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## Material Information- Title:
- Estimation of a housing demand model with interdependent choices about owning or renting /
- Creator:
- Trost, Robert Patrick, 1946-
- Publication Date:
- 1977
- Copyright Date:
- 1977
- Language:
- English
- Physical Description:
- viii, 74 leaves : ; 28 cm.
## Subjects- Subjects / Keywords:
- Analytical estimating ( jstor )
Consistent estimators ( jstor ) Estimation methods ( jstor ) Housing ( jstor ) Housing demand ( jstor ) Income estimates ( jstor ) Mathematical variables ( jstor ) Maximum likelihood estimations ( jstor ) Modeling ( jstor ) Price elasticity ( jstor ) Dissertations, Academic -- Economics -- UF Economics thesis Ph. D Home ownership -- Mathematical models ( lcsh ) House buying -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 72-73.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Robert P. Trost.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000185179 ( AlephBibNum )
03330186 ( OCLC ) AAV1761 ( NOTIS )
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[; T T il..'r t: O lin.', !i I i : I LD rI Dl :L i.1 ii lii.Lf PGDEfI liLif C-iOIC' S ,P".SOUT OUllIJG OR Lro !TIiG Pobert P. Trost A DISSCRT/,TIO[tJ PF''SFrJTLD TO TIIE GRADUATE COUNCIL OF TI! U;!I\L.ESIT'I OF FLOr'iDA Ill PARTIAL. FULFI LL t-1JT OF THE PEQ,.U 1 REILTS FOP THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OFT FLORIDA 1.977 UNIVERSITY OF FLORIDA 3li 1i262 08552 5276 11111I 3 1262 08552 5276 To my parents AC FI P-NILEDGE1 I El iTS The author '.'ishes to thank his Chairman, Professor G. S. Maddala, for his guidance throughout the course of this invest igation. Special thanks also is given to Professor P.. B. Roberts for his help and friendship. The author also .*:ishes to express his sincere apprccia- tion to Professors R. D. Emerson, J. W. tilliman, and D. G. Tay.lor for their contributions to his education at tie Unl"ersity of Florida. Appreciation is also extended to Hs. Candy Caputo, v-.lo demonstrated not only expertise as a typist, but infinite patience and cooperation. Financial support from the National Science Founda- t-ion under cirant 5OC-76-04356 to the University of Florida is gratefully acknow..ledged. Finally, the author's parents ha-.e been an important influence on his life, and they are deeply thanked for their love and urnderstandinci. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . . . . . . . . iii LIST OF TABLES . . . . . . . ... vi ABSTRACT . .. . . . . . . . vii CHAPTER I INTRODUCTION .. . . . . 1 CHAPTER II A CRITIQUE OF PREVIOUS STUDIES . 3 CHAPTER III A MODEL OF INTERDEPENDENT CHOICES . . . . . . . . . 7 1. An Interdependent Choice Model Based on Utility Maximization . . . . 7 2. SpcciEcatiiocn of the Model . . . 19 CHAPTER I'.' TIO0 STAGE ESTIMATION . . . 24 1. Estimation Problem? . . . . . 24 2. Discussion of the Results . . . 30 CHAPTER "' A::iA IflUlM LIKELIHOOD ESTItlATION . 39 Int1rod auction . . . . . . . 39 1. Amremicya 's T'.-.'o Step Ilaximum Likelihood Procedure . . . . . . . 40 2. A Comlpi. iion of OLS and Ila,: iIum Li l:e1 lihood Ltimates and a Test for Simultaneity . . . . . 16 CI: ',PTE' R VI CO CLUSI S . . . . . . 56 -PPE'i' JDI.X 1 DISCUSSIONJ, OF THE DATA . . . 60 APPENrDI:: 2 E:iTEIJSIOi OF THE MODEL TO THE 11ULTI-PERIOD CASE . . . . . . . 65 TABLE OF C01TlSEN'['TS CO1TIIUED Pa e PEFEPENCES . BI OGPJRPHl TCA L S ETCH . . 74 LIST OF TABLES CHAPTER IV TABLE 1 Probit Coefficients for Owners TABLE 2 Comparison of Two Stage and OLS Estimates for Renters . . . TABLE 3 Comparison of Two Stage and OLS Estimates for Owners ... . .. CHAPTER V TABLE 1 Probit Estimates of the Decision Function . . . . . . TABLE 2 '..o Step fla:-:imum Lifelihcood sti- mntes of the Decision Function . TABLE 3 - iiCuising E:-penditut-e Equation of the O -.ner . . . . . . TABLE 4 lio.iusinr.g E::end iture Equation of th-e P enters . . TABLE 5 - TABLE 6( - Iousin q Ex:penditLure Equation of the OwC'.n eer . . . . . . i ising L:: end ture Equation cf. the F]entcrs . . . . . .PF-ENDI:T 1 TABLE t1 Comrnpar Lson of Cetnsus and .F'C Data: To-al Famil, Income by Pace, for all rain i e es . . . . . . T,.LLF 2 Coipar-a1so of cenri us. and S:.' Data: race b- '.'ar io',s D remiogq graphic '.'ari- ables, fo. r Fariliesv '.-'ith T'..'o -r lore tlembers . . . . . . Page 31 S 47 Abstract of Dissertation Prosented to the Graduate. Council of the Unive.rsity of Florida in Partial l'ulfillment of the Pequirements [or the Degree of Doctor of Philosophy ESTE"] ATIOIl OF A, HOUSING[ DlEfl.'D riODEL 1WITH 1 IITERDEPCFrDIT.IJT CHOICES /,BOUT OI'[IIJG OR RPE[TING F.'y Pobert P. Trost flarch, 1977 Cha i rmn : G. S. a addala Major Department: Economics The first limited dependent variable model was proposed by Tobin. His approach is now known as Tobit analysis. Tobit analysis uses a maximiumn likelihood procedure to esti- mate models with a truncated -- that is, a limited -- depend- ent variable ".'hen a large number of observations take on the truncating threshold. Tobin's model has recently been ex- tended to handle more complicated situations. One example of- these e::tensions is a switching regression model where the s'..witchinc function is endogeneous to the model. In this thesis the switching regression model is used to study hous- ing demand equations. The thesis argues that previous studies on housing only emphasize one part of a complete model. Some of these past studies only consider demand equations for owners and renters, vi' while others only consider the consumer's rent versus own decision. The thesis proposes and estimates a complete model which allows for the simultaneous determination of whether or not to own and how much to spend. In this model estimation of the housing demand equations by OLS will be biased if there is simultaneity between these equations and the switching function. To avoid this bias the parameters are estimated by a two step technique and a two step maximum likelihood procedure. Both these pro- cedures are asymptotically consistent and the two step procedure also is asymptotically efficient. In order to test for simultaneity the demand equations also are esti- mated by ordinary least squares (OLS). A maximum likeli- hood ratio test then is used to compare the two step and OLS estimates. The results of this test show that simul- taneity is present. Therefore, the thesis concludes that 'OL cstimrrates of the demand ecquations are bi.aed .nd the ti..'o stage and tL'...c step prLocedures are more appropriate. CHAPTER I INTRODUCTION I lany models of housing demand have been presented in the literature. The earlier models ..ere estimated with aggregate tjme series or cross-sectional data. For example, see Muth (60), Reid (62) or Winger (68). More recent studies such as Carliