied by maximum likelihood by simulating the choice probabilities or parameter probability distributions. The parameter estimates are obtained from the simulated distributions rather than from the direct maximizati on of the likelihood function. McMillen (1992) extended the EM algorithm for spatial specifications of binary probit model s The EM algorithm consists of an iterative two step procedure based on the likelihood function of the continuous latent variable 4 The E step takes the 4 The EM algorithm was introduced by Dempster et al. (1977) for time series models. Amemiya (1985) and Ruud (1991) provide an overview of the method and applications. The algorithm has been proposed for use in spatial models (Flowerdew and Green, 1989) and discrete choice models (Greene, 1990).
74 expectation of the likelihood function conditional on the observed discrete variable and an initial set of para meter values. In the M step, the expectation of the likelihood function is maximized. The procedure is then repeated until convergence An advantage of this estimator is that it avoids the direct evaluation of the multidimensional integral in the likelihoo d function of the spatial probit model. The procedure has essentially two drawbacks. First, it does not produce standard errors for parameter estimates. These precision measures have to be obtained from the N dimensional dependence structure 5 More importa ntly, the procedure requires repeated inversion of matrices, thus it is computationally intensive in moderate to large samples. Computationally simpler alternatives are explored in McMillen (1995b). LeSage (2000) adopted a Bayesian approach based on Gibbs sampling (Geman and Geman, 1984) to estimat e a spatial probit model 6 Gibbs sampling is a Markov Chain Monte Carlo (MCMC) technique that uses sampling from a set of conditional posterior distributions of model parameters to create a Markov chain that converges in the limit to the true posterior dis tribution of the model parameters. This approach has some advantages over the EM estimator in McMillen (1992). The standard errors of the parameters can be directly obtained from the conditional distributions. In addition, it accommodates for g eneral forms of heteroskedasticity and more c omplex likelihood functions than a probit model. Due to its flexibility, the Bayesian framework has been adopted in several recent studies that incorporate spatial depe ndence in 5 McMillen (1992) provides a covariance matrix by interpreting the probit estimator as a weighted nonlinear least squares estimator conditional on the spatial parameters. 6 The Bayesian Gibbs sampl er in LeSage (2000) is an extension to the Bayesian Gibbs sampling for non spatial discrete choice modes in Albert and Chib (1993) and spatial models with continuous dependent variables in LeSage (1997b).
75 multinomial and dynamic settings 7 (e.g. Kakamu and Wago 200 5; Chakir and Parent, 2009 ; Wang and Kockelman 2009 ) Akin to the EM algorithm, this is an iterative procedure that requires recurring matrix inversion Simulation methods have also been used to evaluate the dependent probit likelihood function. Beron and Vijverberg (2003, 2004) proposed a recursive importance sampling (RIS) procedure, which computes the likelihood function by sim ulating the parameter probability distributions 8 A major advantage of this procedure is that, unlike the previous estimation procedures, it directly evaluates the likelihood function of the spatial probit model so LR tests can be administered to test for model specification. In addition, standard errors for estimates can be obtained from the sampling distributions. A major drawback is the high computational costs associated with the simulator as made apparent in the Monte Carlo study conducted in Beron an d Vijverberg (2003). Bolduc et al. (1997) compared the Gibbs sampler and the RIS simulator in the context of a multinomial probit model with an SAE structure. They concluded that while both approaches yield similar results, the Gibbs sampler is conceptuall y and computationally simpler. An alternative to the ML type estimators is to interpret the spatial model as a weighted nonlinear version of a linear probability model with a general covariance matrix (e.g. Amemiya, 1985; Judge et al., 1985). These estima tors are formulated as GMM estimators but can be viewed as weighted nonlinear forms of TSLS (for the SA L 7 LeSage and Pace (2009) provide an extensive over view of the framework and applications in spatial discrete choice models and other limited dependent variable models. 8 The RIS simulator is a more general version of the more familiar GHK simulator (Geweke, 1989; Hajivassiliou, 1990; Keane, 1994). See Vi jverberg (1997) and Beron and Vijverberg (2003) for a detailed treatment of the RIS simulator.
76 specification) and FGLS (for the SAE specification) 9 Proposed spatial estimators for probit models with dependent observations (but homoskedastic erro rs) based on this approach are found in Avery et al., (1983) and Poirier and Ruud (1988). Fleming (2004) discusses in some detail the GMM methodology for SA L and SAE specifications of a binary choice model based on the GMM approach developed for spatial li near models in Kelejian and Prucha ( 1998, 1999). The GMM approach avoids the multidimensional integral in the likelihood function, but the parameter estimates are typically obtained through an iterative procedure a s in Pinkse and Slade (1998). Klier and Mc Millen (2008) proposed a spatial GMM estimator for a SAL specification of a binary logit model b ased on the G MM estimator devel oped in Pinkse and Slade (1998) To avoid the inversion of large matrices required for estimation t hey propose d to linearize the spatial model around a convenient point of parameter values for which the initial value of the spatial parameter is set equal to zero The linearization r educe s the model estimation into two steps: a standard logit model followed by two stage least s quares Klier and McMillen (2008) investigated the small sample properties of their linearized spatial estimator using Monte Carlo simulations. The results from the Monte Carlo study indicate that the linearized spatial estimator performs remarkably well i n identifying the degree of spatial dependence induced in the simulated data ( as measured by the spatial parameter ) for a sensible range of spatial parameter values Because of its promising performance and ease of estimation, a n extension of the Klier and McMillen (2008) methodology to polychotomous choice 9 See Fleming (2004) for more details.
77 models would be valuable to analyze various economic decisi on s with multiple choice alternatives at a micro lev el. This chapter fills this gap in the context of logit models. In the next section it is shown that the key result that enables the linearization of the spatial model in the binary choice case in Klier and McMillen (2008) holds in the multinomia l choice case and th is extended t o a multinomial setting. Furthermore, the extension in this chapter covers various polychotomous logit models all of which have differe nt empirical applications. An important contribution of this research to the spatial modeling literature is that the proposed spatial multinomial choice estimator s are computationally feasible even in very large samples. S patial Logit Estimators for Large Samples The estimation methodology developed in Klier and McMillen (2008) for a binary logit model can be readily extended to a multinomial setting. First, consider a model with unordered choice alternatives. To set up the spatial discrete choice model, c onsider an individual who makes a choice among alternatives As point of departure from the RUM model discussed in the previous chapter, t alternative is now a function of a actors: with (3.1) where is a latent dependent variable representing the und e rlying ut ility from a particular alternative ; denotes the spatial weight s relating neighboring observations and ; and denote alternative invariant and alternative varying covariates, respectively; denotes the vector of errors fo r alternative ; and and
78 a re the model parameters. The vectors of errors for all alternatives are assumed to be independent and identically distributed The spatial weigh t s matrix is typically row standardized such that for and The spatial autoregressive parameter measures th e degree of spatial dependence. A positive (negative) implies that high values of the latent variable for observations neighboring observat ion increase (decrease) the value of for observation The alternative with the highest latent utility is the one chosen by the decision maker which is the choice observed in The dependent variable can be defined in a bina ry form as : The model in ( 3 1 ) forms the basis for a multinomial mixed logit (MXL) model with a spati ally lagged dependent variable. Analogous SAL specifications for the conditional logit (CL) and multinomial logit (MNL) models ar e obtained as special cases of (3.1) with the proper subset of explanatory variables. T he reduced form of the spatial model in (3.1) is given by : w ith, (3.2) where are the elements of the spatial matrix The presence of the latter becomes more evident by writing the reduced model (3.2) i n matrix notation as follows : (3.3) with the resultant covariance matrix: The model covariance structure implies that the error terms are both autocorrelated and heteroskedasti c, which is the reason for the standard discrete ch oice model to be
79 inconsistent. Following the notation in Klier and McMillen (2008), d enote by the variance of the error terms given by the diagonal elements of A variance normalized versi on of the model is obtained by rescaling the model as : w ith, (3.4) The model in (3.4) i mplies the following choice probabilities : (3.5) w here ; , ; and If the error terms are assumed to be distributed IID type I extreme value the difference of the error terms is distributed logistic. These distributional assumptions give rise to the spatial mixed logit (SMXL) model and the choice probabilities take the functional form: (3.6) The present model can in principle be estimated using GMM (as in Pinkse and Slade, 1998) or nonlinear TSLS employing a set of instrument al variables and the gradients The spatial GMM estimator is the set of parameter values that minimizes the objective function where denotes the vector of generalized model residuals ; is a matrix of instruments ; and is a positive definite matrix ( and will be discussed later) The generalized model residuals are simply If the model reduces to nonlinear TSLS.
80 The gradients f or the SMXL model are 10 : (3.7) w here , and I n this approach the estimates are obtained through an iterative optimization process with the following parameter updating rule: where is an initial set of parameter values ; are t he predicted values of the gradient terms from the regression of the gradients on the set of instruments ; and are the generalized model residuals. The algorithm converges when approaches zero. Importantly, e ach step of th is iterati ve procedure requires the i nversion of the matrix ). The main estimation insight from Klier and McMillen (2008) is to avoid the repeated inversion of ) by linearizing the model which in itself can be argued to be an approximatio n to the true model around a particular point of initial parameter values. A convenient choice is for which consistent estimates of and are obtained from the standard MXL model. The linearization of the sp atial model around is possible because, once we set the initial value of equal to zero the parameter is still identified f rom the remaining term s and ( both a function of ) 10 The derivation of the gradient expressions for all the models considered in this section is relegated to Appendix B.
81 in the corresponding gradient in (3. 7 ) At the linearization p oint, t he gradients simplify greatly since ; and become and respectively; and the terms containing vanish because and by construction Inversion of ) is no longer needed as The gradient terms of the linearized SMXL model are: (3.8) The linearization of the spatial model i s carried out by linearizing the generalized residuals around the initial parameter values as : The GMM estimator of the linearized model is the set of parameters that minimizes the objective function where This estimation procedure is no longer iterative and involves two simple steps. First, the model is estimated with standard mixed logit, implicitly linearizing around the estimated mixed logit pa rameters and the initial valu e of equal to zero. The initial parameter values are used to estimate the generalized model residuals and the model gradients given by (3.8) In the second step, each estimated gradient and is regressed on the set of instruments and fitted values ( , and ) are cons tructed. Finally, the coefficients in the regression of ( ) on , and are the estimated parameter values of interest
82 Similar spatial specifications for the multinomial logit (MNL) and conditional logit (CL) models are obtained as special cases of (3.1) with the appropriate subset of explanatory variables 11 A MNL model with a spatially lagged dependent variable is obtai ned by setting in ( 3.1 ) so the model allows only for alternative varying covariates The ensuing choice probabilities for the spatial multinomial logit (SMNL) model are: (3.9) Given these choice prob abilities, the gradient terms for the SMNL model are : (3.10) where and The linearization of the SMNL model around is possible because the gradient of in (3. 10 ) is nonzero at due to the term (a function of which does not vanish but reduces to The gradients of the linerized SMNL model are : (3.11) Similarly, a SAL specification for a CL model is obtained by setting in ( 3.1 ) to allow for individual specific but alternative invariant covariates The choice probabilities for the spatial con ditional logit ( SCL ) model are: 11 Refer to Appendix B for the set up of the SMNL and SCL models and the corresponding gradient derivations.
83 (3.12) These choice probabilities result in the following model gradient terms: (3.13) where and The linearization of the SCL mo del around is also possible since the spatial parameter is identified f rom the remaining terms (a function of ) in the corresponding gradient in (3.13), which simplif y to T he gradients of the linearized SCL model are : (3.14) The linearization of the SMNL and SCL models reduces their estimation to a two step procedure analogous to the procedure described for the SMXL model with the corresponding initial set of parameters values and respective model gradients ; and for the SMNL model and a nd for the SCL model The parameter estimates from the two step procedure are and respectively. For a SAL specification of a discrete choice model with ordered choice alternatives, th can be modeled as follows :
84 with, (3.15) where denotes the latent d ependent variable with observable counterpart ; denotes the explanatory variable s ; denotes spatial weight s for observations and ; is the spatial lag parameter ; is a vector of model parameters ; and is a vector of d isturbances. The natural ordering of the alternatives is a result of the latent variable falling into various mutually exclusive and collectively exhaustive ranges given by the threshold parameters and with and The reduced from of the model is given by: with, (3.16) where are the elements of the inverse matrix In matrix notation, the reduced form can be written as: (3.17) with resulting covariance matrix: This covariance matrix also implies autocorrelated a nd heteroskedastic disturbances. As before, d enote by t he variances of the errors given by the diag onal elements of the covariance matrix Normalizing the model for heteroskedastic variances we have : w ith, (3.18)
85 Assuming that the model error term is distributed logistic, the choice probabilities for the spatial ordered log it (SOL) model are given by: (3.19) where ; ; and to make the scaling of the threshold parameters explicit. The choice probabilities in (3.19) result in the following model gradients: (3.20) The linearization methodology proposed in Klier and McMillen (2008) to avoid the inversion of can be used to estimate th is SOL model only if the gradient with respect to the spatial lag parameter in (3. 20 ) is nonzero once we linearize the model around At the linearization point, and all the terms containing become zero. The gradient of however, remains identified from which re duces to Hence, the gradients for the linearized model are:
86 (3.21) T he estimation procedure for the linearized SOL model consists of the following two steps. In the first step the model is estimated with standard OL to obtain the initial parameter values which are used to compute the generalized residuals and the model gradients in (3.21) In the second step, each estimated gradient is regr essed on the instruments and fitted values are obtained Finally, is regressed on , and t o obtain t he estimated parameter values of the SOL model The lineariz ation approach described in this chapter provide s a good approximation to the underlying spatial model s in ( 3 1 ) and (3.1 5 ) provided the latter are correctly specified and is relatively small ( i.e. close r to the linearization point). Applications of th ese spatial models are present ed in subsequent chapters along with Monte Carlo evidence of the small sample prope rties of some of the proposed spatial estimators Besides the empirical studies provided in this manuscript a nother application of one of these spatial estimators, namely the SMNL estimator can be found in Li ( 2010).
87 In order to carry out these estimati on procedure s a remaining issue is the specification of the instrumental variables in For a SAL model, it is c ommon practice to specify to contain the linearly independent columns of as suggested by Kelejian and Prucha ( 1998 ) We employ the KP instruments for the Monte Carlo analysis and the two empirica l studies that follow
88 CHAPTER 4 SPATIAL ANALYSIS OF LAND USE CONVERSION AT TH E RURAL URBAN FRINGE Background Urban decentralization and dispersion trends have led to increased conversion of rural lands in many urban peripheries and exurban regions of t he U nited States (U .S ) Moreover growth in exurban areas is outpacing gro wth in urban and suburban areas, substantially changing the composition of the rural landscape The growth pressure is particularly high at the rural urban fringe 1 Land use changes occurring in the fringe and beyond warrant examination for the following reasons. First, these areas are under continuous land use c onversion which is typically low density and land intensive An analysis of the 1997 American Housing Survey Data conducte d by the United States Department of Agriculture Economic Research Service f ound that, s ince 1994, 55 % of the total land developed in the U.S. has been developed as housing lots greater than 10 acres and 90% as lots greater than 1 acre In addition, 80% of all new development has occurred outside existing urban areas or in nonmetropolitan areas (Heimlich and Anderson, 2001). This form of land development referred to as is often argued to lead to i nefficient and costly development patterns 2 Second direct connection s exist between individual land use conversion decisions and aggregate impacts of land use c hanges (Bell and Irwin, 1 The 1990 U.S. Census defines urban fringe as rural areas that are located in metropolitan counties. These areas are characterized by development t hat takes the form of new buildings (residential, commercial) and infrastructure, but does not meet the density requirement for urban classification. Exurban growth occurs in the rural areas beyond the urban fringe. It takes the form of large lot developme nt and consists of scattered single family houses on large parcels of land. 2 Although urbanization follows stages of growth that are well understood, urban analysts debate whether a natural outcome of well functioning housing and land m arket s or a result of market failure s Settling this argument is beyond the scope of this work. Thus, when referring to urban development or growth
89 2002). Thus, l and use conversion analysis at a micro level (decision making level) provide s valuable insights in to the formation of land use patterns in terms of timing of conversion and the spatial distribution of land uses. D evelopment costs such as local provision of public services, are shown to be a function of both the development pattern s and the rate at whi ch the conversion of land occurs (Frank, 1989; Ladd, 1992 ; Burchell et al., 1998). Lastly, i incentives. T heir decisions however, may inflict economic, social, and e nvironmental costs upon local co mmunities that need to be addressed in public policy formulations. As the urban growth pressure keep s rising imposing continuous restraint s on open space and resources it necessitates further examination of the land use conversion process in order to ob tain a better understanding of this process and the driving factors behind the land use conversion decisions The economic literature on land use has been predominantly concerned with the efficient allocation of land as a scarce resource among competitiv e uses One strand of this literature has examined the determinants and trends of land use changes ( e.g. Brown et al., 2005 ; Lubowski et al., 2008 ) Another strand has analyzed urban sprawl and its consequences (e.g. Ottensmann, 1977; Peiser, 1989; Ewing 1 997; Burchell et al., 1998; Rusk, 1999; Brueckner, 2000). A third area has focused on the evaluation of existing land conservation programs (e.g., Kline and Alig, 1999; Newburn et al., 2006; Towe et al., 2008) and growth management controls (e.g. Fischel, 1990; Navaro and Carson, 1991; Feiltson, 1993; Nelson and Moore, 1996; Nelson, 1999; Kline, 2000) In the light of rising environmental concerns, more recent work has shifted focus towards quantifying the linkages between land use change s and ecosystem cha nge s with the
90 purpose of forecasting such changes ( e.g. Lewis, 2010). W hile recognizing that land use conversion is a spatial process associated with spatial externalit ies ( and spillovers) th is literature has only occasionally explicitly modeled the spati al dependence that characterizes the land use conversion process Because of the estimation complexity of the spatial models s patial dependence has either been overlooked (e.g. McMillen, 1989) resulting in potentially inconsistent estimation or spatiall y disperse data have been aggregated, resulting in artificially sharp intraregional differences or unrealistic interregional uniformity (Bockstael, 1996). The purpose of this study is to examine the factors that drive land use conversion decis ions while inv estigating the possibility that their decisions are spatially interdependent A decision to convert a given parcel of land to a particular land use is hypothesized to depend among other factors, on the propensity of landowners of nearby parcels to choose that particular use. This suggests that the type of land conversion that is likely to occur in o ne area may influence (or may be influence d by) the type of land conversion in adjacent areas. Accounting for the potential presence o f spatial interaction among individual landowners is important for two reasons. From an empirical perspective estimating a standard model with spatially dependent observations will likely result in biased estimates and invalid statistical inference. From a policy perspective, the presence of spatial interactions among landowners who constitute the supply side of local land markets suggests that land parcels do not develop in seclusion As such, uncoordinated local land use policies designed at a small scal e such as subdivision regulation and zoning while attempting to
91 manage growth at a local level may fragment urban development and result in suboptimal land use patterns at a regional level use conversion decisions, w e borrow an economic model of land use conversion from the optimal time to development literature. The base of the economic model is a landowner who is assumed to make an inter temporal, profit maximizing choice regarding the conversion of a parcel of land to some alternative use. Since only a number of factors that affect the stream of returns and the cost of conversion are observable to the analyst, the net returns are modeled as having a systematic observable portion and a random unobservable portion. Thi s treatment of the net returns allows for a latent variable formulation of the optimal land use decision and allows for probabilistic statements about the land use choice. Since the data on land use is typically categorical and the choice of la nd use is mutually exclusive, the theoretical model leads naturally to an empirical discrete choice framework. The empirical model used in this study is analogous to other spatially explicit land use change models estimated at the parcel level within a dis crete choice framework (e.g. Landis and Zhang, 1998; Irwin, 2002; Carrin Flores and Irwin, 2004). U nlike the preceding analyses that only consider two land uses, our spatial analysis encompasses f our land use categories agr icultural, residential, indust rial, and commercial to accommodate for the possibility that land conversion to different urban uses may involve different development processes The hypothesized spatial interaction among landowner s regarding land conversion decisions is represented as a spatial lag Th us w e estimate a multinomial logit model with a spatially lagged dependent variable using the linearization approach developed in the previous chapter. This novel estimation
92 method is feasible to use with parcel level data allowing us to model the individual land use conversion decisions at the appropriate scale while exploiting the richness of parcel level data The latter enables addressing not only the spatial dependence on the land use change decisions but also the sp atia l heterogeneity of the landscape and land use policies The remainder of th is chapter is organized as follows. First, some of the external costs and benefits that make the urbanization phenomenon policy w orthy are reviewed Next a model of land use conve rsion is developed to account for the assumed spatial interaction among landowners of neighboring parcels in the form of a spatial lag. The subsequent section t ransitions the economic model into an econome tric model to be estimated using the SMNL estimator developed in the previous chapter. This is followed by a Monte Carlo study of the finite sample properties of the spatial estimator A description of the data and the geographic area of study Medina County, Ohio along with a discussion of the empirica l results succeed Conclusions are drawn in the f inal section. Impact s of Urban Growth The outward expansion of urban areas into rural areas and t he resultant sprawling development patterns are an outcome of the interplay between private decisions and p ublic policies. A fundamental aspect of the dynamics of the American society is that as individuals and businesses achieve economic success, they relocate (Bier, 2001). located in less dense areas is driven by a preference for more spacious living arrangements, natural amenities such as scenic views and recreational opportunities quality of public services, etc. Potential benefits from low density development include neighborhoods with lower crime rates, f lexible
93 transportation, access to more open space, more privacy, a better sense of place and community, cheaper land and fewer taxes, better air quality and a geographic separation of residences from commercial and industrial activities (Gordon and Richardson, 1 997). Firms on the other hand, come as a response to economic opportunities such as agglomeration economies, better access to input and output markets, favoring public policies (e.g. lower taxes or less stringent environmental regula tions ) etc. S ome government al policies have also encouraged exurban developmen t. E xamples of these policies are the expansion of highways (e.g. Heavner, 2000) income tax subsidies for housing at a federal and state level, extension of public utilities an d zoning at a local level ( e.g. Moss, 1977; Anas et al. 1998; Pasha, 1996 ; Anas, 2001 ) In addition, the spatial distribution of urban growth is largely shaped by technology advancements that continue to lower the cost of transportation and communication facilitating a trend of a more dispersed economy. Nelson (1992) summarizes the drivers of urban growth in to four main categories: the continued deconcentration of employment and the rise of exurban industrialization, the latent antiurban and rural locatio n preferences of U.S. households, technology advancements that make exurban living possible, and development policy biases favoring exurban development over compact development. S prawling urban development is associated with regional economic, environment al and social impacts that are both positive and negative. From an economic perspective, while urban growth enhances economic activity in the areas in which it is occurring it also decentralizes economic growth. Positive impacts on rural communities invo lve opportunities for off farm employment niche market access,
94 increase d values of rural lands etc. Ahearn et al., (1993) report that a n average farm household earns more off farm income than on farm Moreover, off farm employment opportunities have been crucial to the survival of many farms (Alig and Ahearn, 2006). However, urban development also results in increased land rental rates, localized declines in agricultural econom ies and loss or fragmentation of agricultural land s Importantly, prime agricu ltural land is continuously being lost to urbanization (Gardner, 1977). Heimlich and Anderson (2001 ) assert that low density and fragmented development patterns do not threaten the national food and fiber production, but change the agricultural product and service mix and may result in reduced production of some high value or specialty crops. Potential benefits from low density development like b etter housing choices, improved infrastructure, freedom from public transportation, diverse location options for businesses and households involve higher costs of provision of public services, higher transportation costs from traveling over longer distances, etc. In addition, they incur social c ost s such as potential conflicts between farmers and homeowner s altera tion of the rural character compositional change in rural population, and increased separation of the urban poor. Porter (1997) shows that changes in community attributes, such as the mix of residents or loss of open space, are often perceived as costs by long term residents of the community. Land use changes are also responsible for a growing number of environmental concerns. They are considered a major driver of changes occurring in natural ecosystems (Naiman and Turner, 2000; Armsworth et al., 2004). L and use and land cover c hanges are ranked second after climate change for their effects on the
95 functionality of these ecosystems (Grimm et al., 2008), the urbanized land contributing a major share (Alberti, 2005; Collins et al., 2000). The distribution of the population at a lower density results in loss of open space, increased traffic congestion and pollution, degradation of water quality, and fragmentation or loss of natural areas. Various ecological studies conducted on exurban development suggest signi ficant impacts on biodiversity (Hansen et al., 2005). A long term result of continuous land conversion could be declines in populations of desirable plant and wildlife species and increases in populations of opportunistic species (Maestas, 2007). Furthermo re, regional changes in land cover and land use have been found to have cumulative effects on global climate change. Many of the aforementioned impacts take the form of externalities and incur costs that are not internalized (Burchell, 1998) Thus, ineffi ciencies in land use patterns emerge. In the absence of market imperfections, an individual landowner use choice coincides with the socially optimal land use choice. However, in the presence of externalities social action is needed to manage social and environmental resources calling for growth managemen t and smart growth initiatives Hence, s tate and local governments have a dopt ed preservation of open space programs, conservation and retention programs, l and use planning policies and growt h controls The role of these policies is to alter the private incentives so as to narrow the divergence between private and socially optimal outcomes and direct urban growth toward more sustainable areas A better understanding of the land use conversion process would help policymakers design more effective regional environmental, growth, and development policies.
96 E conomic Model of Land Use Conversion To frame use conversion decisions, w e borrow an economic model of land use conversion from the optimal time to development literature ( e.g. Arnott and Lewis 1979; Fujita 1982 ; Capozza, Helsley, and Mills 1986 ; Capozza and Helsley, 1989 ) C onsider a risk neutral price taking economic agent (the landowner) that owns parcels of land of uni form quality 3 The landowner chooses an optimal land use for a parcel such as to maximize the present discounted sum of the expected stream of future net land returns ( net of conversion costs ) 4 T he landowner builds his expectations of future net returns b ased on information about historic and current return values 5 In addition, the net returns are a function of land attributes which can be classified in three categories: biophysical characteristics, location (neighborhood) characteristics, and land use r egulation s For instance, land attributes associated with the agricultural land use may include land biophysical characteristics such as soil type, slope, fertility level, water holding capacity, etc. ; location characteristics such as proximity to market, transportation routes, public infrastructure, etc; and land use regulations such as subdivision ordinances and zoning. In the case of residential or commercial land use, the expected returns may be a function of the distance to urban centers, employment, s hopping sites, neighborhood amenities, public services, zoning etc. Lastly some 3 The unifor m quality assumption allows land use choices for a heterogeneous parcel to be treated as the sum of land use choices on constituent uniform quality parcels when returns and conversion costs are approximately linear in land quality (Lubowski et al, 2008). 4 An efficient land market produces a gap between the price of land at the boundary of an urban area (minus conversion cost) and the value of agricultural land rent because of expected future rent increases due to development (Capozza and Helsley, 1989). 5 Schatzki (2003) provides some empirical evidence in support of the assumption that land returns tend to hold over time (i.e. follow a random walk)
97 factors that affect the stream of returns and the cost of conversion a re not observed by the analyst, so net returns are composed of a systematic observable portion and a ran dom unobservable portion. This treatment of the net returns entails a latent variable formulation of the optimal land use decision and allows for probabilistic statements about the land use choice by the landowner. Since the data on land use is typically c ategorical and the choice of land use is mutually exclusive, the theoretical model leads naturally to an empirical discrete choice framework. Note that the assumptions above have to be modified to model land use conversion under a growth management scenari o, since the decision of the landowner is conditioned upon objectives. L et denote the returns from a parcel of land currently in use at time and denote all observed attributes associated with parcel Let where denote s the attributes that directl y affect land returns and denote s those parcel attributes that affect returns through conversion costs Note that represents the benchmark land use at time ; land could be at an undeveloped state with the potential to convert to a develope d state, or it could be at a developed state with the potential to convert to an alternative developed state 6 Th e land returns for parcel in use a re g iven by: (4.1) 6 C onversion of land typically occurs in one direction, from an u ndeveloped state to a developed state, an d generally from a rural land use (e.g. agricultural, forest) to an urban land use (e.g. residential, commercial, industrial). The reverse conversion from urban uses to other land uses is, in comparison, much more costly making the land use conversion deci sions potentially irreversible. The model presented here, however, does not impose any restrictions on the direction of land conversion.
98 where represent s the uno bserved p arcel attributes Let denote the net returns from a parcel of land in use at time Then, t he net returns for parcel in use are obtained as : (4.2) where is the expected marginal cost of conversion. Under the model assumptions, the dynamic optimization problem simplifies to a one period optimization problem with the decision rule to choose the land use with the highest expected net return. Hence, t he landowner will choose to convert parcel in use to an alternative use if and only if : (4.3) where and are the expected net returns of parcel in competing uses and Thus, parcel will be converted to use if the expected net returns from use exceed the expected returns from the current use or any other alternative land use in the choice set. If the inequality does not hold for any of the uses in the choice s et, parcel will remain in the original state Th e decomposition of the net returns into a systematic observable portion and a random unobservable portion entail s a latent variable representation of the problem. pa rcel under current land use : (4.4) where is observed but not Then, parcel will be convert ed to use if and only if : (4.5) This reformulation of the optim al land use decision can be interpreted in the context of pressure for conversion and allows for probabilistic statements about the land use choice by the landowner. The probability that the landowner chooses land use over land use is the probability that the unobserved factors when combined with the
99 observed factors are such that the are greater than under land use result ing in th at particular outcome. Thus, t he probability that a parcel will be converted f rom use to use is given by: (4.6) In addition to the parcel attributes, it is plausible that t he decision to convert a given parcel of land from one use to another will also depend on the propensity of landowners of nearby parcels to choose a particular use (ei ther strategically or collaboratively). For example, a parcel in agricultural use may be less likely to convert to residential use if it is surrounded by parcels that are likely to remain in agricultural use. This feature of the land use decision s naturally leads to dependence over space. In this case, the net returns ( for parcel ) from a given alternative are a function of the observed attributes and unobserved attributes b ut now is c omprised of own parcel attributes and nearby parcel s attributes (owned by a different landowner) that for the given alternative : (4.7) As before, parcel will convert to use if and only if: (4.8) The current model implies th e following probabilities of choosing alternative : (4.9) Equivalently, the probability that the landowner chooses land use is the expected value of the following indicator function over all possible values of unobserved factors: (4.10)
100 where Thus, the probability that a parcel of land will convert to an alternative land use depends among other things on the underlying factors determining the probabilities of neighboring parcels to convert (or remain) in that particular land u se. Th e model presented in this section is a simplified representation of the land use conversion process and has been widely used to model landowners behavior (e.g. McMillen, 1989; Bockstael and Bell, 1997; Landis and Zhang, 1998; Irwin and Bockstael, 2 002). The model can be made dynamic by recognizing that the returns of a parcel of land in different uses thus its probability of conversion may depend on the initial state parcel changes over time, and changes in the surrounding landscape ( Bockstael, 1996). Econometric Model Th e choice probabilities for the land use conversion model in (4.10) can be empirically estimated once a functional form is assigned to the observed portion of the latent net returns and a distributional assumption is m ade for the unobserved portion. A common approach for dealing with spatially interdependent decisions i s to estimate a spatial autoregressive lag model after quantifying the spatial interactions using a spatial weight s matrix W e specify a s patial autoregressive lag model in the context of a multinomial logit model since the observed parcel attributes are location specific T he spatial multinomial logit (SMNL) model takes the form : with, (4.11) where ( represent s the l atent land net returns of parcel from land use ( ) ; denotes the spatial weights relating parcels in locations and ; denotes observed
101 parcel attributes ; is a binary variable that takes a value of 1 if alternative is chosen by the landowner of parcel and zero otherwise; represents unobserved parcel attrib utes associated with land use ; and and a re the parameters of interest. For a multinomial logit model, t he disturbances are assumed to follow an IID type I extreme value distribution. The spatial weight s matrix is row standardized such that and The alternative with highest latent net returns is the one chosen by the landowner which is the land use choice observed and indicated by in (4.11) The parameter measures th e strength o f the spatial interaction between landowners of adjacent land parcels regarding their land use change decisions. In this context, a positive means that a high propensity of a given landowner to convert his land parcel to a particular land use positively affects the propensity of a landowner of a nearby parcel to conver t the latter to the land same use, thus increasing the probability of conversion of both parcels to the given land use. conversion decisions under posi tive spatial dependence results in clusters of land parcels in similar use. Moreover, a higher implies a stronger mutual influence of landowners of adjacent land parcels in their respective land conversion decisions A spatially lagged dependent variable has been incorporated successfully in models where the dependent variable is continuous (e.g Case et al., 1993; Brueckner, 1998; Brett and Pinkse, 2000; Saavedra, 2000; Fredriksson and Millimet, 2002) T he estimation of a s patially autoregressive lag model however, is challenging in a discrete choice framework To estimate the model we use the linearization methodology developed in the previous chapter which reduces the model estimation to two simple
102 steps: a standard multinomial logit model with no spatial dependence followed by TSLS estimation of the linearized model which accounts for the sp atial dependence. More specifically, the estimation procedure consists of: Step 1 T he model is estimated with standard multinomial logit to obtain initial parameter values for and the initial value of is set equal to zero. T he initial paramete r values are then used to compute the generalized model residuals and model gradients At the linearization point the gradient terms f or the SMNL model are given by: (4.12) Step 2 In the first stage of the TSLS procedure, e ach estimated gradient in is regressed on the set of instruments (KP instruments) and fitted values are constructed ( and ). In the second stage generalized residuals are computed as ( ) which are then regressed on and The coefficients obtained from the last regression are the estima ted parameter values of interest Monte Carlo Analysis Prior to estimating the land use conversion model, we conduct a Monte Carlo study i n order to assess the finite sample performance of the SMNL estimato r 7 The adjustments for the mult inomial choice extension. We consider four choice alternatives 7 Similar simulation studies were also conducted to examine the finite sample properties of the SCL and SMXL estimator s. Those studies generated qualitatively similar results.
103 to make the simulations comparable to the land use application and examine two scenarios In the first scenario, all four alternatives have the same choice probabilit y of 0.25. To generate this data we s tart with the reduced form of the model (4.11) written in matrix notation as: (4.13) with covariance matrix We generate a single explanatory variable u niformly distributed in the interval Next, we specify (see below) and set the value of between and varying it in increments of Having specified and a value for t he explanatory variable is first transformed to obtain as where is the square root of the error variances given by the diagonal terms of A second transformation follows to obtain as For simplicity, each of the parameters is set equal to Equal parameter values for the different choice alternatives and a fixed (thus, a fixed ) ensure equal choice probabilities. Subsequently, the simulated probabilities are obtained as: (4.14) To generate the observed individual choices based on the simulated probabilities, we generate a uniform random variable ( ) and set for if : w ith (4.15) In the second scenario, we modify the way we generate the data to accommodate for different choice probabilities. This is the case that is more likely to represent a real data situation. To generate this data we first fix the choice probabilities and sol ve for the parameters that correspond to the desired probabilities (Appendix C). Specifically, we set to resemble to some
104 extent to the proportion of land parcels in each land use category in our dataset The explanat ory variable is generated uniformly distributed in the interval and transformed as described above to ob tain and The parameters are now a function of the expected value of this is the reason for changing the interval of to avoid a mean of zero and as a result a function of hence denoted by ( Having obtained and the parameters t he simulated probabilities are generated as in (4.14) and the observed individual choices are assigned based on the simulated probabilities using the assignment rule in (4.15) The spatial weight s matrix in both cases is specified as a row standardized first order contiguity matrix setting the non zero elements to : with endpoints This specification pr e serves the spatial structure of the model but simplifies considerably the model estimation The matrix of instruments is specified to contain the linearly independent vectors in A nave estimation procedure a standard multinomial logit model which ignores the spatial effects (MNL) is used as an estimation benchmark. To summarize the simulation results, w e report the average bias and root mean squared er ror (RMSE) of the three estimated slopes (one is normalized to zero) and the spatial parameter estimated by the SMNL model In general, we expect that if the true spatial structure in the data is accurately represented by the specified s patial model, the l inearization of th e latter will provide accurate estimates as long as the value of spatial dependence as measured by is small with more accurate estimates corresponding to smaller values of In addition, we expect the SMNL model to produce better estimates ( with smaller bias and RMSE ) than MNL in the presence of spatial dependence.
105 Two sets of simulation results for the equal probability choice scenario based on two different sample sizes are presented in Table 4 1 and Table 4 2 Table 4 1 reports the simulation results based on 1,000 replications for a sample of 1 000 observations Table 4 2 report s results based on 500 replications for a sample of 5,000 observations Both sets of results indicate that the performance of the MNL model in the estimation of the slope coefficients as measured by the average bias and RMSE of the corresponding estimates progressive ly deteriorates as the spatial dependence induced in the data increases. The SMNL estimator has a relatively similar performance to the MNL estimator regarding the estimation of the slopes, both in terms of average bias and RMSE for the sample size of 1,0 00 observations (Table 4 1 ). Our expectation of the performance of SMNL estimator relative to MNL estimator regarding the estimation of the slopes as spatial dependence increases is only weakly confirmed by the results from the larger sample (Table 4 2). I n this case, the SMNL estimator produces estimates that have smaller bias and RMSE compared to the corresponding estimates from the MNL model however the difference in the average bias and RMSE of the slope coefficients produced by the two models is negli gible An obvious disadvantage of the MNL model is that it does not produce an estimate of the spatial lag parameter This parameter is in itself of interest in practice as it indicates the presence of spatial interactions or spatial spillover effects. Regarding this parameter, t he results exhibit the anticipated trend that as the spatial dependence induced in the dat a increases (i.e. for higher values of true ) so does the bias and RMSE of the estimated This trend is a byproduct of the model linearization. More specifically, o ur results indicate that is accurately estimated by SMNL model when the
106 v alue of tru e is zero which is important since it correctly identifies the absence of spatial dependence but also for values of that are closer to zero ( the linearization point ) In particular, the SMNL estimator performs remarkably well in capturing the deg ree of spatial dependence induced in the simulated data for values of true between and A nother way to evaluate the performance of the SMNL estimator regarding the estimation of the spatial parameter is to compute the relative percentage bias w hich expresses the average bias as a percentage of the true value of the spatial parameter 8 In both sets of simulations the relative percentage bias of the estimated when the value of true is between 0.1 and 0.5 is relatively low and ranges betwee n % (for ) and % (for ) The relative percentage bias becomes much higher for true values of larger than 0.5, reaching up to % when is set to 0.9 The simulation results for the different choice probabilities scenario based on th e sample size s of 1,000 observation s and 5,000 observations are presented in Table 4 3 and Table 4 4 respectively These results exhibit similar general trends in terms of the deterioration of estimated slopes for both models as the induced spatial depend ence ( ) increases. However, T he SMNL estimator outperforms the MNL estimator regarding the estimation of the slopes, both in terms of average bias and RMSE. I n some instances, the average bias and RMSE of the estimates obtained by the SMNL model amount to less than half of the magnitude of the respective bias and RMSE measures for the MNL model. Thus, the estimation of the slopes deteriorates much faster when the model is estimated using MNL. These results confirm our expectations o f the performance of SMNL est imator in the presence of spatial effects. We note however, that in some 8
107 cases the slope coefficients estimated using the SMNL model still h ave considerable bias In terms of the estimation of the spatial parameter ( these results exhibit similar trend s to the previous scenario. A s the induced spatial dependence increases so does the bias and RMSE of the estimated As before, the SMNL model produces quite accurate estimate s of when the true value of is set equal to zero and for small values of true In particular, the SMNL estimator performs well in capturing the degree of spatial dependence for true values of between and In this case, t he relative percentage bias of the estimated r anges between % and % when the true value of the spatial parameter is set between 0.1 and 0.5. The relative percentage bias of the estimated increases up to % when the true value of is set to 0.9 Higher accuracy (smaller bias) and precision (smaller RMSE) of the estimates can be obtained by increasing the sample size. In general the results from this Monte Carlo analysis indicate that the SMNL estimator captures well the degree of s patial dependence in the simulated data as long as the valu e of the induced spatial dependence as measured by is small hence closer to the linearization point We further note that t he SMNL model generally outperforms the MNL model with regard to the estimation of the slope coefficients although the latter are still sometimes estimated with considerable bias Judging from these simulation results, the linearization approach appears successful. Determinants of Land Use Choice In this section, we follow the spatial ly explicit approach described in the previous sections to analyze the factors that drive the land use conversion process We are
108 particularly interested in those factors that affect the pattern of new urban development as it occurs in rural urban fringe areas. The typical pattern of new development in these areas thr oughout the U.S. is low density and land intensive resulting in sprawling urban patterns The area considered for this study is a rural urban fringe county within the Cleveland, Ohio metropolitan area Medina County located just south of the c i ty of Cleveland and its suburbs is typical of such development (Figure 4 1) The parcel database for Medina County is comprised of data from the Medina information on land uses, parcel characteristics, major roads, and socio economic indicators. The dis aggregation scale of the data at the parcel level is appropriate for modeling the economic decision s of the individual landowners. In addition, the parcel level data enables addressing the spatial heterogeneity of the landscape (e.g. soil type and quality) as well as local land use policie s (e.g. zoning) To generate the parcel level data, we use a Geographic Information System (GIS) that allows generating the relevant set of variables using the geocoded parcels a nd additional GIS data layers. The data lay ers include 1990 land use (Medina County and Cleveland State University), major roads (Ohio Department of Transportation), soil type (STATSGO), Census block group boundaries and data from the U.S. Census of Population (U.S. Census Bureau). In addition, we use buffers to obtain neighborhood attributes that are likely to affect land conversion such as proportion of land in surrounding land uses population and housing densit ies etc Figure 4 2 shows changes in land use patterns in Medina County from 1970 t o 2000. The prevalent land development in this region is residenti al, accounting for more
109 than 85% of land development in the last 30 years. The pattern of urban development in Medina Count y is comprised of clusters of land tracts in similar land use, prov iding some preliminary visual evidence of positive spatial dependence. This type of evidence nonetheless, is helpful for conceptualizing the structure of spatial dependence that need s to be specified and included in the model. Mo reover, Figure 4 2 suggest s that u rban development in Medina County may involve divergent development processes for different urban land uses. For instance, residential development seems to have become more fragmented and dispersed whereas the commercial and industrial land develo pment has become more clustered over time. For this reason, we employ a model that encompasses multiple urban land uses. The major urban center of our study area is Cleveland so we measure proximity from each parcel centroid to the center of Cleveland vi a the major roads network (Totdiscle). Local markets are important for urban land uses; therefore, distance to nearest city (Disttonear) is included. To capture the potential disamenity effects of population on urban development, we measure the density of population in 1990 within the local neighborhood of each parcel (Popdens). The localized housing demand from the in migration of urban residents in the region is captured by the proportion of houses in 1990 (Housedens). To investigate whether surrounding land uses confer either positive or negative spillovers, we include three variables that measure the proportion of the surrounding land in residential (Reside), commercial (Commarea) and agricultural (Agarea) land uses respectively We also capture const raints to the density of development from large lot zoning with a dummy variable that is assigned a value of one if the minimum lot size
110 is zoned as three acres or greater and zero otherwise (Largelot). T hese variables measure, to an extent, some of the po tential spatial dependence. Whether they suffice for that purpose can be seen from the statistical significance of the estimate of the spatial lag parameter in the SMNL model. To control for differences in township specific characteristics such as local iz ed land use policies we construct dummy variables for each township and normalize estimation to the Homer and Sharon Township s which are both v ery rural T able 4 5 through Table 4 7 provide a description of the variables used in the model, descriptive st atistics for those variables, and a summary of the proportion of the parcels in each land use category. The data set contains 9,760 parcels of which 5,991 parcels (61 %) are in agricultural use; 2,917 parcels (30%) are in residential use; 572 parcels (6%) a re in commercial use; and, 280 parcels (3%) are in industrial use. The average parcel is about 18 acres in size and is located about 100 miles from Cleveland. Because the true specification of is unknown, for our spatial analysis we construct four different weight s matrices that impose varying assumptions about the extent and gradient of spatial dependence. All of them set the non zero elements of the matrix to : which is the inverse Euclidean distance between locations and and parameter. The first two specifications of set the friction parameter to one and consider maximum cut off distances beyond which at 800 and 1600 m eters. The next two specifications set the friction parameter to two (inverse of the squared Euclidean distance) and employ the same cut off distances All matrices set and are row standardized (rows sum to one).
111 The e stimated parameters for t he MNL model our benchmark model and the various specifications of the SMNL model are presented in Table 4 8 in which the base category chosen is the industrial land use. In general, t he estimates from these models align w ith expectations In addition the SMNL estimates are reasonably robust across the different specific ations However, the sign of few of the coefficients changes occasionally across some of the SMNL model specifications This point needs further investigation. A possible explanation f or this discrepancy could be that some of the spatial weights matrices define neighborhoods that are rather large and include parc els in mixed land uses. The estimates show that as distance to Cleveland (Totdiscle) increases the relative probabilities of parcels in agricultural, residential, and commercial uses decrease. So distance to the major urban center is an important factor. L ocal markets are also important to these land uses as measured by the distance to the nearest town (Disttonear). The e stimat es indicate that the probability of residential and agricultural land uses is higher relative to the industrial land use as distance to nearest town decreases. In contrast, the relative probability of the commercial land use increases with this distance. A nother measure of proximity to local markets is per capita income (Percpinc). We would expect that the relative probability of agricultural land use decrease with a higher per capita income since agricultur al land development tends to occur in economically depressed areas. However, our estimates do not suggest a significant per capita income effect. Land characteristics of the parcel and characteristics of the surrounding area strongly affect the probability of agricultural use. For example, parcel size ( A cres ) has
112 the expected sign indicating that if parcel size increases, the probability of agricultural land uses is higher relative to the probability of industrial land use. The same holds for the commercial use. On the other hand, this relative probabilit y is lower for the r esidential use as the parcel size increases. In addition, these estimates suggest that the relative probability between agricultural and industrial land use is higher when the majority of the area surrounding the parcel is in agricultur al use (Agarea). Conversely, when urban uses surround the parcel (Reside and Commarea), the relative probability of agricultural use is lower. The p opulation density (Popdens) has a positive effect on the probability of commercial use relative to industri al use. Housing density (Housedens) is found to increase the relative probability of agricultural and commercial land use ( relative to i ndustrial use) but t he magnitude of this effect is fairly small. In addition, the housing density negatively affects th e relative probability of residential use, indicating a preference for lower density areas for this type of urban development. Interestingly, the effect of population and housing density on residential and commercial land uses becomes insignificant in some of the spatial model specifications perhaps an indication that these neighborhood effects are adequately captured by the spatial specification of the model. A binary variable representing large minimum lot zoning (Largelot) is introduced as a land use p olicy variable. Estimates suggest t hat the minimum lot size policy m inimum lot size zoned as three or more acres de creases the relative probability of residential land use relative to the other urban land uses, implying that if the parcel is
113 subject t o a restriction of a minimum lot size of three or more acres, the residential land use is less likely to occur. Our estimates also include indicator variables for the township the parcel is located in, although their estimated parameters are not presented in the table for readability. The township parameter estimates indicat e that a parcel located in any one of these townships is more likely to experience urban development than a parcel located in the township s to which we normalized the results: Homer Tow nship and Sharon Township. This is expected since Homer Township and Sharon Township unlike most of the other townships, are very rural and ha ve experienced almost no urban growth. Finally and most importantly the estimates of the spatial lag parameter ( ) produced by the SMNL model vary from 0. 26 to 0. 49 (depending on the specification of ), and they are all highly statistically significant. This strongly suggests the presence of positive spatial interaction in land use conversion decisions of landown ers of neighboring parcels More specifically, a high propensity of a given landowner to convert his land parcel to a particular land use positively affects the propensity of a landowner of an adjacent parcel to convert the latter to th at specific land use thus increasing the probability that the two neighboring parcels will convert to the same land use This, in turn, implies clustering of land tracts that are in similar land use a result that is in accordance with the spatial pattern of land uses for Me dina County observed in Figure 4 2 Since the discrete choice models are nonlinear in parameters, the estimated coefficients for both the MNL and SMNL models do not have marginal effects
114 interpretation. Therefore, the marginal effects need to be computed. Note that in our spatial model the spatial effects are global i.e. a change in the th observation of the th explanatory variable will have an impact on the expected probability of the event of interest (i.e the probability of conversion) not only for own region (parcel ) but also the other regions (neighborin g parcel ). This has implications when co mputing marginal effects as the marginal effects are no longer point estimates, but rather matrices with the diagonal elements representing the direct effects of a change in the explanatory variable and th e off diagonal elements representing the indirect (or spillover) effects (LeSage and Pace 2009). The marginal effects for our SMNL model are given by: (4.16) where denotes an vector of ones and represents element by element (Hadamard) multiplication. To provide scalar measures of the marginal effects, LeSage and Pace (2009) suggest to average over the diagonal elements to obtain a measure of the direct eff ects, average the row (or column) sums to produce a measure of the total effect, and take the difference between the se two effects to get a measure of the indirect effects. More specifically, the marginal effects for the SMNL model are obtained as follows: (4.17) The estimates of the marginal effects for the MNL model and th e different specifications of the SMNL model are presented in Table 4 9 The interpretation of a
115 marginal effect in a discrete choice model is straightforward ; i t measures the change in the event probability associated with a change in the average observat ion for a given explanatory variable. The interpretation of the marginal effects produced by a spatial discrete choice model is similar with the exception that now a change in a particular explanatory variable for a given observation generates multiple effects which are summarized into a direct effect, indirect effect, and a total effect. The direct effect shows the i mpact of a one unit change in a covariate on the event probability associated with the spatial unit of interest Th e indirect effect measures the i mpact of a one unit change in a covariate on the event probability associated with spatial unit s neighboring the spatial unit of interest. Naturally, the total effect as the sum of these two effects, shows the average total impact in the event probability associated with a change in a particular observation (i.e. the spatial unit of interest). Note that a marginal effect produced from the MNL model for a specific choice alternative would be interpreted as a direct effect, wh ich in this case also equal s the total effect since the indirect effect in a non spatial model is zero (LeSage and Pace, 2009) Comparing the marginal effects produced by the MNL model with the corresponding direct effects from the SMNL model the SMNL mo del produces marginal effec ts for all four land use categories that are considerably larger in magnitude for about half of the explanatory variables Note that the total effect of a marginal change in any of these explanatory variables in the probability o f conversion of a parcel of land to each of these land use categories is even larger if we consider the indirect effects. For the remaining half of the explanatory variables, the SMNL model produces larger direct effects (and total effects) for the commerc ial and industrial land use categories Thus,
116 the probabilistic impacts of the determinants of land use changes obtained under the MNL model especially for the commercial and industrial land use categories are likely understated A major disadvantage of the MNL model is that it does not provide estimates of indirect impacts, which as the estimates from the SMNL model specifications show are in general statistically different from z ero Inter estingly, the SMNL model specifications corresponding to the spat ial weights matrices that define the neighborhood using a 1600 meter buffer produce estimates of indirect effects that are of nearly the same magnitude as the direct effects. In this case, changes occurring in a parcel affect as much the probability of con version of surrounding parcels as they affect the probability of conversion of the own parcel. This, in turn, makes land conversion of neighboring land parcels to similar land uses much more likely. The interpretation of the estimates of the marginal eff ects complies with those of the estimated coefficients. The size of the parcel has a negative effect on the probability of conversion to residential use, but a positive effect in the probability of conversion to agricultural, commercial, and industrial use s. This is expected since r esidential development generally occurs in smaller parcels. Distance to Cleveland negatively affects the probability of conversion of a parcel to residential and commercial use and positively affects the probability of conversio n to i ndustrial use Proximity to markets seems to matter more to the agricultural and the commercial use s The marginal effects associated with the proportion of land parcels in surrounding land uses correctly show that land parcels are likely to convert (or remain) to the land use that dominates the neighborhood. However, we would expect these effects to become insignificant in the spatial model specifications since the neighborhood is
117 modeled explicitly Population density and housing density have a fai rly small effect on the probability of conversion to all land uses. It seems like the spatial model is adequately capturing these neighborhood effects. The marginal effects for our policy variable suggest that if parcels are zoned to three or more acres th ey are more likely to stay in agricultural use. Lastly, a cross all specifications of the SMNL model, the direct effects and indirect effects have the same sign, a result that is consistent with the presence of positive spatial dependence. On a final note, the SMNL model produces a matrix of marginal effects for each of the determinants of land use conversion to a given land use alternative providing abundant information regarding the direct impacts of each of the determinants of land use conversion on the probability of conv ersion of a particular land parcel as well as the indirect impacts that spill over from (or to) surrounding land parcels T he scalar measures reported in the section summarize these effects as averages over the whole sample. But, this is only one way to an alyze these effects. As LeSage and Pace (2009) point out, i f interest lies in a particular area ( neighborhood ) within the region of study rather than in the entire region the matrices of marginal effects can be appropriately partitioned to represent different neighborhoods and the marginal effects can be summarized in a similar manner. This is particularly helpful to identify hot spots of urban development. Final Remark s In this chapter, we estimate a spatially explicit model of land use conversion by employing a spatial multinomial logit model that explicitly accounts for spatial dependence in the land use conversion decisions in the form of a spatial lag W e also asses s the performance of the spatial multinomial logit (SMNL) estimator used to
118 estimate the model empirically through a Monte Carlo study. Simulation results indicate that the linearized spatial model provides a good approximation to the original spatial mode l and produces fairly accurate estimates of the degree of spatial dependence in the simulated data provided the original spatial model is correctly specified and the spatial dependence is fairly low (i.e. the value of the spatial parameter is close r to th e linearization point) Judging from these results, the linearization approach appears successful. There are several advantages to the estimation methodology used in this chapter compared to other spatial methods and standard modeling methods. First, unli ke the majority of spatial methodologies in the literature that are designed to model binary choice decisions this methodology is designed to analyz e spatially dependent multinomial choice decisions. Second, it is easy to implement even in large samples allowing us to estimate the model with disaggregated data at the parcel level, a scale that is appropriate for modeling i ndividual landowners Third, it enables the estimation of the direct and indirect (spillover) effects of the determinants of land use conversion providing abundant information about the impacts of the se determinants not only on t he probability of conversion of a particular land parcel but also on the probability of conversion of other land parcels located within a given proxi mity to the parcel of interest. In addition, it produces an estimate of the spatial lag parameter, which measures the strength of the interaction in land use change decisions of l andowners of nearby parcels. Knowledge of these effects as will become more evident shortly, is of value to inform policy. Finally, as indicated by the results from the Monte Carlo study, the SMNL estimator performs well in capturing the spatial dependence in the data A
119 drawback of this methodology is that since it is based on a linear approximation of the original spatial model it does not perform equally well for empirical settings characterized by especially high spatial dependence. The e mpirical results suggest that the location of new urban development is guided by a prefer ence for lower density areas but in proximity to current urban development. This development trend is sometimes deterred by zoning policies The most important insight from this application comes from the estimates of the spatial lag parameter, indicating that spatial dependence is an important feature of the land use conversion process that needs to be taken into account w hen analyzing land use conversion decisions for both empirical and policy reasons. Empirically, a land use change model that ignores spa tial dependence is likely to suffer from model mis specification. F rom a policy perspective this result suggests that parcels do not develop in isolation, thus the land conversion that is likely to occur in a specific area influence s land conversion in adj acent areas For this reason uncoordinated local land use policies that are designed at a small scale such as subdivision regulation and zoning while attempting to locally manage growth may fragment urban development and result in suboptimal land use pat terns regionally This is of major concern since it can be argued that the very policies that are designed to discourage sprawling urban development could be at worst causing it and at best enabling it (Attkisson, 2009; Batchis, 2010) The power to de sign and enact land use regulation in Ohio, like many other states that have not adopted a growth management plan is delegated to the local governments (i.e. counties, townships, and municipalities) under the assumption that
120 they have a better unde rstandi ng of problems of local concern The local governments typically make land use regulations following the statutory basis provided by the Standard State Zoning Enabling Act (SZEA) th e purpose of promoting public heal th, safety, morals, or the general welfare of the community although planning is not mandatory and t he Standard Planning Act (SPA) that followed in 1928 was nev er integrated with the SZEA ( Attkisson, 2009). The statutory provisions in the Ohio Revised Code that deal with the zoning powers of counties and townships are formulated in a virtually identical language (Hunt, 20 01 ). In addition to zoning powers the Ohi o Revised Code grants similar powers to local governments to enact subdivision regulation. With such legislation in place, individual communities are justified to enact land use regulations based entirely in their own self interest such as to solve proble ms of a very local nature or advance their narrowly defined development goals. B ecause of the spatial interactions that characterize the land use conversion decisions as well as the spatial spillover effects associated with the land use determinants inclu ding land use regulation, the results from this study suggest that individual communities that act in self interest fail to internalize the effects of their policies in the neighboring communities resulting in suboptimal urbanization patterns at a regiona l level. For instance, a community that uses land use controls (e.g. exclusionary zoning) to manage its urban areas will push inevitable urban growth to adjacent communit ies that do not have such regulation in plac e causing t he region to sprawl For this r eason coordination of local land use policies at a regional level may
121 provide a way to internalize the externalities associated with urban growth and manage it more efficiently. In this regard knowledge of the presence and the extent of spatial spillover effects can help inform the design of l and use polic ies at an appropriate scale so as to achieve some v ertical consistency between local and regional land use policies and development goals
122 Figure 4 1. Medina County, Ohio Figure 4 2. Land use c hanges in Medina County, Ohio (1970 2000)
123 Table 4 1. Simulation r esults for the SMNL m odel with e qual c hoice p robabilities (Sample s ize: 1,000 o bservations) Standard Multinomial Logit Spatial Multinomial Logit 0 Bias 0.0009 0.0006 0.0052 0.0010 0.0006 0.0052 0.0020 RMSE 0.1591 0.1573 0.1634 0.1591 0.1573 0.1634 0.1949 0.1 Bias 0.0027 0.0021 0.0020 0.0029 0.0024 0.0017 0.0048 RMSE 0.1614 0.1576 0.1640 0.1614 0.1576 0.1640 0.1963 0.2 Bias 0.0113 0.0101 0.0071 0.0119 0.0107 0.0077 0.0040 RMSE 0.1616 0.1573 0.1646 0.1615 0.1573 0.1645 0.2002 0.3 Bias 0.0265 0.0253 0.0222 0.0274 0.0262 0.0231 0.0028 RMSE 0.1623 0.1585 0.1657 0.1623 0.1585 0.1656 0.2072 0.4 Bias 0.0486 0.0482 0.0444 0.0497 0.0494 0.0456 0.0055 RMSE 0.1651 0.1615 0.1707 0.1652 0.1617 0.1708 0.2183 0.5 Bias 0.0769 0.0774 0.0730 0.0783 0.0788 0.0745 0.0206 RMSE 0.1737 0.1722 0.1790 0.1741 0.1727 0.1794 0.2286 0.6 Bias 0.1180 0.1171 0.1136 0.1196 0.1187 0.1153 0.0502 RMSE 0.1942 0.1927 0.1993 0.1950 0.1935 0.2000 0.2514 0.7 Bias 0.1712 0.1709 0.1676 0.1730 0.1726 0.1694 0.0974 RMSE 0.2311 0 .2290 0.2344 0.2323 0.2302 0.2355 0.2926 0.8 Bias 0.2500 0.2471 0.2444 0.2518 0.2489 0.2463 0.1781 RMSE 0.2943 0.2899 0.2936 0.2957 0.2914 0.2950 0.3651 0.9 Bias 0.3749 0.3718 0.3688 0.3766 0.3736 0.3706 0.3383 RMSE 0. 4055 0.4018 0.4024 0.4070 0.4033 0.4039 0.5424 Note: Results are based on a sample with 1,000 observation s and 1,000 replications. The slope s true value is 1 in all cases.
124 Table 4 2. Simulation r esults for the SMNL m odel with e qual c hoice p robabili ties (Sample s ize: 5,000 o bservations) Standard Multinomial Logit Spatial Multinomial Logit 0 Bias 0.0018 0.0005 0.0016 0.0018 0.0005 0.0016 0.0008 RMSE 0.0713 0.0731 0.0725 0.0713 0.0731 0.0725 0.0794 0.1 Bias 0.0058 0.0026 0.0013 0.0056 0.0024 0.0011 0.0005 RMSE 0.0709 0.0736 0.0725 0.070 9 0.0736 0.0726 0.0798 0.2 Bias 0.0136 0.0101 0.0097 0.0132 0.0097 0.0093 0.0027 RMSE 0.0737 0.0748 0.0733 0.0736 0.0747 0.0732 0.0831 0.3 Bias 0.0276 0.0241 0.0243 0.0271 0.0235 0.0238 0.0075 RMSE 0.0778 0.0777 0.0762 0.0777 0.0775 0.0761 0.0867 0.4 Bias 0.0495 0.0455 0.0461 0.0488 0.0448 0.0454 0.0195 RMSE 0.0881 0.0866 0.0858 0.0877 0.0862 0.0854 0.0912 0.5 Bias 0.0774 0.0743 0.0750 0.0765 0.0733 0.0741 0.0373 RMSE 0.1056 0.1053 0 .1039 0.1050 0.1047 0.1032 0.1012 0.6 Bias 0.1165 0.1131 0.1141 0.1154 0.1119 0.1130 0.0678 RMSE 0.1368 0.1352 0.1348 0.1359 0.1343 0.1339 0.1220 0.7 Bias 0.1687 0.1662 0.1672 0.1674 0.1649 0.1659 0.1160 RMSE 0.1830 0. 1821 0.1818 0.1819 0.1810 0.1806 0.1617 0.8 Bias 0.2462 0.2412 0.2434 0.2448 0.2398 0.2420 0.1980 RMSE 0.2562 0.2521 0.2534 0.2550 0.2508 0.2520 0.2379 0.9 Bias 0.3676 0.3635 0.3656 0.3662 0.3620 0.3642 0.3491 RMSE 0.3 742 0.3706 0.3723 0.3728 0.3692 0.3710 0.3899 Note: Results are based on a sample with 5 000 observation s and 5 00 replications. The slope s true value is 1 in all cases.
125 Table 4 3. Simulation r esults for the SMNL m odel with d ifferent c hoice p robabil ities (Sample s ize: 1,000 o bservations) Standard Multinomial Logit Spatial Multinomial Logit 0.0 True Coeff 1.5980 0.9098 0.4026 Bias 0.0033 0.0015 0.0006 0.0070 0.0009 0.0011 0.0109 RMSE 0.1148 0.1264 0.1400 0.1828 0.1324 0.1401 0.1792 0.1 True Coeff 1.4490 0.8250 0.3650 Bias 0.1467 0.0873 0.0386 0.0543 0.0611 0.0281 0.0118 RMSE 0.1861 0.1528 0.1450 0.1933 0.1437 0.1409 0.1805 0.2 True Coeff 1.3181 0.7504 0.3321 Bias 0.2716 0.1614 0.0728 0.1084 0.1145 0.0535 0.0005 RMSE 0.2948 0.2035 0.1582 0.2154 0.1713 0.1470 0.1807 0.3 True Coe ff 1.2005 0.6835 0.3025 Bias 0.3832 0.2273 0.1035 0.1566 0.1618 0.0761 0.0212 RMSE 0.3999 0.2592 0.1741 0.2452 0.2058 0.1552 0.1850 0.4 True Coeff 1.0928 0.6221 0.2753 Bias 0.4855 0.2879 0.1312 0.1980 0.2043 0.0962 0.0446 RMSE 0.498 6 0.3136 0.1912 0.2739 0.2399 0.1642 0.1907 0.5 True Coeff 0.9915 0.5645 0.2498 Bias 0.5798 0.3436 0.1566 0.2366 0.2434 0.1144 0.0732 RMSE 0.5908 0.3655 0.2095 0.3034 0.2735 0.1746 0.2005 0.6 True Coeff 0.8935 0.5087 0.2251 Bias 0.670 4 0.3964 0.1805 0.2758 0.2809 0.1317 0.1064 RMSE 0.6801 0.4158 0.2285 0.3354 0.3074 0.1863 0.2152 0.7 True Coeff 0.7946 0.4524 0.2002 Bias 0.7605 0.4489 0.2041 0.3189 0.3192 0.1490 0.1440 RMSE 0.7690 0.4661 0.2473 0.3721 0.3425 0.1979 0.2 384 0.8 True Coeff 0.6881 0.3917 0.1733 Bias 0.8574 0.5053 0.2289 0.3700 0.3619 0.1677 0.1825 RMSE 0.8649 0.5207 0.2681 0.4171 0.3822 0.2117 0.2656 0.9 True Coeff 0.5559 0.3165 0.1400 Bias 0.9781 0.5751 0.2600 0.4451 0.4180 0.1926 0. 2191 RMSE 0.9846 0.5886 0.2946 0.4856 0.4356 0.2309 0.2946 Note: Results are based on a sample with 1,000 observations and 1,000 replications. The slope s true value changes with The choice probabilities are set as follows: , and
126 Table 4 4. Simulation r esults for the SMNL m odel with d ifferent c hoice p robabilities (Sample s ize: 5 ,000 o bservations) St andard Multinomial Logit Spatial Multinomial Logit 0.0 True Coeff 1.6097 0.9164 0.4055 Bias 0.0026 0.0021 0.0032 0.0014 0.0018 0.0030 0.0012 RMSE 0.0500 0.0539 0.0587 0.0791 0.0570 0.0589 0.0742 0.1 True Coeff 1. 4597 0.8310 0.3677 Bias 0.1469 0.0883 0.0426 0.0643 0.0643 0.0324 0.0000 RMSE 0.1550 0.1034 0.0723 0.1028 0.0855 0.0663 0.0762 0.2 True Coeff 1.3278 0.7559 0.3345 Bias 0.2724 0.1632 0.0765 0.1177 0.1181 0.0572 0.0114 RMSE 0.2769 0.17 19 0.0968 0.1432 0.1305 0.0814 0.0783 0.3 True Coeff 1.2094 0.6885 0.3047 Bias 0.3843 0.2294 0.1065 0.1645 0.1650 0.0788 0.0307 RMSE 0.3876 0.2357 0.1223 0.1839 0.1740 0.0980 0.0838 0.4 True Coeff 1.1008 0.6267 0.2773 Bias 0.4854 0.28 85 0.1332 0.2057 0.2064 0.0979 0.0557 RMSE 0.4881 0.2936 0.1463 0.2216 0.2136 0.1137 0.0964 0.5 True Coeff 0.9987 0.5686 0.2516 Bias 0.5802 0.3445 0.1581 0.2457 0.2460 0.1157 0.0862 RMSE 0.5824 0.3488 0.1691 0.2592 0.2519 0.1290 0.1172 0 .6 True Coeff 0.8999 0.5123 0.2267 Bias 0.6711 0.3976 0.1822 0.2849 0.2836 0.1331 0.1197 RMSE 0.6730 0.4013 0.1919 0.2970 0.2886 0.1447 0.1447 0.7 True Coeff 0.8002 0.4556 0.2016 Bias 0.7623 0.4509 0.2064 0.3273 0.3222 0.1508 0.1558 RMSE 0.7640 0.4541 0.2151 0.3381 0.3266 0.1611 0.1766 0.8 True Coeff 0.6928 0.3944 0.1745 Bias 0.8601 0.5078 0.2319 0.3788 0.3651 0.1700 0.1939 RMSE 0.8616 0.5106 0.2395 0.3882 0.3688 0.1789 0.2117 0.9 True Coeff 0.5595 0.3185 0.1410 Bias 0.9821 0.5784 0.2641 0.4548 0.4218 0.1958 0.2311 RMSE 0.9834 0.5808 0.2708 0.4628 0.4251 0.2035 0.2468 Note: Results are based on a sample with 5 0 00 observations and 500 replications. The slope s true value changes with The choice probabilities are set as follows: , and
127 Table 4 5. Land u se c onversion m odel : v ariable d escription Variable Description Landuse 1990 land use (=0 if agricultur al, 1 if residential, 2 if commercial, and 3 if industrial ) Acres P arcel size in acres Percpinc P er capita income Totdiscle D istance of each parcel cent er in meters from the center of Cleveland Disttonear D istance of each parcel center in meters to t he nearest city Popdens 1990 population density within each parcel s neighborhood Housedens P roportion of houses in 1990 Reside P roportion of land in residential use Commarea P roportion of land in commercial use Agarea P roportion of land in agricult ural use Largelot L argelot (=1 if min lot size zoned as >= 3 acres, 0 otherwise) Table 4 6. Land u se c onversion m odel : v ariable d escriptive s tatistics Variable Mean Std. Dev. Min Max Landuse 0.50 0.73 0 .00 3 .00 Acres 17.60 28.55 <1 .00 1 150.56 Per cpinc 14 130 .06 2,29 8.59 5,831 .00 19,952 .00 Totdistcle 166 294 .10 38 839 .08 222 .14 261 608 .00 Disttonear 17 66 3.67 8,494 .32 192 .25 40,377 .11 Popdens 25 6.77 3 79.57 4.81 3 ,7 98.63 Housden 8 6.91 126 .47 1.95 1 ,31 5.74 Reside 0.28 0.17 0.00 0.99 Comarea 0. 04 0.07 0.00 0.51 Agarea 0.64 0.21 0.00 1.00 Largelot 0.16 0.37 0 .00 1 .00 Table 4 7. Proportion of p arcels in e ach l and u se c ategory LandUse Parcels Share (%) Agricultural 5,991 61.38 Residential 2,917 29.89 Commercial 572 5.86 Industrial 280 2 .87 Total 9,760 100 .00
128 Table 4 8. Estimated c oefficients of the l and u se c hange m odel SMNL Models MNL Model W_800_f=1 W_ 800 _f= 2 W_ 1600 _f= 1 W_1600_f=2 Variables Est. St. Err. Est. St. Err. Est. St. Err. Est. St. Err. Est. St. Err. a cres: ag 0.080 0.009 0.052 0.007 0.048 0.007 0.025 0.005 0.032 0.005 0.253 0.014 0.819 0.095 0.762 0.101 0.432 0.107 0.988 0.089 com 0.068 0.009 0.061 0.009 0.068 0.008 0.083 0.007 0.044 0.009 totdiscle: ag 0.230 0.061 0.242 0.374 0.408 0.391 0.095 0.287 0.013 0.335 0.235 0.067 1.328 0.623 1.988 0.660 4.488 0.593 3.122 0.578 com 0.319 0.067 2.499 0.682 1.444 0.679 0.837 0.73 4.658 1.043 disttonear: (x10,000) 0.425 0.163 6.232 1.138 8.044 1.322 0.218 0.987 4.865 0.907 0.278 0.170 1.821 1.221 0.607 1.320 1.903 1.381 3.323 1.046 com 1.060 0.192 9.315 4.177 3.163 4.428 2.848 2.902 11.701 2.709 agarea: ag 1.797 0.807 5.277 0.966 5.702 1.020 1.757 0.902 1.651 0.9 99 1.848 0.834 1.695 1.340 1.604 1.358 3.455 1.061 6.145 1.216 com 2.074 1.008 11.898 4.779 6.018 5.334 5.811 3.809 1.260 3.180 reside: ag 1.906 0.929 5.436 2.233 12.814 2.245 6.886 1.694 13.678 1.561 4.234 0.954 7.542 3.016 0.6 33 2.912 7.856 1.955 14.525 2.288 com 0.283 1.141 12.219 4.006 10.628 4.532 5.295 3.669 27.136 4.419 commarea: ag 0.016 1.495 10.169 2.742 13.140 3.038 0.027 1.982 2.075 2.199 3.753 1.509 3.953 3.710 7.230 3.678 5.812 3.338 6 .997 3.598 com 4.470 1.683 0.357 9.851 10.467 11.174 13.157 5.021 14.925 4.417
129 Table 4 8. C ontinued SMNL Model s MNL Model W_800_f=1 W_1 800 _f= 2 W_ 1600 _f= 1 W_1600_f=2 Variables Est. St. Err. Est. St. Err. Est. St. Err. Est. St. Err. Est. St Err. popdens: ag 0.006 0.001 0.004 0.004 0.018 0.005 0.003 0.003 0.018 0.004 0.003 0.001 0.000 0.004 0.004 0.004 0.017 0.003 0.004 0.004 com 0.011 0.002 0.023 0.011 0.016 0.013 0.039 0.012 0.041 0.010 housedens: ag 0 .018 0.003 0.016 0.012 0.048 0.015 0.005 0.008 0.050 0.011 0.011 0.003 0.009 0.011 0.013 0.012 0.061 0.01 0 0.021 0.011 com 0.034 0.005 0.017 0.030 0.008 0.037 0.052 0.035 0.088 0.028 percpinc: (x10,000) 0.383 0.418 0.93 6 1.765 2.198 1.653 4.909 1.445 8.785 1.612 0.011 0.429 2.051 1.865 0.519 1.780 0.139 1.431 0.469 1.561 com 1.271 0.494 6.162 4.366 0.963 4.291 5.091 4.976 46.482 8.486 largelot: ag 0.758 0.233 0.661 0.352 1.571 0.392 1.459 0.2 79 2.511 0.310 0.284 0.245 0.544 0.299 0.549 0.302 1.018 0.36 0 0.404 0.332 com 0.881 0.261 3.990 1.031 2.279 0.932 1.215 0.783 1.391 0.858 rho --0.261 0.027 0.331 0.027 0.487 0.04 0.400 0.036 Note: Sample size is 9,760 parcels. All models include indicator variables for the township in which the parcel resides. The columns for the SMNL model correspond to different specifications of the spatial weight s matrix W that vary the cut off distance (800 and 1600) and the friction param eter (f=1 or 2). See text for details.
130 Table 4 9 Marginal e ffects of the e stimate d c oefficients of the l and u se c hange m odel SMNL Models MNL Model W_800_f=1 W_800_f=2 W_1600_f=1 W_1600_f=2 Marginal Effects Direct Indirect Total Direct Indirect Tota l Direct Indirect Total Direct Indirect Total acres: ag 0.0273 0.0079 0.0016 0.0095 0.0048 0.0015 0.0063 0.0033 0.0025 0.0058 0.0049 0.0027 0.0075 (0.0007) (0.0011) (0.0003) (0.0014) (0.0010) (0.0003) (0.0012) (0.0003) (0.0002) (0.0005) (0 .0003) (0.0002) (0.0005) res 0.0322 0.0167 0.0033 0.0201 0.0142 0.0039 0.0181 0.0061 0.0046 0.0107 0.0089 0.0049 0.0138 (0.0009) (0.0016) (0.0004) (0.0020) (0.0016) (0.0004) (0.0021) (0.0002) (0.0002) (0.0004) (0.0003) (0.0002) (0.0005) co m 0.0034 0.0051 0.0010 0.0062 0.0057 0.0015 0.0072 0.0022 0.0016 0.0038 0.0019 0.0010 0.0029 (0.0002) (0.0004) (0.0001) (0.0004) (0.0006) (0.0002) (0.0007) (0.0002) (0.0002) (0.0004) (0.0002) (0.0002) (0.0003) ind 0.0015 0.0037 0.0007 0.0044 0.0036 0.00 09 0.0046 0.0006 0.0005 0.0011 0.0022 0.0012 0.0034 (0.0002) (0.0009) (0.0002) (0.0011) (0.0004) (0.0001) (0.0005) (0.0001) (0.0001) (0.0001) (0.0002) (0.0002) (0.0003) totdiscle (x 10,000): ag 0.0005 0.0223 0.0054 0.0277 0.0157 0.0052 0. 0209 0.0406 0.0303 0.0708 0.0284 0.0160 0.0444 (0.0032) (0.0195) (0.0057) (0.0245) (0.0243) (0.0077) (0.0318) (0.0411) (0.0314) (0.0724) (0.0249) (0.0155) (0.0403) res 0.0020 0.0103 0.0022 0.0125 0.0239 0.0068 0.0307 0.0567 0.0421 0.0988 0.01 97 0.0110 0.0307 (0.0033) (0.0021) (0.0005) (0.0025) (0.0039) (0.0011) (0.0049) (0.0022) (0.0017) (0.0039) (0.0068) (0.0029) (0.0094) com 0.0048 0.0350 0.0090 0.0439 0.0147 0.0056 0.0203 0.0005 0.0006 0.0011 0.0308 0.0179 0.0487 (0.0016 ) (0.0042) (0.0012) (0.0053) (0.0017) (0.0007) (0.0023) (0.0053) (0.0041) (0.0094) (0.0029) (0.0015) (0.0044) ind 0.0063 0.0229 0.0057 0.0287 0.0229 0.0071 0.0300 0.0166 0.0125 0.0291 0.0221 0.0129 0.0350 (0.0018) (0.0028) (0.0007) (0.0034) (0.0016) (0. 0005) (0.0021) (0.0023) (0.0018) (0.0041) (0.0013) (0.0008) (0.0021) disttonear (x 10,000): ag 0.0070 0.1015 0.0220 0.1235 0.1612 0.0543 0.2155 0.0067 0.0054 0.0121 0.1385 0.0793 0.2177 (0.0092) (0.0126) (0.0030) (0.0156) (0.0105) (0. 0037) (0.0140) (0.0055) (0.0041) (0.0095) (0.0659) (0.0416) (0.1072) res 0.0174 0.1395 0.0274 0.1669 0.0766 0.0221 0.0987 0.0344 0.0255 0.0600 0.0317 0.0176 0.0493 (0.0083) (0.1080) (0.2990) (1.3591) (0.1245) (0.0357) (0.1601) (0.0287) (0.0215 ) (0.0501) (0.0244) (0.0183) (0.0418) com 0.0349 0.1315 0.0308 0.1624 0.0055 0.0047 0.0101 0.0476 0.0360 0.0836 0.1003 0.0579 0.1582 (0.0061) (0.0099) (0.0032) (0.0129) (0.0949) (0.0314) (0.1256) (0.0611) (0.0461) (0.1071) (0.0071) (0.0036) (0.0106) ind 0.0106 0.0935 0.0254 0.1189 0.0791 0.0275 0.1066 0.0065 0.0051 0.0115 0.0065 0.0038 0.0103 (0.0041) (0.0052) (0.0013) (0.0065) (0.0386) (0.0135) (0.0519) (0.0177) (0.0133) (0.0310) (0.0039) (0.0023) (0.0062)
131 Table 4 9 Continued SMNL Mo dels MNL Model W_800_f=1 W_800_f=2 W_1600_f=1 W_1600_f=2 Marginal Effects Direct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total agarea: ag 0.0112 0.0313 0.0059 0.0372 0.0664 0.0219 0.0884 0.1400 0.1055 0.2455 0.0611 0.0343 0.0953 (0.0494) (0.0075) (0.0017) (0.0091) (0.0062) (0.0022) (0.0084) (0.1092) (0.0822) (0.1912) (0.0343) (0.0198) (0.0539) res 0.0216 0.0767 0.0147 0.0915 0.0417 0.0115 0.0532 0.0445 0.0331 0.0776 0.0595 0.0327 0.0922 (0 .0435) (0.0664) (0.0156) (0.0782) (0.0767) (0.0214) (0.0977) (0.0037) (0.0028) (0.0066) (0.0218) (0.0149) (0.0363) com 0.0161 0.1641 0.0400 0.2042 0.0643 0.0199 0.0842 0.0999 0.0761 0.1760 0.0047 0.0031 0.0078 (0.0337) (0.0140) (0.0043) (0.0181) (0.0100 ) (0.0035) (0.0134) (0.1144) (0.0842) (0.1985) (0.0307) (0.0201) (0.0507) ind 0.0489 0.1187 0.0312 0.1499 0.0891 0.0302 0.1193 0.0044 0.0037 0.0081 0.0031 0.0016 0.0047 (0.0211) (0.0939) (0.0253) (0.1185) (0.0459) (0.0180) (0.0635) (0.0029) (0.0023) (0.0052) (0.0021) (0.0012) (0.0034) reside: ag 0.5831 0.1453 0.0405 0.1858 0.3974 0.1375 0.5349 0.2064 0.1562 0.3627 0.2638 0.1539 0.4178 (0.0557) (0.0097) (0.0032) (0.0126) (0.0313) (0.0108) (0.0416) (0.0133) (0.0097) (0.0229) (0.0126) (0.0085) (0.0210) res 0.5643 0.1661 0.0329 0.1991 0.0037 0.0030 0.0067 0.0643 0.0476 0.1118 0.1115 0.0611 0.1726 (0.0479) (0.0101) (0.0035) (0.0133) (0.0145) (0.0055) (0.0200) (0.0058) (0.0042) (0.0100) (0.0076) (0.0068) (0.0142) com 0.0383 0 .2910 0.0704 0.3614 0.3541 0.1191 0.4732 0.1725 0.1307 0.3033 0.2771 0.1584 0.4354 (0.0337) (0.0218) (0.0070) (0.0285) (0.0272) (0.0100) (0.0364) (0.2112) (0.1548) (0.3658) (0.0141) (0.0061) (0.0199) ind 0.0195 0.0204 0.0030 0.0234 0.047 1 0.0154 0.0624 0.0981 0.0731 0.1712 0.0983 0.0566 0.1549 (0.0238) (0.0165) (0.0047) (0.0210) (0.0166) (0.0067) (0.0233) (0.0142) (0.0109) (0.0251) (0.0817) (0.0371) (0.1177) commarea: ag 0.4619 0.2776 0.0657 0.3433 0.4486 0.1534 0 .6021 0.1760 0.1332 0.3092 0.0218 0.0113 0.0331 (0.0796) (0.0259) (0.0071) (0.0329) (0.0342) (0.0117) (0.0453) (0.2239) (0.1666) (0.3903) (0.0788) (0.0500) (0.1287) res 0.3224 0.1627 0.0323 0.1950 0.1385 0.0423 0.1808 0.0426 0.0318 0.0744 0. 0958 0.0523 0.1481 (0.0667) (0.0211) (0.0049) (0.0260) (0.0305) (0.0093) (0.0398) (0.0789) (0.0588) (0.1376) (0.0569) (0.0417) (0.0970) com 0.1865 0.0526 0.0158 0.0684 0.3049 0.1069 0.4118 0.1915 0.1457 0.3372 0.1207 0.0688 0.1895 (0.0467) (0.1672) (0.0427) (0.2091) (0.3219) (0.1144) (0.4309) (0.0205) (0.0148) (0.0353) (0.0663) (0.0358) (0.0985) ind 0.0471 0.0623 0.0176 0.0799 0.0053 0.0042 0.0094 0.0580 0.0443 0.1024 0.0467 0.0278 0.0745 (0.0389) (0.0859) (0.0254) (0.1111) (0.020 4) (0.0076) (0.0280) (0.0106) (0.0081) (0.0186) (0.0030) (0.0019) (0.0048)
132 Table 4 9 Continued SMNL Models MNL Model W_800_f=1 W_800_f=2 W_1600_f=1 W_1600_f=2 Marginal Effects Direct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total popdens: ag 0.0001 0.0002 0.0001 0.0003 0.0006 0.0002 0.0008 0.0006 0.0005 0.0011 0.0002 0.0001 0.0003 (0.0001) (0.0002) (0.0001) (0.0002) (0.0004) (0.0001) (0.0005) (0.0006) (0.0005) (0.0011) (0.0002) (0.0001) (0. 0003) res 0.0003 0.0001 0.0000 0.0001 0.0001 0.0000 0.0001 0.0001 0.0001 0.0003 0.0001 0.0001 0.0002 (0.0001) (0.0001) (0.0000) (0.0002) (0.0003) (0.0001) (0.0004) (0.0002) (0.0001) (0.0003) (0.0001) (0.0001) (0.0001) com 0.0003 0.0004 0.0001 0. 0005 0.0005 0.0002 0.0007 0.0006 0.0005 0.0010 0.0003 0.0001 0.0004 (0.0001) (0.0000) (0.0000) (0.0001) (0.0004) (0.0002) (0.0006) (0.0001) (0.0000) (0.0001) (0.0001) (0.0001) (0.0002) ind 0.0001 0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0001 0. 0001 0.0002 0.0003 0.0002 0.0005 (0.0000) (0.0000) (0.0000) (0.0000) (0.0003) (0.0001) (0.0004) (0.0000) (0.0000) (0.0000) (0.0001) (0.0001) (0.0001) housedens: ag 0.0002 0.0006 0.0001 0.0007 0.0013 0.0004 0.0017 0.0011 0.0009 0.0020 0.0005 0.0003 0.0008 (0.0003) (0.0005) (0.0001) (0.0006) (0.0008) (0.0003) (0.0011) (0.0011) (0.0008) (0.0020) (0.0004) (0.0003) (0.0007) res 0.0007 0.0002 0.0000 0.0003 0.0005 0.0001 0.0006 0.0007 0.0005 0.0012 0.0002 0.0001 0.0003 (0.0002 ) (0.0004) (0.0001) (0.0005) (0.0009) (0.0003) (0.0012) (0.0000) (0.0000) (0.0001) (0.0002) (0.0001) (0.0003) com 0.0009 0.0004 0.0001 0.0005 0.0006 0.0002 0.0008 0.0007 0.0005 0.0012 0.0005 0.0003 0.0008 (0.0002) (0.0007) (0.0002) (0.0009) (0.0008) (0.0003) (0.0011) (0.0010) (0.0007) (0.0017) (0.0003) (0.0002) (0.0005) ind 0.0004 0.0000 0.0000 0.0000 0.0003 0.0001 0.0004 0.0003 0.0002 0.0005 0.0008 0.0005 0.0013 (0.0001) (0.0004) (0.0001) (0.0005) (0.0005) (0.0002) (0.0006) (0.0004) ( 0.0003) (0.0008) (0.0002) (0.0002) (0.0004) percpinc (x 10,000): ag 0.0000 0.0456 0.0130 0.0586 0.0451 0.0151 0.0603 0.2053 0.1548 0.3600 0.3003 0.1733 0.4736 (0.0235) (0.0681) (0.0174) (0.0830) (0.0574) (0.0196) (0.0768) (0.1156) (0.0829 ) (0.1982) (0.2091) (0.1381) (0.3468) res 0.0431 0.0652 0.0128 0.0780 0.0278 0.0079 0.0357 0.0270 0.0201 0.0471 0.0418 0.0222 0.0640 (0.0206) (0.0560) (0.0174) (0.0723) (0.0609) (0.0168) (0.0777) (0.0540) (0.0398) (0.0938) (0.0707) (0.0682) (0.134 9) com 0.0505 0.1253 0.0304 0.1557 0.0028 0.0000 0.0028 0.1287 0.0978 0.2266 0.3882 0.2229 0.6111 (0.0149) (0.1467) (0.0422) (0.1880) (0.0085) (0.0028) (0.0113) (0.1979) (0.1451) (0.3429) (0.0237) (0.0108) (0.0343) ind 0.0074 0.0145 0.0046 0. 0191 0.0201 0.0072 0.0273 0.0496 0.0369 0.0864 0.0461 0.0274 0.0735 (0.0109) (0.0405) (0.0118) (0.0520) (0.0259) (0.0091) (0.0350) (0.6615) (0.5077) (1.1692) (0.0172) (0.0097) (0.2688)
133 Table 4 9 Continued SMNL Models MNL Model W_800_f=1 W_80 0_f=2 W_1600_f=1 W_1600_f=2 Marginal Effects Direct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total largelot: ag 0.0426 0.0156 0.0043 0.0199 0.0118 0.0037 0.0155 0.0497 0.0376 0.0873 0.0423 0.0242 0 .0665 (0.0127) (0.0191) (0.0048) (0.0235) (0.0015) (0.0005) (0.0020) (0.0258) (0.0185) (0.0443) (0.0 2 26) (0.0 1 71) (0.0 4 97) res 0.0402 0.0166 0.0032 0.0198 0.0135 0.0036 0.0171 0.0053 0.0039 0.0091 0.0172 0.0094 0.0265 (0.0122) (0.0216) (0.00 47) (0.0252) (0.0212) (0.0061) (0.0271) (0.0011) (0.0008) (0.0020) (0.0756) (0.0503) (0.1246) com 0.0129 0.0626 0.0155 0.0780 0.0304 0.0096 0.0400 0.0363 0.0276 0.0639 0.0036 0.0020 0.0056 (0.0067) (0.0052) (0.0016) (0.0068) (0.0031) (0.0013) (0.0043 ) (0.0447) (0.0326) (0.0773) (0.0094) (0.0073) (0.0166) ind 0.0152 0.0303 0.0080 0.0383 0.0287 0.0097 0.0384 0.0187 0.0139 0.0326 0.0287 0.0168 0.0455 (0.0062) (0.0353) (0.0102) (0.0453) (0.0200) (0.0079) (0.0278) (0.0029) (0.0022) (0.0051) (0.01 11) (0.0070) (0.0180) Note: The s tandard errors of the estimated marginal effects presented in parentheses are obtained using the Delta method.
134 CHAPTER 5 NEIGHBORHO O D INFLUENCE AND PUBL IC POLICY: ADOPTION OF OPEN ENROLLMENT POLICIES BY SCHOOL DISTRIC TS IN OHIO Background Public school reform has been at the heart of public debate in the United States (U.S.) since the early 1980s concerning issues of student academic performance and dropout rates national academic standards, school governance and fund ing, equity and diversity, safety and desegregation (Crepage, 1999). Of many strategies considered to bring about school reform in the public school system, public school choice r eceived considerable attention Public school choice refers to policies that p rovide parents with an opportunity to select an educational program or a public school for their children to attend tuition free other than the neighborhood school to which they would be normally assigned based on the district of residence 1 E i ght basic t ypes of school choice policies have been proposed and implemented by various states to address p ublic education challenges ranging from desegregation of schools to improvement of education practices : magnet schools, post secondary enrollment programs, drop out prevention programs, intra district open enrollment, interdistrict open enrollment, voucher programs tuition agreement programs, and charter schools 2 (Ysseldyke et al., 1992). 1 Some parents have played a role in selecting the public school their children attend as part of their enrolling their children int o private schools. Thus, parental choice has always existed to some extent. Public school choice policies, however, extends this parental role. 2 Magnet schools are schools with a particular pedagogical theme designed to attract minority and white student s to attend the same school; postsecondary programs allow students to enroll in college courses and receive high school credit; dropout prevention programs allow eligible students that have not succeeded in a school and students at risk to attend an altern ative school; intradistrict open enrollment allows students to transfer to a different school within their district of residence ; interdistrict open enrollment allows students to transfer to a school outside their district of residence; voucher programs al low students to attend a private school with tuition paid from public education funds ; tuition agreement programs allow students to attend tuition free any school of their choice if they are not assigned to a
135 P ublic school choice is advocated as a strategy to reform the public school system based on rational choice theory arguments that originate in Friedman (1955) and Chubb and Moe (1988) 3 The principal argument asserts that, s ubjecting schools to market forces induces competition and forces the otherwise monopolized public school s ystem to be come more efficient In addition, it improve s the quality of education because the schools that fail to compet e a re likely to be forced out of the public school system. This, in turn, changes the incentives of schools to embrace change and provi de high quality programs in response to market pressures. Inevitably, school choice as a strategy for public school reform has generate d much debate regarding issue s of government involvement and public education as a public good. Advocates of public school choice attribute the lack of accountability, efficiency, and quality of public schools to the political governance of the education system They argue that the administration of the public scho ol system offers school districts little incentive for i mprovement (Chubb and Moe, 1990), arguing further that s chool choice promotes educational accountability (Young and Clinchy, 1992), equity and diversity (Young and Clinchy, 1992 ; Nelson et al., 1993 ), autonomy and competition that leads to general school system improvements (Cookson, 1994; Liebe rman, 1990 ), encourages experimentation and risk taking provides disadvant ag ed and minority students with better school options (Lieberman, 1990) etc. school in their town; and charter schools are p ublicly sponsored but autonomous schools that employ outcome (Ysseldyke et al., 1992). 3 Friedman was the first economist to call for an educational reform and proposed a voucher system that would allow parents to opt out of the public school system and enroll their children in private schools.
136 Critic s question the appropriateness of a market based reform because p ublic education is a common good (Henig, 1994). Thus, t hey contest the implementation of public school choice in a market like setting on grounds of equity and diversity S c hool choice may de epen the socioeconomic stratification in public schools as parents and students tend to separate themselves from those who are different 4 (Lutz, 1996) P arents ma y be un informed to make choices that are in th e best interest of their children (Cookson, 1994 ; Smith and Meier, 1995 ) or s imply make choices based on convenience or other nonacademic factors such as religious beliefs or social class (Smith, 1995) Furthermore s chools that become f inancial ly distress ed under c ompetition for student enrollment are unlikely to provide equitable educa tion to the remaining students Thus, it is of concern that s chool choice may allocate the most disadvantaged students to the wors t schools. Despite the debate, t he concept of s chool choice seems to appeal to the general public A Gallup poll of 1987 reported that 71% of the public in the nation supported school choice (Smith, 1995). Re sul ts from a more recent poll the 1997 Phi Delta Kappa/Gallup Poll of the Public Attitu d es Towards the Public Schools showed that although respondents were generally satisfied with the ir neighborhood school 47% of the people surveyed gave their local school a grade of B or higher and 7 8 % a grade of C or higher 73% of the responders beli eved that school choice would improve the efforts to 4 For this reason, an alternative to school choice that is currently being discussed (though not yet implemented ) to address the stratification of the American school districts along the wealth and racial dimensions is to periodically redraw the school district boundaries (much like with the electoral districts) to ensure an economic and racial balance (e.g. Saiger, 2010).
137 pass some form of school choice legislation in every state and Washington D.C. (Cookson, 1994) The empirical evidence from studies of various choice programs implemented th roughout the U.S. regarding the m ain arguments brought forth by s upporters and opponents of public school choice is debatable Studies that examine the impact of school choice on educational outcomes altho ugh generally do not report negative impacts report effects t hat range f rom no change in educational outcomes to significant improvements in educational outcomes H oxby (2000, 2003) provides evidence that suggests that competition among public schools du e to school choice substantially improves educational achievement, school productivity, and efficiency. Rapp (2000) analyses the influence of school choice on the behavior of school teachers. He is unable to establish a general influence of competitiveness on teacher effort, but his results suggest that one form of school choice, namely intradistrict enrollment leads to more teacher effort. Belfield and Levin (2002 ) examined the evidence of over 41 empirical studies on the effects of competition on educati onal outcomes such as student test scores, graduation rates, educational expenditures, and teacher quality They find that the majority of the studies are able to establish a link between competition ( due to school choice ) and e ducational outcomes and repo rt beneficial effects of competition However, the authors conclude that the magnitudes of these effects are only modest with respect to changes in the levels of competition Belfie l d and Levin (2002) analysis did not include empirical studies on voucher programs. However, t he latter M being one of the most extensively researched also report effects anywhere from no gains in student
138 academic achievement to large gains (e.g. Witte et al., 1995; Green et al., 199 9 ; Rouse, 1998) Similar effects are reported for o pen enrollment programs (e.g. O Murdoch 2000 ; Cullen et al., 2005 ; Carlson et al., 2011 ) M agnet schools seem to be more consistently associated with improved student outcomes (e.g. Metz, 1986; Crain et al., 1 992). The empirical e vidence on the type of students that take advantage of choice programs is mixed as well While s ome studies have found that children that attend choice programs are more able and advantaged suggesting that school choice increases sort ing along the social class and ability dimension (e.g. Coleman et al., 1993; Witte, 1993 ; Buddin et al., 1998; Goldhaber et al., 1999) o ther studies find evidence of higher participation among minorities and low income students ( e.g. Duax, 1988; Lee et al ., 1994; Schneider et al., 1996). The discrepanc y in the effects of school choice programs on educational outcomes could be due to difference s in policy design and the public school environment i n which these policies are implemented or perhaps due to meth odological differences i n e stimating competitive pressures and educational outcomes. The f indings of these studies however, point to the necessity of further analyzing different choice programs to obtain a better understanding of the operational aspects a nd the effectiveness of these market based mechanisms. The purpose of this study is to analyze the adoption decisions of school choice policies interdistrict open enrollment being our policy of interest from the standpoint of school districts. T his fo rm of educational choice is utilized in the U.S. by more than 40 states and serves to more students than any other type of school choice programs (Reback, 2008). In addition, unlike other school choice programs, it affects just about
139 every school district in a state. More spe cifically, we are interested in examin ing t he determinants of adoption of interdistrict open enrollment policies by school districts Interdistrict open enrollment was first introduced in Ohio in 1989 with open enrollment options extended in 1997. The bill that passed in of following three policies: no open enrollment (i.e. school district chooses to accept only resident students), adjacent open enrollme nt (i.e. school district chooses to accept nonresident students only from adjacent districts) or statewide open enrollment (i.e. school district chooses to accept nonresident students from any district) combined fu nding efforts from the state and school districts through local property taxes. Therefore, t he implementation of interdistrict open enrollment policies carries financial ramifications for both the sending and receiving school districts. On the one hand, th e sending school district loses funds because state funds allocated to a student follow the student across district lines. On the other hand, the receiving school district may need additional funds to educate nonresident students which are hard to obtain l ocally through increases in property taxes. Because the survival of schools and the quality of educational services they provide are directly linked to the school funding school districts are faced with difficult decisions. Districts can choose not to par ticipate, but any district can lose students to a nearby participating district. I n terdistrict open enrollment policies in Ohio public schools have been examined in terms of operational outcomes such as a dministrative operability and technical effective ness (e.g. Farrell 1994) Other studies have focused on whether or not there are quantitative differences in demographic characteristics, financial indicators, and
140 academic performance be tween school districts that decide to be open or closed to interdistr ict enrollment ( e.g. Metzler, 199 6 ; Fowler, 1996 ; Crepage, 1999 ) In addition these studies have collected descriptive evidence as to why districts decide to adopt one enrollment policy over another Our study builds upon these studies to p rovide a quanti tative analysis of the determinants of the adoption of interdistrict open enrollment policies by school districts and examine their relative importance in the probability of a school district to adopt any of the three interdistrict open enrollment alternat ives We estimate a spatially explicit model of adoption decisions drawing from the literature on social interaction s (e.g. Case, 1992; Besley and Case, 1995; Brett and Pinkse, 1997; Brueckner and Saavedra, 2001) that suggests that when f aced with difficult decisions economic agents are influenced by decision makers in reference groups and either mimic their decisions or respond strategically to them. If t his holds, we expect to find positive neighborhood influence on ado ption decisions an aspect of the school district s adoption decision s t hat has been little explored. Rincke (2006) examined the neighborhood influence in the adoption of open enrollment policies using school district data from five states in the U.S. (Ark ansas, California, Idaho, Massachusets, and Ohio) and found significant spatial effects suggesting that school district decisions are indeed heavily influenced by other school districts in reference groups. Our study provides further evidence that suppor ts this view but differs from this study in the following aspects First, binary ; a school district decides to adopt a new policy, namely open enrollment, or not. Ou r set of policy altern atives includes all policy options provided by the interdistrict open enrollment statute. Second, because Rincke
141 (2006) was interested in neighborhood influence in diffusion of policy innovations in local jurisdictions, in his study spatial dependence is a ssumed to arise between school district s that belong to the same county However, i t may well be the case that s chool districts that are located in the periphery of a count y may be influenced by nearby school districts that belong to an adjacent county. O u r specification of the spatial weights matri x is more flexible and accommodates for this possibility. The remainder of this chapter is organized as follows. The next section briefly describes the context, design and implementation of interdistrict open e nrollment The subsequent section describe s our estimation framework followed by a section that describes the data set used to empirically estimate a spatially explicit model of policy adoption deci sion s T he last section presents the estimation results and concludes. Interdistrict Open Enrollment in Ohio Following the recommendations of the State Board of Education, the Ohio Education 2000 Commission, and the Gillmor Commission, the Ohio legislatur e int roduced school choice to school system through the Omnibus Educational Reform Act of 1989, Amended Senate Bill 140 (Ohi o Department of Education, 1991) The Senate Bill 1 40 included three school choice policies, two of which were mandate d intradistrict open enrollment and post secondary enrollment and the third option interdistrict open enrollment was voluntary The i ntradistrict open enrollment option allow ed students to enroll in a school within their residential district other than the neighborhood school assigned based on their residence. The post secondary enrollment option allowed the 11 th and 12 th grade students to receive credit for post secondary courses that would s atisfy both high school graduation
142 requirem ents and coll ege requirements. The third policy, interdistrict open enrollment gave the school districts an option to accept nonresident students provided the students resided in adjacent district s T he interdistrict open enrollment statute set forth a time line for i mplementing the program and provided a list of procedures and requirements for applications and admissions to guide the school districts as well as parents and students that were interested in utilizing this option The law prohibited school districts to s elect students for admission based on race, disability, academic perform ance, athletic talent, and disciplinary record ; h owever, some provisions were put in place to prioritize enrollment (e.g. resident students or previously enrolled students over first t ime applicants) maintain a racial balance, accommodate for district enrollment capacity etc. The implementation of interdistrict open enrollment started with a three year pilot program in the school year 1990 1991 and initially involved 3 (out of 611) school districts and 23 high school and elementary students Participation of s chool district s and student s in the remaining two years of the pilot program increased to 10 school districts and 115 students for the school year 1991 1992 and 49 school distri cts and 551 students for the school year 199 2 199 3 ( Ohio Department of Education, 1993 ) F ull implementation of the interdistrict open enrollment option was authorized for the fall of 1993 P rior to the beginning of the school year 1993 1994 t he board of education of any school district had to adopt a resolution that either entirely prohibited the enrollment of students from adjacent districts or permitted the enrollment of students from all adjacent districts (Ohio Department of Education, 1991) Particip ation of school districts in the interdistrict open enrollment program i ncreased to 301 in the school year 1993
143 1994, 341 in the school year 1994 1995, 353 in the school year 1995 1996, and 376 in the school year 1996 1997 (Crepage, 1999). T h e interdistri ct open enrollment program was expanded with the passage of the Senate Bill 55 (SB 55) in July 1997 which allowed school districts to adopt on or after July 1998 one of the following policies that either : e ntirely prohibits interdistrict open enrollment from any other school district (except for the stu dents for whom tuition is paid) ; permits open enrollment of students from adjacent school districts, as under current law; or permits the open enrollment of students from any city, exempted village, or loca l school district that is not part of the joint vocational school district (Crepage, 1999). SB 55 also expanded the post secondary option to students in 9 th and 10 th grade. Participation of school districts in the expanded interdis trict enrollment program increased as well Current participation for the school year 2010 2011 is 429 school districts ( city, local, exempted village, joint vocational, and career centers) are open to statewide enrollment, 90 school districts are open to adjacent enrollment and 144 school districts are closed (i.e. accepting only resident students) (Oh io Department of Education, 2010 ). has raised many concerns over school funding. jointly funded by the state and school districts through local property taxes. Under t he interdistrict open enrollment program the amount of public funds education follows the student as he/she transfers across district lines. The refore, the implementation of interdistrict open enrollment policies carries financial ramifications for both the sending and the receiving districts Since t he first year of the pilot program, a
144 formula has been used to transfer funds from the sending sch ool district to the receiving school district 5 In t he 1 997 1998 school year, Ohio introduced a figure to be used in the (Crepage, 1999). Th is figure represents the combined state and district funds allocated to a student for the school year adjusted by a county equalization factor ( to allocate more funds to districts in high cost counties ) and amounts to about 60 62% of the state average expenditure per student (Crepage, 1999). If a st udent enrolls in a school outside the residing district, the entire guarantee amount ( computed based on the residing district ) is deducted from the state aid the sending district receives for the school year and is added to the state aid of the receiving d istrict. Because the educational cost per student differs across school districts, a receiving district does not always benefit from open enrollment. The amount of money that each participating student brings might not cover all educational costs, in which case additional funds need to be generated locally to educate nonresident students Since parents of nonresident students are not part of the additional funds. T he impl ementation of the i nterdistrict open enrollment program has generate d large variations in school funds for all school districts. Ruggles (1997) conducted a financial analysis of the interdistrict open enrollment program for the period starting in t he 1993 1994 school year and ending in the 1996 1997 school year His analysis shows variations in revenue gains and losses to school districts from 5 For the school year 1990 199 1, the school district that accepted a transfer student received $2,636 of doing attended, an amount that was deducted from the funds of the sending distric t (Ohio Department of Education, 1991).
145 participation (or lack thereof) in interdistrict open enrollment that range f rom a loss of about $7,333,000 for the Akron City School District to a gain of about $6,485,000 for the Coventry Local School District These sizable changes in school funds suggest that any school district could take financial advantage of this option by improving their academic progra ms facilities, implementing better marketing strategies etc. a main goal of the open enrollment program is to set up a more competitive atmosphere among school districts however, it might create perverse financial incentives. Fowler (1996) finds tha t all else equal, school leaders seem willing to compete for students and state funds but under different conditions (e.g. when having to compete with wealthier district s ) competition is less likely Thus, t he respon se of school districts to the policy n eed s to be evaluated within the financial and political constraints that each school district faces. T he literature that has examined the supply side of enrollment options have found systematic differences in characteristics betw een school dis tricts that decide to adopt interdi strict open enrollment (open school districts), and those that forgo the option (closed school districts) Fowler (1996) examined the ment option for the school year 1993 1994 its first year of full implementation. The author reported that open districts tended to be those with low student enrollment, declining enrollment trends of previous five years, low minority enrollment (less tha t 1%), rural location, and below average per pupil expenditure ($3,501 $4,500) In contrast, closed districts tended to have high enrollment with increasing trends, suburban locations, minority enrollment between 11% and 20%, suburban locations, and above average per pupil expenditure ( over $5,501 ) A study by Metzler (1996) for the school year 1994 1995
146 reported similar district differences. Open school districts had lower average daily membership, median family income, average class size, percentage of bl ack and Asian students, percentage of students passing the proficiency tests, revenue and expenditure per pupil, average teacher salary and staff attendance, and higher percentage of economically and academically disadvantaged students. These findings were also corroborated by Crepage (1999) for the school districts that participated in the interdistrict open enrollment program for the school year 1996 1997. The difference s in the profiles of school di stricts that chose to adopt or forego interdistrict ope n enrollment options suggest s that their adoption decision s are motivated by different factors. Fowler (1996) also surveyed school district leaders and gather ed some descriptive evidence as to what were the major reasons that led them to the decision to ad opt (or not) i nterdistrict open enrollment The survey results indicate that the primary reasons for superintendents of open districts to choose open enrollment were concern s about losing student enrollment and state funds. C losed districts on the other h and, cho se to be closed mainly because of insufficient space (crowded buildings, large class sizes, maximum pupil teacher ratio) and financial concern over subsidizing education costs of nonresident students with local funds Interestingly, about a third o f the superintendents of open districts indicated a major reason to be the influence from the decision of adjacent districts to be open in some cases indicating that the decision was made collaboratively with adjacent districts Similar evidence is provid ed in Crepage (1999). Based on this evidence, we next model the policy adoption decisions in a manner that is consistent with their economic behavior explic itly
147 accounting for spatial interdependence in decision making among neighboring school districts. We further note that, if spatial dependence is not accounted for, the policy adoption model is likely to produce misleading results. Estimation Framework We can start modeling the decision s within a RUM frame work b y assuming that there are factors that jointly determine a choice of a policy alternative. S ome of these factors are observed by the researcher ; for instance, c haracteristics of school districts such as student enrollment class siz e pupil teacher ratio; financial factor s such as revenues generated from property taxes and instruction expenses; and academic factors such student attendance, student performance etc. Other factors are unobserved; for instance philosophical beliefs or perception s of school leaders regarding the different policy alternatives etc. In addition alternative d epend s as made evident in the preceding discussion on the propensity of nearby school districts to choose that particular policy either s trategically or collaboratively Finally, i t is reasonable to assume t hat t he school district makes a decision to adopt a policy by choosing the policy alternative that generates the highe st expected benefits In this case, the term benefit needs to be interpreted within the context of the sc hool district policy. For instance, if a school district is considering adopting an open enrollment policy because of low student enrollment, the optimal policy alternative would be the one that maximizes student enrollment, thus, the alternative actually adopted by the d istrict and the choice observed by the researcher Because some of the factors that determine the expected benefits a school district derives from various policy alternatives which in turn, determine a
148 actual adoption of a policy alternative are un observed we can set up a latent variable model of Co nsider a school district that faces a decision to adopt one of policy alternatives in the choice set and let denote a general alternative. Let denote a latent variable representing the expected benefits that school district generates from policy alternati ve ; depends among other factors, expected benefits as follows: (5.1) w here and represent s school district observed and un observed factors associated with policy alternative respectively Then, under the model assumptions, the school district will choose to adopt a particular policy alternative, say if and only if the p olicy alternative generates the highest exp ected benefits when compared to all other policy alternatives : (5.2) E quivalently, the researcher observes the adoption of policy alternative if: (5.3) In this for mulation, the policy adoption decision can be interpreted in the context of the propensity for adoption t hus, we can make p robabilistic statements about the adoption decision. The probability that a school district chooses policy alterna tive over policy alternative is the probability that the combination of observed and unobserved factors generate s higher expected benefits under policy alternative than under alternative result ing in th at particular outcome. T he probability that sc hool district adopt s policy alternative is given by:
149 (5.4) Thus, t he probability that a school district will adopt a particu lar policy alternative depends among other things on the underlying factors determining the probabilities of neighboring school districts to adopt th e given policy alternative The choice probabilities for the policy adoption model in ( 5 4) can be empir ically estimated once a functional form is specified for the expected benefits in (5.1) and a distributional assumption is made for the unobserved factors. The specification of the functional form of (5.1) requires specification of spatial interactions amo ng school districts. We follow common practice and specify a spatial weights matrix that quantifies the spatial relationships among neighboring school districts and estimate the following s patial autoregressive lag model : with, (5.5) w here denotes the model covariates ; are the spatial weights assigned to s chool districts in locations and ; and and a re the parameters of interest. T he parameter measures th e degree of neighborhood influence. A positive means that a high propensity of adoption of a particular policy by a given school district positively affects the propensity of adoption of that alternati ve by a neighboring district, thus, increasing the probability of adoption of that policy alternative for both neighbors. Lastly, we assume that the error terms in (5.5) follow an IID type I extreme value distribution and estimate a spatial multinomial log it (SMNL) model using the estimation methodology developed in C hapter 3
150 Data To estimate the policy adoption model described in the previous section, we utilize data on for the school year 2000 2001. The data set for this applicati on is compiled fr om various sources and c ontains information on district demographic characteristics (Ohio Department of Education), income and racial heterogeneity indices (National C enter for Education Statistics), property tax information (Ohio Departme nt of Taxation), and geographic (latitude and longitude) information of location of the school districts 6 Our sample is comprised of 609 school districts (city, local, and exempted village) of which 365 are open to statewide enrollment, 90 are open to adjacent enrollment, and 154 have chosen not to participate in the open enrollment program (Table 5 1) T he spatial distribution of the adoption of these enrollment policies is given by Figure 5 1 F igure 5 1 shows c lustering of school districts that have adopted the same open enrollment policies an indication of positive n eighborhood influence in adoption decisions among neighboring districts This influence seems to arise mainly between school districts in adjacent locations and spills over county lines. This visual evidence is useful for defining the spatial relationships between school districts (i.e. the type of the spatial weights matrix ) to be used in the model The districts that have chosen to be closed are primarily located in the suburbs of Cinci nnati, Columbus, and Cleveland metropolitan areas These areas are characterized by high student enrollment and high er than average median income In contrast, s chool districts that are open are 6 The dataset for this study was compiled and made available by Dr. David M. Brasington, University of Cincinnati.
151 located mainly in rural areas and urban area s that are charac terized by low student population and lower than average median income ( Figure 5 2). A description of the variables used in the model along with descriptive statistics is provided in Table 5 2 and Table 5 3. The dependent variable in our model is an indic ator variable for the school districts (Openenroll), with three policy alternatives: 1) statewide open enrollment, 2) adjacent open enrollment, and 3) no open enrollment. To explain the adoption decisions we includ e the following covariates First, we include the fall school district enrollment for the school year 2000 2001 (Falladm00) computed as kindergarten through grade 12 enrollment adjusted by adding non attending pupils and subtracting any unauthorized atten dance. The mean student enrollment for the school year 2000 2001 is about 2,850 pupils. Because many school districts that have decided to adopt interdistrict open enrollment are characterized by low minority enrollment, we include the percentage of studen t enrollment that is non white (Pctmin00). The average percentage of minority student enrollment in our sample is 7.2%. Other important demographic characteristics that are income and r acial heterogene ity so we u se racial heterogeneity (Racehet00) and income heterogeneity (Inchet00) measured by a Leik (1966) index for the school district in the school year 2000 2001. A typical school district in our sample is characterized by relatively low racial hete rogeneity but relatively high income heterogeneity. We are also interested in whether changes in these measures affect the choice probability of a particular policy alternative, so the change in racial heterogeneity (Racehet_ch9000) and income heterogeneit y (Inchet_ch9000) in the school district between 1990 and 2000 are also
152 included. The average change in the racial heterogeneity index i s positive indicating an increase in racial heterogeneity from the previous decade for the typical school district. In c ontrast, the average change in income heterogeneity is negative indicating a reduction in in come heterogeneity. The decision of a school district to be open or closed to interdistrict open enrollment also largely depends on s cho ol district enrollment capac ity Among various measures of capacity (e.g. class size, building capacity, student teacher ratio), we use the change in the student teacher ratio in the school district from the previous decade (Tadmprtc_ch9000) to investigate whether or not a change in capacity strongly effects the probability of adoption of any of the three policy alternatives The average change in the student teacher ratio is negative showing a decrease in the number of students assigned to a teacher compared to the previous decade T he percentage of student s in each school district proficient and above in math based on a 9 th grade proficiency test (Math00) is included as a measure of academic performance. Another important aspect of the adoption deci sion s is the financial aspect. School districts may choose to be open or closed depending on whether or not the state funds that accompany the transfer students cover all the educational expenses Thus, decision s will highly depend on current availability of local funds as well as their ability to generate additional educational funds locally. For this reason, w e include the local school revenues per pupil for the school year 2000 2001(Localxpa00) and the change in these revenues from the previous decade (Localxpa_ch9000). Local school revenues are obtained by dividing Class 1 (agricultural and residential) tax collections by the fall
153 school district enrollment. For the school year 2000 2001 the average per pupil rev enue generated locally through property taxes is about $2,000 This figure has increased by an average of $890 from the previous decade In addition, we include the C lass 1 property tax rate in effective mills (Ptaxrate00), the percentage of real property in the school district that is of agricultural value (Pctag00), the change in the percent of real property in the school district that is of agricultural value (Pctag_ch9000), and Class 2 property value per pupil (Cl2valpera00) to examine the relative impo rtance of the factors in the probability of adoption of the three policy alternatives Finally, to carry out our analysis we need to define the spatial relationships between the school d istricts and specify a spatial weights matrix Because of the heterogeneity in the size of the different school districts (Figure 5 1) specifying a distance based spatial weights matrix would result in a large number of neighbors for small districts an d few ( or perhaps no) neighbors for large districts (Anselin, 2002) So we specify the spatial weights matrix based on a fixed number of nearest neighbors to ensure there will be some neighbors for every school district. A school district in Ohio has on av erage 6 nearest neighbors; however, the number of neighbors varies from 1 (or zero for the few island districts) to 15 first order neighbors. U sing the school we specify four spatial weights matrices based on 3 6 8, and 10 nearest neighbors. Note that the specification of the spatial weights matrix based on 3 nearest neighbors, while it may capture well the neighborhood structure for the larger districts, it is likely to misrepresent the neighborhood for the moderate to small size districts. On the other hand the specification of the spatial weights matrix
154 based on 10 nearest neighbors is likely to mis represent the neighborhood for the large districts as it will i nclud e second order, third order or even higher order neig hbors between which little to no spatial interaction is expected to be present The remaining two specifica tions (based on 6 and 8 nearest neighbors) are more likely to capture the neighborhood structure of the typical school district. Following common pra ctice, a ll matrices set and are row standardized (rows sum to one). Estimation R esults The results for the different specifications of the SMNL model (based on the different specifications of ) along with MNL estimates ( as benchmark estimates ) are presented in Tabl e 5 4 T he base category chosen for the analysis is the statewide open enrollment policy In general, t he estimates from these models align w ith expectations In addition, the SMNL estimates are robust across the different spatial specific ations ; occasiona l changes in sign occur only for estimates that are not statistically different from zero. Because the MNL and SMNL mode ls are nonlinear in parameters the estimated coefficients only indicate the direction of the effect of each hypothesized factor i n the relative probability of adoption of adjacent open enrollment and no open enrollment alternatives (relative to statewide open enrollment) M arginal effects on the other hand, allow interpretation of the magnitude of each factor in terms of its importance i n the probability of adoption of the three policy alternatives Because the spatial effects in our spatial model specifications are global a change in a covariate for a given school distinct will affect not only the probability of policy adoption for the own district but also the policy adoption probability for the remaining school districts the marg inal effects
155 are no t point estimates, but matrices with the diagonal elements representing the direct (own) effects of a change in the explanatory va riable and the off diagonal elements representing the indirect (spillover) effects The marginal effects for our SMNL model are given by: (5.6) where denotes an vector of ones and represents element by element (Hadamard) multiplication. To provide scalar measures of the marginal effects, we to obtain a measure of t he direct effects, average the row sums to produce a measure of the total effect, and take the difference between the two measures to get a measure of the indirect effects as follows : (5.7) The estimates of the marginal effects for the MNL model and the different specifications of the SMNL model are presented in Table 5 5 The se estimates are also fairly robust across the different specifications of the SMNL model a s well as across the spatial and non spatial specifications A marginal effect produced by the M NL model measures the change in the probability of adoption of a given policy alternative associated with a change in the average observation (school district) for a particular explanatory variable. The interpretation of the marginal effects produced by a spatial discrete choice model is analogous with the exception that a change in a particular explanatory variable for a given school district generates multiple effects The direct
156 effect measures the i mpact of a one unit change in a covariate on the probability of adoption of a given policy alternative for the school district of interest The indirect effect measures the i mpact of a one unit change in a covariate on the probability of adoption of a given policy alternative for neighboring school district s T he total effect as the sum of the preceding two effects, sho ws the average total impact of a change in a covariate for a given school district o n the probability of adoption of a given policy alternative for the school district of interest. For comparison purposes a marginal effect produced by the MNL model for a specific policy alternative would be interpreted as a direct effect since the indirect effect s in non spatial model s are zero (LeSage and Pace, 2009). Prior to interpreting the estimates of the marginal effects, it is worth remarking that for the majorit y of the explanatory variables, the m agnitudes of the marginal effects for the MNL model are comparable to the magnitudes of the direct effects produced by the SMNL model specifications. However, when we look at the total effects, the latter are much large r due to the indirect effects pointing to the importance of the neighborhood a decision making aspect that the MNL model fails to capture We note further, that across all specifications of the SMN L model, the direct effects and indirect effects have the same sign, a result that is consistent with the presence of positive neighborhood influence. estimated coefficients for fall student enrollment (Falladm00) are not statistically significant, thus there is uncertainty about the direction of the effect of this factor on the probability of adoption of adjacent and no open enrollment alternatives ( rel ative to statewide enro llment ) possibly due to being fairly contemporaneous As expected, t he
157 estimates of the marginal effects with respect to this factor are also imprecisely estimated in both the MNL and SMNL specifications The effect of the percentage of minority student en rollment (Pctmin00) is statistically significant in all model specifications A n increase in the percentage of minority enrollment is associated with a decrease in the probability of adoption of statewide open enrollment and an increase in the probability of adoption of the other two alternatives. This is expected since statewide open enrollment is typically adopted in school districts with low minority population. The Leik (1966) index measure of racial heterogeneity (Racehet00) negatively affects the adop tion probability of adjacent open enrollment and no open enrollment relative to statewide open enrollment. We would expect that an increase in racial heterogeneity be associated with a decrease in the adoption probability of statewide open enrollment since the adoption of this policy alternative occurs in school districts that have a rather homogeneous racial composition Our expectation s are confirmed by the effects of the change in the racial index from the previous decade A positive change in the racial heterogeneity index from the previous decade (Racehet_ch9000) denoting an increase in racial heterogeneity neg atively affects th e probability of adoption of statewide enrollment and increase s the adoption probability of the no open enrollment alternat ive Similar effects are found for the income heterogeneity index (Inchet00). A n increase in income heterogeneity increases the probability of adoption of statewide open enrollment and decreases the adoption probability of the no open enrollment policy. Th e effect of income heterogeneity in the probability of adoption of the adjacent open enrollment alternative and the effects of the change in income heterogeneity from the previous decade (Inchet_ch9000) are unclear since their estimates are not
158 statistical ly significant. The effect of the capacity variable the change in student t eacher ratio (Tadmprtc_ch9000) is positive for the adoption probability of adjacent enrollment and negative for the statewide open enrollment alternative The average change in the student teacher ratio from the previous school decade is negative, indicating an average decrease in the number of students per teacher. This could explain the positive effect in the probability of adoption of adjacent open enrollment as more students can be accommodated if school districts have not reached the maximum student teacher ratio Th e academic student performance meas ured by the percentage of students in each school district proficient and above in math (Math00) in creases the probability of a school district choosing not to participate in open enrollment while decreases the probability of adoption of the statewide enrollment policy S chool districts that choose to be closed are generally characterized by higher student performance (higher percentage of students that pass proficiency tests) than open enrollment school districts. Regarding the effects of the financial factors, the local school revenue per pupil (Localxpa00) is expected to positively affect the probability of a school distric t not participating in open enrollment relative to statewide enrollment ; however, this is only confirmed by the MNL marginal effect estimate of the no open enrollment alternative. Local revenues are especially important to those school districts with above average per student expenses which are typically the school districts that do not participate in the open enrollment options. However, when school districts are able to generate more funds locally they are more likely to participate in open enrollment as shown by (Localxpa_ch9000). This is also a result confirmed by the MNL marginal effect estimate
159 of the no open enrollment policy alternative. The effect of the Class 1 property tax rate (Ptaxrate00) is uncertain because the marginal effect estimates acros s all model specifications are not statistically different from zero. T he amount of property with agricultural value (Pctag00) ha s a negative effect in the probability of a school district to be closed but a positive effect on the probability of adoption o f adjacent open enrollment. This is expected since the school districts that are open to interdistrict open enrollment are mainly located in rural areas. The effect of the Class 2 property value per pupil in the probability of adoption of open enrollment p olicies is negative for the MNL model, but this effect becomes statistically insignificant in the specifications of the spatial models. Most importantly the estimates of the spatial lag parameter ( ) produced by the SMNL model vary from 0. 32 to 0. 63, de pending on the specification of The estimate of the spatial lag parameter shows weak presence of spatial dependence for the specification of based on 3 nearest neighbors likely indicat ing that the spatial weights matrix based on 3 nearest neighbor s does not represent well the range of interactions among school districts but this dependence becomes stronger as the neighborhood size increases As a result, the indirect effects obtained for the spatial model specification based on 3 nearest neighbor s are generally not statistically significant. As anticipated, the neighborhood influence is strongest for the neighborhood specification based on 8 nearest neighbors. This influence decreases as the size of the neighborhood becomes larger to include 10 ne arest neighbors. Interestingly, the indirect effects of some of the school districts decision factors such as the percentage of minority enrollment, income heterogeneity, and academic performance (based on the
160 math proficiency tests) are larger than the re spective direct effects for the last two spatial model specifications indicating a n especially strong mutual influence of neighboring school districts along these enrollment dimensions. These results confirm our expectations that there is positive neighb orhood decisions are influenced by other scho ol districts with similar demographic characteristics and financial constraints (i.e. school districts in reference groups) Th us, a high propensity of a given school district to adopt a particular open enrollment policy alternative positively affects the propensity of a nearby school district to adopt the given alternative, increasing the probability that the two neighboring dist ricts will adopt the same policy This, in turn, implies clustering of school districts with similar enrollment policies which complies wit h the spatial pattern of open enrollment policies observed in Figure 5 1 Finally the matri ces of marginal effects obtained by the SMNL model for each of the provid e information not only regarding th e direct impacts of each of these factors o n the probability of adoption of open enrollment policy alter natives, but also the indirect impacts that spill over from (or to) the surrounding school districts As it was made evident in the previous chapter, t he scalar measures reported in the section are only one way to summarize and analyze these effects. T he matrices of marginal effects can be p artitioned to represent different neighborhoods of interest and the marginal effects can be analyzed similarity The latter is particularly helpful to identify the locations with the most influence (i.e. the refe rence groups).
161 Final Remarks This chapter provide s a quantitative analysis of the determinants of the adoption of interdistrict open enrollment policies by school districts particularly focusing on the neighborhood influence adoption decisions It estimates a spatially explicit model of policy adoption decisions school districts to examine the importance of these determinants in the adoption probability of three interdistrict open enrollment alternatives; statewide o pen enrollment, adjacent open enrollment, and no open enrollment The empirical results substantiate extant descriptive evidence regarding the determinants of adoption of open enrollment policies. Among the most influential factors are minority population racial and income heterogeneity, change s in student teacher ratio, student academic performance and financial factors that indicate a school funds locally. More importantly, the results show strong neighborhood influence in policy adoption decisions The results from our analysis have the following empirical and policy implications. Empirically, a policy adoption model that ignores the potential neighborhood inf luence is likely to suffer from specification error and produce potentially misleading results. From a policy viewpoint, the presence of neighborhood effects provides further support to the view that public policy adoption is stimulated by interaction amon g decision makers at a local level Thus the adoption of new public polic ies much like the diffusion of technology innovations is largely influenced by reference groups. A spatial analysis of these neighborhood effects can help identify the most influenti al locations (i.e. reference groups).
162 Figure 5 1. Interdistrict open enrollment policies in Ohio Sour ce: Ohio Department of Education Figure 5 No open enrollment Adjacent enrollment Statewide open enrollment
163 T able 5 1. School d istricts in e ach e nrollment c ategory Open Enrollment Policy School Districts Share (%) Statewide 365 .00 59.93 Adjacent 90 .00 14.78 No open 154 .00 25.29 Total 609 .00 100.00 Table 5 2 Policy a doption m odel: v ariable d escription Variable Description Openenroll Indicator variable for school district s open enrollment status (1 statewide enrollment, 2 adjacent enrollment, 3 no open enrollment) Falladm00 Fall school district enrollment (in thousands of students) for school year 2000 2001 Pctmin00 Percent of student enrollment that is non white for school year 2000 2001 Tadmprtc_ch9000 Change in student teacher ratio in school district between 1990 and 2000 Ptaxrate00 Class 1 property tax rate in effective mills in school district for school year 2000 2001 Pctag00 Percent of real proper ty in school district that is agricultural value for 2000 Pctag_ch9000 Change in percent of real property in school district that is agricultural value (1990 2000) Racehet00 Racial heterogeneity in school district in 2000, measured by Leik (1966) index Racehet_ch9000 Change in racial heterogeneity in school district between 1990 and 2000 Inchet00 Income heterogeneity in school district in 2000, measured by Leik (1966) index Inchet_ch9000 Change in income heterogeneity in school district between 1990 an d 2000 Localxpa00 Local school revenues per pupil (in thousands of dollars) for school year 2000 2001 Localxpa_ch9000 Change in local school revenues per pupil between 1990 and 2000 Math00 Percent of students in each school district proficient or abov e in math section of the school year 2000 2001 9 th grade proficiency test Cl2valpera00 Class 2 property value per pupil (in thousands of dollars)
164 Table 5 3 Policy a doption m odel : v ariable d escriptive s tatistics Variable Mean Std. Dev. Min Max open enroll 1.65 0.86 1 .00 3 .00 falladm00 2.85 4.98 0.04 72.28 pctmin00 7.23 13.77 0.00 100.00 tadmprtc_ch9000 0.25 1.98 10.20 8.10 ptaxrate00 26.88 5.23 18.35 61.93 pctag00 13.04 12.79 0.00 56.07 pctag_ch9000 2.50 3.05 14.30 3.63 racehet00 0.0 6 0.05 0.00 0.45 racehet_ch9000 0.02 0.03 0.15 0.18 inchet00 0.71 0.10 0.49 1.21 inchet_ch9000 0.35 0.10 0.72 0.09 localxpa00 2.02 1.43 0.36 19.42 localxpa_ch9000 0.89 0.66 0.55 7.11 math00 75.22 11.38 28.50 98.70 cl2valpera00 17.55 21.71 0.50 232.31
165 Table 5 4. Estimated c oefficients of the p olicy a doption m odel SMN L Models MNL Model W_3nn W_6nn W_8nn W_10nn Est. St. Err. Est. Est. St. Err. St. Err. St. Err. St. Err. St. Err. St. Err. falladm00: 0.058 0.072 0.051 0.170 0.066 0.174 0.029 0.177 0.109 0.180 0.002 0.029 0.014 0.029 0.028 0.031 0.034 0.031 0.025 0.030 pctmin00: 0.064 0.024 0.073 0.027 0.078 0.027 0.072 0.027 0.080 0.027 0.091 0. 025 0.103 0.030 0.109 0.031 0.104 0.030 0.105 0.031 racehet00: 20.503 7.238 23.735 11.890 25.303 11.878 22.823 11.743 24.878 11.846 17.426 7.217 23.868 11.017 23.974 11.495 22.658 11.157 23.937 11.101 racehet _ch9000: 16.875 8.125 19.328 12.314 22.160 12.203 19.196 12.155 21.298 12.267 15.874 8.467 18.820 12.213 15.540 12.914 14.439 12.548 18.456 12.656 inchet00: 2.538 1.656 2.076 1.887 2.003 1.864 2.2 96 1.858 2.362 1.873 11.190 2.413 11.978 4.005 11.538 4.060 10.704 4.100 9.976 4.103 inchet_ch9000: 0.786 1.499 0.637 1.677 0.296 1.656 0.895 1.660 0.788 1.669 1.767 1.841 1.825 2.457 1.167 2.506 0.799 2.528 0.872 2.536 tadmprtc_ch9000: 0.145 0.064 0.136 0.069 0.122 0.070 0.146 0.070 0.145 0.070 0.023 0.079 0.119 0.102 0.076 0.101 0.085 0.101 0.071 0.102 math00: 0.010 0.015 0.008 0.017 0.003 0.017 0.012 0.018 0.009 0.018 0.063 0.020 0.081 0.031 0.090 0.031 0.094 0.031 0.086 0.031 localxpa00: 0.483 0.500 0.096 1.024 0.680 0.966 0.148 0.971 0.040 0.997 1.601 0.396 0.850 0.846 0.533 0.808 0.191 0.849 0.525 0.90 4
166 Table 5 4. Continued SMN L Models MNL Model W_3nn W_6nn W_8nn W_10nn Est. St. Err. Est. Est. St. Err. St. Err. St. Err. St. Err. St. Err. St. Err. localxpa_ch9000: 0.664 0.773 0.424 1.401 0.581 1.346 0.4 26 1.335 0.210 1.345 1.766 0.613 1.056 1.025 0.659 0.988 0.322 1.034 0.661 1.084 ptaxrate00: 0.052 0.041 0.043 0.051 0.021 0.051 0.031 0.052 0.031 0.053 0.001 0.040 0.007 0.049 0.003 0.049 0.004 0.048 0.005 0.049 pctag00: 0.016 0.013 0.005 0.017 0.014 0.017 0.014 0.017 0.014 0.017 0.093 0.026 0.134 0.054 0.164 0.053 0.160 0.052 0.137 0.052 pctag_ch9000: 0.090 0.047 0.093 0.048 0.075 0.049 0.090 0.0 48 0.096 0.048 0.023 0.076 0.072 0.119 0.122 0.116 0.103 0.117 0.083 0.117 cl2valpera00: 0.022 0.019 0.053 0.043 0.011 0.042 0.027 0.042 0.017 0.043 0.031 0.013 0.028 0.023 0.017 0.022 0.011 0.022 0.019 0.023 rho --0.322 0.188 0.418 0.217 0.626 0.247 0.524 0.253 Note: The columns for the SMNL model correspond to different specifications of the spatial weight s matrix W that vary the number of nearest neighbors to 3, 6, 8, and 10, respectively. See text for details.
167 Table 5 5. Marginal e ffects for the e stimated c oefficients of the p olicy a doption m odel SMNL Model (W_3nn) SMNL Model (W_6nn) SMNL Model (W_8nn) SMNL Model (W_10nn) MNL Model Direct Indirect Total Direct In direct Total Direct Indirect Total Direct Indirect Total falladm00: stw 0.0062 0.0048 0.0020 0.0068 0.0064 0.0041 0.0105 0.0030 0.0046 0.0076 0.0075 0.0074 0.0149 (0.0083) (0.0135) (0.0057) (0.0190) (0.0133) (0.0089) (0.0218) (0.0080) (0.01 17) (0.0193) (0.0109) (0.0109) (0.0205) adj 0.0069 0.0044 0.0018 0.0063 0.0056 0.0036 0.0091 0.0013 0.0017 0.0030 0.0067 0.0066 0.0133 (0.0085) (0.0152) (0.0063) (0.0214) (0.0160) (0.0106) (0.0264) (0.0087) (0.0122) (0.0209) (0.0121) (0.011 5) (0.0227) no 0.0007 0.0004 0.0002 0.0005 0.0008 0.0006 0.0014 0.0017 0.0029 0.0046 0.0008 0.0009 0.0016 (0.0031) (0.0027) (0.0013) (0.0039) (0.0035) (0.0027) (0.0062) (0.0018) (0.0045) (0.0060) (0.0022) (0.0029) (0.0050) pctmin00: stw 0.0148 0.0116 0.0050 0.0166 0.0114 0.0077 0.0191 0.0083 0.0131 0.0213 0.0100 0.0106 0.0205 (0.0041) (0.0030) (0.0035) (0.0039) (0.0033) (0.0023) (0.0048) (0.0041) (0.0052) (0.0083) (0.0041) (0.0051) (0.0083) adj 0.0061 0.0054 0.00 22 0.0076 0.0054 0.0033 0.0087 0.0030 0.0041 0.0072 0.0044 0.0042 0.0086 (0.0027) (0.0025) (0.0016) (0.0033) (0.0025) (0.0015) (0.0041) (0.0019) (0.0021) (0.0040) (0.0031) (0.0024) (0.0037) no 0.0087 0.0061 0.0029 0.0090 0.0060 0.0044 0.0104 0.0052 0.00 89 0.0142 0.0056 0.0064 0.0119 (0.0026) (0.0019) (0.0022) (0.0031) (0.0021) (0.0024) (0.0040) (0.0020) (0.0044) (0.0064) (0.0019) (0.0034) (0.0053) racehet00: stw 3.6687 3.2156 1.3803 4.5959 3.1265 2.0826 5.2091 2.1204 3.2901 5.4104 2.6475 2.7693 5.4167 (1.1441) (1.2216) (0.9573) (1.5407) (1.2892) (0.7163) (1.7588) (1.3834) (1.9837) (2.0009) (1.1111) (1.3201) (1.8327) adj 2.1497 1.8645 0.7500 2.6146 1.9332 1.2180 3.1513 1.0160 1.4004 2.4164 1.4213 1.3689 2.7902 (0.8445) (1. 1112) (0.5975) (1.4529) (1.2228) (0.9971) (1.8359) (1.0646) (0.9084) (1.6875) (0.1538) (0.0907) (0.1595) no 1.5191 1.3511 0.6303 1.9813 1.1932 0.8646 2.0578 1.1044 1.8897 2.9940 1.2262 1.4003 2.6266 (0.7646) (0.7525) (0.5392) (1.0956) (0.83 70) (0.8013) (1.4011) (0.6709) (1.7727) (1.9941) (0.5948) (1.3209) (1.6090) racehet_ch9000: stw 3.1575 2.5836 1.1077 3.6913 2.4616 1.6235 4.0852 1.5581 2.3817 3.9399 2.1617 2.2510 4.4127 (1.2940) (1.2541) (0.8381) (1.6666) (1.1 936) (1.1806) (1.8544) (1.2636) (1.6182) (2.1286) (1.0365) (1.1943) (1.9697) adj 1.7434 1.5248 0.6137 2.1386 1.7823 1.1320 2.9143 0.8762 1.2150 2.0913 1.2296 1.1866 4.4127 (0.9534) (1.1313) (0.5665) (1.5277) (1.2552) (1.0264) (1.9681) (0.9784) (0.9413) (1.6886) (1.0815) (0.9049) (1.6290) no 1.4141 1.0588 0.4940 1.5528 0.6794 0.4916 1.1709 0.6819 1.1667 1.8486 0.9321 1.0644 1.9965 (0.9155) (0.8475) (0.5041) (1.2321) (0.9286) (0.6965) (1.5458) (0.7122) (1.4156) (1.9243) (0.7727) (1.1708) (1.6905)
168 Tabl e 5 5. Continued SMNL Model (W_3nn) SMNL Model (W_6nn) SMNL Model (W_8nn) SMNL Model (W_10nn) MNL Model Direct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total inchet00: stw 1.2685 0.8405 0.3816 1.2222 0.7361 0.5189 1.2550 0.6242 1.0340 1.6583 0.6479 0.7157 1.3636 (0.2919) (0.2523) (0.2775) (0.3665) (0.2683) (0.1703) (0.4139) (0.2933) (0.5280) (0.6228) (0.2922) (0.3067) (0.4543) adj 0.1139 0.0579 0.0177 0.0757 0.0045 0.0189 0.0234 0.0625 0.0730 0.1355 0.0864 0.0743 0.1608 (0.1960) (0.1786) (0.0750) (0.2525) (0.2373) (0.1627) (0.3995) (0.1024) (0.1404) (0.2332) (0.1256) (0.1297) (0.2434) no 1.1546 0.7826 0.3639 1.1465 0.7406 0.5378 1.2784 0.5617 0.9610 1.5227 0.5614 0.6414 1.2 028 (0.2567) (0.2316) (0.2615) (0.3347) (0.2639) (0.1916) (0.4361) (0.2579) (0.4032) (0.6233) (0.2587) (0.2616) (0.4270) inchet_ch9000: stw 0.2397 0.1532 0.0683 0.2215 0.0815 0.0569 0.1384 0.0788 0.1216 0.2003 0.0897 0.0944 0.1 842 (0.2405) (0.1942) (0.0858) (0.2680) (0.1818) (0.1161) (0.2924) (0.1602) (0.2172) (0.3659) (0.1739) (0.1618) (0.3246) adj 0.0639 0.0374 0.0144 0.0518 0.0081 0.0036 0.0118 0.0403 0.0556 0.0959 0.0443 0.0426 0.0869 (0.1770) (0.1519) (0.0634) (0.2145) (0.1591) (0.1044) (0.2634) (0.0863) (0.1171) (0.1981) (0.1072) (0.1066) (0.2096) no 0.1758 0.1158 0.0539 0.1697 0.0734 0.0533 0.1267 0.0385 0.0659 0.1044 0.0454 0.0519 0.0973 (0.1956) (0.1666) (0.0744) (0.2326) (0.1720) (0.1140) (0.2812) (0.1373) (0.21 35) (0.3473) (0.1489) (0.1533) (0.2987) tadmprtc_ch9000: stw 0.0169 0.0174 0.0074 0.0249 0.0130 0.0086 0.0216 0.0106 0.0160 0.0267 0.0119 0.0121 0.0240 (0.0103) (0.0077) (0.0067) (0.0123) (0.0073) (0.0079) (0.0128) (0.0073) (0. 0134) 0.015896 (0.0073) (0.0108) (0.0140) adj 0.0169 0.0109 0.0044 0.0153 0.0100 0.0063 0.0163 0.0068 0.0095 0.0162 0.0087 0.0084 0.0171 (0.0075) (0.0064) (0.0035) (0.0084) (0.0066) (0.0055) (0.0103) (0.0035) (0.0040) (0.0073) (0.0036) (0.0031) (0.0076) no 0.0000 0.0065 0.0031 0.0096 0.0031 0.0022 0.0053 0.0038 0.0066 0.0104 0.0032 0.0037 0.0069 (0.0083) (0.0070) (0.0046) (0.0111) (0.0068) (0.0056) (0.0122) (0.0054) (0.0124) (0.0173) (0.0059) (0.0084) (0.0140) math00: stw 0.0067 0.005 2 0.0024 0.0076 0.0048 0.0034 0.0082 0.0051 0.0086 0.0137 0.0049 0.0055 0.0105 (0.0025) (0.0021) (0.0023) (0.0038) (0.0021) (0.0033) (0.0047) (0.0018) (0.0042) (0.0065) (0.0018) (0.0057) (0.0063) adj 0.0001 0.0001 0.0001 0.0002 0.0012 0. 0009 0.0021 0.0002 0.0001 0.0002 0.0000 0.0001 0.0001 (0.0018) (0.0017) (0.0007) (0.0023) (0.0024) (0.0018) (0.0041) (0.0010) (0.0014) (0.0024) (0.0013) (0.0013) (0.0025) no 0.0066 0.0053 0.0025 0.0078 0.0060 0.0043 0.0103 0.0050 0.0085 0.0134 0.004 9 0.0056 0.0106 (0.0020) (0.0019) (0.0023) (0.0037) (0.0022) (0.0022) (0.0045) (0.0017) (0.0043) (0.0075) (0.0026) (0.0026) (0.0061)
169 Table 5 5. Continued SMNL Model (W_3nn) SMNL Model (W_6nn) SMNL Model (W_8nn) SMNL Model (W_10nn) MNL Model Dire ct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total localxpa00: stw 0.0948 0.0405 0.0194 0.0599 0.0784 0.0519 0.1302 0.0027 0.0069 0.0096 0.0247 0.0287 0.0534 (0.0723) (0.0902) (0.0353) (0.1236) (0.0898) (0.0540) (0.1341) (0.0572) (0.0850) (0.1420) (0.0721) (0.0648) (0.1349) adj 0.0846 0.0176 0.0076 0.0252 0.0538 0.0341 0.0878 0.0082 0.0118 0.0200 0.0059 0.0063 0.0122 (0.0538) (0.0888) (0.0360) (0.1245) (0.0959) (0.0673) (0.1592) (0.0 466) (0.0645) (0.1108) (0.0614) (0.0595) (0.1208) no 0.1794 0.0582 0.0270 0.0852 0.0246 0.0178 0.0424 0.0109 0.0187 0.0296 0.0306 0.0350 0.0656 (0.0549) (0.0561) (0.0198) (0.0696) (0.0581) (0.0355) (0.0922) (0.0465) (0.0673) (0.1132) (0.0549) (0.0428) ( 0.0938) localxpa_ch9000: stw 0.0912 0.0264 0.0145 0.0409 0.0773 0.0518 0.1291 0.0033 0.0007 0.0026 0.0221 0.0274 0.0494 (0.1091) (0.1193) (0.0487) (0.16714) (0.1160) (0.0740) (0.1825) (0.0721) (0.1090) (0.1808) (0.0908) (0.0878) (0.1770) adj 0.1090 0.0491 0.0206 0.0696 0.0426 0.0267 0.0693 0.0225 0.0321 0.0546 0.0174 0.0178 0.0352 (0.0861) (0.1212) (0.0500) (0.1694) (0.1270) (0.0845) (0.2096) (0.0645) (0.0865) (0.1489) (0.0827) (0.0797) (0.1616) no 0.2002 0.0755 0.0350 0.1105 0 .0347 0.0252 0.0598 0.0192 0.0329 0.0521 0.0395 0.0451 0.0846 (0.0764) (0.0673) (0.0304) (0.0886) (0.0693) (0.0461) (0.1127) (0.0566) (0.0819) (0.1365) (0.0649) (0.0580) (0.1164) ptaxrate00: stw 0.0052 0.0037 0.0015 0.0052 0.00 18 0.0011 0.0029 0.0016 0.0022 0.0038 0.0021 0.0020 0.0041 (0.0059) (0.0048) (0.0024) (0.0070) (0.0045) (0.0047) (0.0034) (0.0034) (0.0052) (0.0083) (0.0039) (0.0042) (0.0078) adj 0.0062 0.0037 0.0015 0.0053 0.0019 0.0012 0.0031 0.0015 0.0021 0. 0036 0.0020 0.0019 0.0039 (0.0047) (0.0046) (0.0020) (0.0064) (0.0031) (0.0031) (0.0024) (0.0028) (0.0036) (0.0062) (0.0035) (0.0032) (0.0065) no 0.0010 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0001 0.0001 0.0002 0.0001 0.0001 0.0002 (0.0 045) (0.0034) (0.0016) (0.0050) (0.0075) (0.0076) (0.0058) (0.0026) (0.0045) (0.0071) (0.0029) (0.0033) (0.0062) pctag00: stw 0.0067 0.0072 0.0034 0.0106 0.0072 0.0053 0.0126 0.0072 0.0125 0.0197 0.0063 0.0073 0.0136 (0.0026) (0.0032) (0.0 029) (0.0052) (0.0033) (0.0047) (0.0064) (0.0030) (0.0122) (0.0128) (0.0029) (0.0071) (0.0082) adj 0.0035 0.0019 0.0008 0.0027 0.0040 0.0028 0.0068 0.0014 0.0022 0.0037 0.0017 0.0018 0.0036 (0.0017) (0.0017) (0.0008) (0.0024) (0.0036) (0.0032) (0.0062) (0.0019) (0.0022) (0.0038) (0.0019) (0.0016) (0.0031) no 0.0102 0.0091 0.0042 0.0133 0.0112 0.0081 0.0193 0.0086 0.0148 0.0234 0.0080 0.0092 0.0172 (0.0023) (0.0034) (0.0034) (0.0054) (0.0043) (0.0040) (0.0084) (0.0037) (0.0091) (0.0120) (0 .0034) (0.0046) (0.0078)
170 Table 5 5. Continued SMNL Model (W_3nn) SMNL Model (W_6nn) SMNL Model (W_8nn) SMNL Model (W_10nn) MNL Model Direct Indirect Total Direct Indirect Total Direct Indirect Total Direct Indirect Total pctag_ch9000: stw 0.0071 0.0032 0.0010 0.0043 0.0005 0.0009 0.0015 0.0010 0.0031 0.0041 0.0012 0.0004 0.0016 (0.0088) (0.0076) (0.0035) (0.0110) (0.0069) (0.0050) (0.0119) (0.0063) (0.0116) (0.0178) (0.0068) (0.0074) (0.0143) adj 0.0110 0.0091 0.0037 0.0128 0.0089 0.0059 0.0148 0.0049 0.0071 0.0120 0.0066 0.0065 0.0131 (0.0057) (0.0048) (0.0028) (0.0062) (0.0057) (0.0053) (0.0093) (0.0051) (0.0042) (0.0078) (0.0053) (0.0039) (0.0069) no 0.0039 0.0058 0.0027 0.0085 0.0094 0.0068 0.0162 0.0060 0.010 2 0.0161 0.0054 0.0062 0.0116 (0.0080) (0.0080) (0.0040) (0.0115) (0.0081) (0.0076) (0.0143) (0.0066) (0.0123) (0.0171) (0.0069) (0.0084) (0.0143) cl2valpera00: stw 0.0006 0.0026 0.0009 0.0036 0.0000 0.0001 0.0001 0.0006 0.0007 0.00 14 0.0000 0.0002 0.0002 (0.0024) (0.0036) (0.0016) (0.0051) (0.0033) (0.0021) (0.0054) (0.0021) (0.0030) (0.0051) (0.0026) (0.0026) (0.0052) adj 0.0031 0.0051 0.0021 0.0071 0.0013 0.0009 0.0022 0.0014 0.0019 0.0033 0.0012 0.0012 0.0024 (0.0021) (0.0040) (0.0020) (0.0054) (0.0037) (0.0024) (0.0061) (0.0022) (0.0027) (0.0047) (0.0027) (0.0026) (0.0051) no 0.0037 0.0025 0.0011 0.0036 0.0013 0.0009 0.0022 0.0007 0.0012 0.0020 0.0012 0.0014 0.0026 (0.0015) (0.0016) (0.0010) (0.0022) (0.00 15) (0.0012) (0.0026) (0.0012) (0.0019) (0.0030) (0.0014) (0.0016) (0.0026) Note: Standard errors of the estimated marginal effects presented in parentheses are obtained using the Delta method
171 CHAPTER 6 CONCLUSIONS AND FUTU RE WORK The main contribut ion of this dissertation is methodological. T he linearization estimation methodology of Klier and McMillen (2008) originally applied to a binary logit model is extended to a multinomial setting for various specifications of logit models all of which have different empirical applications. T he performance of this methodology in finite samples is asse ssed in the context of a spatial multinomial logit (SMNL) model using Monte Carlo methods. Simulation results indicate that the linearized model provides a good approximation to the original spatial model and produces fairly accurate estimates for a reasonable range of induced spatial dependence in the simulated data. Judging from these results, the linearization approach appears successful. We further note that t he finite sample properties of other spatial estimators proposed in this dissertation, namely the spatial conditional logit (SCL) and the mixed logit (MXL) estimators, were assessed as well, although the results were not presented here as they were qualita tive similar to the results from the SMNL estimator. However, f urther examination of the performance of the spatial ordered logit (SOL) estimator is needed because of its distinctive ordered structure There are several advantages to the estimation method ology proposed in this dissertation relative to other spatial methods and standard modeling methods. First, unlike the majority of the spatial estimators in the literature, it is designed to model spatially dependent polychotomous choice decisions. Second, it allows a model with a spatially lagged dependent variable to be estimated within a discrete choice framework even with very large samples which is typical in micro level data. Third the Monte Carlo evidence for the SMNL estimator indicates that this estimator performs well in
172 capturing the spatial dependence in the simulated data. Finally, the spatial estim ation method enable s the estimation of indirect effects (in addition to direct effects) providing abundant information about spatial linkages and the strength of potential interaction between economic agents at different locations. This information is of value to inform policy. T he proposed spatial estimation approach is used in t wo empirical studies. The first study involves a spatial analysis of use conversion decisions. The main objective of this study is to understand what drives land use conversion and identify the factors that play a significant role in the conversion of land to urban use. We estimate a spatially explicit mode l of land use conversion in the rural urban fringe using parcel level data from a rural urban fringe county in Ohio, Medina County ; a data scale that is appropriate for modeling the economic decision of the individual landowners. Unlike previous spatially explicit studies, four land use categories are considered in th e analysis: agricultural, residential, industrial, and commercial. Empirical results from this application corroborate previous research findings s uggesting that the location of new urban deve lopment is guided by a preference for lower density areas but in proximity to current urban development. This development trend is sometimes deterred by zoning policies The main insight gained with this application is that spatial dependence is an importa nt factor to take into account when analyzing land use conversion decisions because it helps us understand the underlying mechanisms of land use changes W e find significant evidence of spatial dependence in land use decisions ; consistent with the notion t hat land use change is a spatial process. The presence of spatial spillover effects suggests that local policies designed at a small scale could lead to sub optimal
173 land use pattern s at a regional level Thus, knowledge of the presence and the extent of sp atial spillover effects can help inform the design of land use policy at an appropriate scale. An important area to extend th is analysis is to look at the predict ions of future changes in land use patterns. An initial assessment of the predictive ability of the model can be done by performing categorical in sample predictions Be tter in sample predictions (relative to the non spatial model) are likely to translate to better predictions for future land use changes. The results from the model can a lso be use d with other spatial analysis tools such as Ar c GIS to perform scenario analysis and examine how a change in one of the f actors that influence s land use conversion changes the predicted l and use pattern s The focus of the second study is neighborhood influe nce in the adoption of public policies. It uses school district data to estimate a spatially explicit model of policy u nder consideration is interdistrict open enrollment with the following policy a lternatives; statewide open enrollment, adjacent open enrollment, and no open enrollment. The empirical results substantiate extant descriptive evidence regarding the determinants of adoption of open enrollment policies. Among the most influential factors are minority enrollment racial and income heterogeneity, changes in student teacher ratio, student academic performance, and financial factors More importantly, the results show strong neighborhood influence in policy adoption decisions. The results from this analysis provide further support to the view that public policy adoption is stimulated by interaction among decision makers at a local level. In addition, the adoption of new public policies
174 much like the diffusion of technology innovations is largel y influenced by reference groups. A spatial analysis of these neighborhood effects can help identify the most influential locations Other directions in which this analysis can be extended is to analyze the school adoption decision s not only ov er space but also over time In addition, other ways in which neighbors may influence adoption decisions can be explored since our specification of the spatial weights matrix assumes that the spatial influence is mutually symmetric. Another interesting venue of future work would be to combine the analyses from both empirical studies to examine t he school housing connection. Finally, t he methodology extended within this dissertation is relevant to myriad empirical problems that require micro level analysis and in volve a discrete choice decision i n which relative location matters For instance, it can be employed to model household location decision, firm location decisions, and occupational choice (based on
175 APPENDIX A LOGIT CHOICE PROBABILITIES AND MA RGINAL EFFECTS Derivation of Logit Probabilities Consider the RUM equation s : (A.1) where is individual indirect utility from alternative ; is the systematic component of the indirect utility for individual associated with alternative ; and is t he unobserved component of utility for individual associated with alternative In this set up, individual chooses a particular alternative from the choice set say if and only if : (A.2) The probability that ind ividual choose s alternative is given by : where Therefore (A.3) L et alternative b e the firs t alternative in the choice set and write the probability more explicitly a s : Since the error terms are assumed to be independently d istributed then:
176 where is the PDF of the error term for alternative Moreov er since the error terms are also assumed to be distributed t ype I e xtreme value (or Gumbel), the PDF takes the following functional form: Thus, N ote that Hence, integrating by subst itution: Note : Moreover, as Thus, evaluating the definite integrals: Evaluating the integrals and simplify ing:
177 To simplify the notation, let can be written more compactly as: can be integrated by substitution by noting that thus Substituting it in the probability formula: because as and as Substituting back the expression for : Replacing alternative with the general alternative we get the logit probability formula: (A.4)
178 Marginal Effects and Elasticities M arginal effect s measure the change in the choice probability for a ceteris paribus change in an observed fa ctor 1 L et de note an alternative varying regressor that enter s the systematic component of the utility in (A.1). From (A. 4 ), the probability that individual chooses a particular alternative is given by: Hence, t he change in the probability that decision maker chooses a lternative given a change in an attribute associated with alternative is given by the following marginal effect: (A.5) If the observed factors enter linearly in the systematic component of the utility, then is constant and equal to the model parameter associated with Note that t his marginal effect is the largest when so the effect of a n attribute change is largest when there is a high degree of uncertainty regarding the choice. This observation is important for policy purposes, because it identifies the choices that c ould be mostly influenced b y policy 2 1 This section derives marginal effects associated with changes in alternative varying regressors. The computation of marginal effects for alternative invariant regressors, however, is straightforward and results in similar expressions. 2 See Train (2007) for policy examples.
179 Sometimes it is of interest to look at the change in the probability of choosing alternative as an a ttribute of another alternative, say changes. In this case, the (cross) marginal effect is given by: (A.6) An interesting aspect of the choice probabilities is that, when an attribute of an alternative changes, the changes in the choice probabilities for all alternatives sum to zero (because the choice probabilities sum to one) Differently put, if the choice probability of an alternative with an improved attribute increases, it does so at the expense of the other alternatives. Generalizing the marginal effect expressions in (A. 5 ) and (A. 6 ) for any alternative in the c hoice set and summing the marginal effects, we get: (A.7) Elasticities measure the percentage change in the choice probabilities for a percentage change in the value of a regressor. The ( own ) elastici ty of a change in the
180 probability that decision maker chooses a lternative given a change in an attribute associated with alternative is: (A.8) I f the observed factors linearly enter in the systematic component of the utility, then in (A. 8 ) is also constant and equal to the model parameter associated wi th Similarly, t he ( cross ) elasticity of a change in the choice probability of alternative as an attribute of alternative changes, is: (A.9) Note that the cross elasticity in (A. 9 ) depends only on alternative Thus, a change in an attribute for alternative changes the probabilities for all other alternatives by the same percent age ( IIA property )
181 A PPENDIX B DERIVATION OF GRADIE NTS FOR SPATIAL LOGI T MODELS This appendix contains the derivations of the gradient terms fo r a spatial autoregressive lag (SAL) specification of the multinomial logit (MNL) conditional logit (CL ) mixed logit (MXL), and ordered logit (OL) models 1 The lineari zation methodology proposed in C hapter 3 for the estimation of these spatial discrete choice models is plausible only if the gradient of the spatial autoregressive parameter ) is nonzero once we linearize the model around a convenient point of initial parameter values which sets t he spatial p arameter equal to zero. Thus, i n addition to obtaining the functional form for each of the gradient s we want to show that at the lin earization point, the spatial parameter remains identified from the corresponding gradient For the spatial logit models with unordered choice alternatives, we start the gradient derivations with the mixed model since it combin es both types of co variates ( individual spec ific and alternative specific ) and derive the spatial MNL and spatial C L model s as special cases Spatial Mixed L ogit (SMXL) Model Consider the following SA L specification of a discrete choice model : with, (B.1) where is a latent dependent variable with observable counterpart ; denotes the sp atial weights relating observations and ; and denote alternative invariant and 1 The difference between the polychotomous models with unordered choice altern atives is as follows. The MNL model accommodates only explanatory variables that vary over observations (individuals) but not over alternatives. T he CL model accommodates only alternative varying explanatory variables. The MXL combines the two preceding mo dels. The terminology for the mixed logit model is not to be confused with the random parameters logit model (RPL) also often referred to as a mixed logit model.
182 alternative varying covariates respectively ; is a vector of IID disturbances for alternative ; is the spatial autoregressive parameter; and and a re the model parameters The alternative with the highest latent utility is the one chosen by the decision maker, which is the choice observed in The model can be written in a reduced form as: with, (B.2) where are the elements of the spatial matrix The reduced model can be written in matrix notation as: (B.3) which results in the error covariance matrix: This covariance structure implies that the error terms are both autocorrelated and heterosked astic Denote by the variance of the error terms given by the diagonal elements of The model can be normalized for heteroskedastic errors as follows: with (B.4) Define the dependent variable in a binary form as : The model in (B.4) implies the following probability of choosing alternative ( by individual ) :
183 Hence, (B.5) where ; , ; and If the error terms are assumed to be distributed IID type I extreme value, the error difference s will possess a logistic distribution. These distributional assumptions, give rise to the SMXL model and the choice probabilities take the following functional form: (B.6) Having speci fied the cho i ce probabilities, the model gradients can be derived by differentiating these probabilities with respect to the model parameters Gradient of : For a general alternative :
184 Let if and zero otherwise. C ombining the resu lts, the gradient term for is given by: (B.7) Gradient of : Hence, (B.8) Gradient of :
185 In order to obtain the partial derivatives , and recall that and Hence, (B.9) where Similarly,
186 (B.10) where To complete the derivation, we need to compute t he partial derivatives and in (B.9) and (B.10). Note that , and which can be written in terms of the error variances as follows: and Hence Similarly, The last partial derivative , can be obtained by differentiating the covariance matrix with respect to and extracting the diagonal terms. Thus,
187 because and are symmetric hence equal to their transpose. Thus (B.11) where Consequently, (B.12) Combining the results in (B. 9 ), (B. 10 ), and (B.1 2 ): (B.13) Substituting (B.13) into the gradient expres sion, t he gradient of becomes: (B.14) where , and
188 At the linearization point ( , and where by construction thus the terms containing in (B.14) vanish. However, the gradient of does not vanish because of the term s and which become and respectively. Thus, this model can be linearized around a point of parameter values which sets because the parameter remains identified from the corresponding gradient. At the linearization point, the gradient s o f the linearized SMXL model simplif y to: (B.15) Spatial Multinomial Logit (SMNL) Model A SAL specification for the MNL model can be obtained by setting in (B.1) as follows: with, (B.16) w here is a latent d ependent variable with observable counterpart ; denotes covariate s that are alternative invariant; denotes the spatial weight s for observations and ; and are the model parameters ; and is a vector of IID disturbances for alternative T hen, from (B. 5 ) t he probability that individual chooses alternative is given by :
189 (B.17) Assuming that and are distributed IID type I extreme value, the choice probabili ties for the SMNL model take the functional form : (B.18) T he gradients with respect to the model parameters ( ) for this model are obtained as follows Gradient of : For alternative Define if and zero otherwise. Then the gradient term for is given by: (B.19) Gradient of :
190 From (B.1 3 ): Hence, the gradient of is: (B.20) where and At the linearization point ( and thus the terms containing in (B.20) vanish. The gradient of and thus, the spatial parameter remain identified because of the term which reduces to Thus, this model can also be linearized around a point of parameter values which sets The gradients for t he linearized SMNL model are : (B.21) Spatial Conditional Logit (SCL) Model A SAL specification for the CL model can be obtained by setting in (B.1) as follows: with, (B.22)
191 w here is a latent d ependent variable with observable counterpart ; denotes covariate s that are alternative varying; denotes the spatial weights for observations and ; is a vector of IID disturbances for alternative ; and and are the model parameters. In this setting, t he probability that individual makes choice is given by: (B.23) These choice pr obabilities result in the following gradients for the model parameters Gradient of : Hence, (B.24) G radient of :
192 From (B.1 3 ): and Hence, the gradient term for is: (B.25) where and At the linearization point ( and thus the terms containing in (B.25) vanish. However, the gradient of does not vanish because of the terms which reduce to Thus, this model can also be linearized around a point of parameter values which sets because the parameter remains identified from the corresponding gradient. At the linearization point, the gradients for the SCL model simplify to : (B.26) S patial Ordered Logit (SOL) Model C onsider a SAL s pecification of an ordered discrete choice model : with, (B.27)
193 where denotes a latent d ependent variable with observable counterpart ; denotes a matrix of explanatory variable s ; denotes the spatial weight s relating observations and ; is the spatial autoregressive parameter ; is a vector of model parameters ; and is a vector of disturbances T he natural ordering of the alternatives is a result of the latent variable falling into various mutually exclusive and exhaustive ranges given by the threshold parameters and with and The dependent variable can be defined in a binary form as: The reduced form of the model is given by : with, (B.28) where are the elements of the spatial matrix The reduced model can be written in matrix notation as follows: (B.29) with resulting covariance matrix: This covariance structure also implies autocorrelated and heteroskedastic disturbances. Denote by t he variances of the errors given by the diag onal elements of the covariance matrix T he model can be normalized for heteroskedastic variances as follows: with, (B.30) T he choice probabilities for this model are given by: (B.31)
194 where is the CDF of , ; ; and to make the scaling of the threshold parameters more explicit. A logistic distribution for the error s gives rise to the spatial ordered logit (SOL) model with the following choice probabilities: (B.32) The gradient s for this model can be obtained by differentiating these choice probabilities with respect to the model parameters ). Gradient of : Hence, (B.33) Gradient of :
195 Thus, (B.34) Gradient of : Hence, (B.35) Gradient of :
196 From (B.1 3 ) : In addition, n ote that which can be written in terms of the error variances as follows: Hence, From ( B.1 1 ) : thus The gradient of is given by: (B.36) where and At the linearization point ( , and thus the terms containing in (B.36) vanish. The gradient of in (B.36) is nonzero when because of the term which becomes thus this model can be linearized as well around Th e gradient terms for the linearized SOL model are:
198 APPENDIX C DESCRIPTION OF MONTE CARLO EXPERIMENTS Scenario 1: SMNL Model with Equal C hoice Probabilities In this scenario, we consider a model with four choice alternatives each with the same choice probability ( To generate the data for this model we s tart with the reduced form of the model written in matrix notation as: (C.1) with covarianc e matrix We generate a single explanatory variable uniformly distributed in the interval Next, we specify t he spatial weights matrix as a row standardized first order contiguity matrix, which sets the non zero elements to: with endpoints and set a value of between and varying it in increments of Having specified and a value for t he explanatory variable is first transformed to obtain as where is the square root of the error variances given by the diagonal of A second transformation follows to obtain as For simplicity, each of the parameters is set equal to Equal parameter values for the different choice alternatives and a fixed (thus, a fixed ) ensure equal choice probabilities. Subsequently, the simulated probabilities are obtained as: (C.2) To generate the observed individual choices based on the simulated probabilities, we generate a uniform random variable ( ) and set for if : w ith (C.3)
199 Scenario 2: S MNL Model with Different C hoice P robab ilities In this scenario, we still consider four choice alternatives, but now each alternative has a different choice probability. We are interested in alternatives with the following choice prob abilities: To generate this data we s tart with the reduced form of the model in (C.1) and generate a single explanatory variable uniformly distributed in the interval Next, we set the desired choice probabilities and solve for the parameters that correspond to those probabilities. Let be the choice alternatives and be implicit. Then, the SMNL model can be written as: (C.4) Given the model in (C. 4 ), the choice probabilities are given by: (C.5) With a fixed we can fix by solving for Starting out with : By symmetry,
200 Normalizing : (C.6) (C.7) (C.8) The solution to , should also satisfy: (C.9) Substituting (C.6) into (C.7) : (C.10) Similarly, substituting (C.6) into (C.8):
201 (C.11) To simplify the notation, let: Then, (C.12) (C.13) Substituting (C.12) into (C.13) and solving for : (C.14) Thus
202 (C.15) Plug ging ( C.14 ) into ( C.12) and solving for : (C.16) Hence (C.17) Finally, p lug ging (C.14) and (C.16) into (C.6) and so lving for : (C.18) Thus (C.19) It becomes evident from (C.15), (C.17), and (C.19) that the parameters are now a function of this is the reason for changing the interval of to ensure a mean of 1 rather than 0 and as a result a function of hence denoted by ( As in the first scenario, we set the value of between and varying it in increments of and specify as a first order contiguity matrix. T he explanatory variable is t ransformed to obtain and as described above and the observed individual choices are assigned based on the simulated probabilities given by (C.2) using the assignment rule in (C.3).
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222 BIOGRAPHICAL SKETCH Ledia Guci was born in Korca, Albania. She received an Associate of Arts degree in a gribusiness m anagement from Dimitris Perrotis College of Agricultural Studies, Thessalon iki, Greece, and a Bachelor of Science degree in a gricultural b usiness from University of Arkansas in 2005. She joined the Food and Resource Economics Department at the University of Florida in the fall of 2006 to pursue graduate studies in a gricultural e c onomics with a concentration in i nternational t rade. After completing the coursework for the first year of the m and while working as a research assistant for the Florida Department of Citrus, she was offered funding to start the doctoral pr ogram in the same department She was awarded a Master of Science degree in 2008 and a Doctor of Philosophy degree in August of 2011. Her research interests are in the areas of applied econometrics, spatial econometrics, natural resource and environmental economics, and urban and regional economics.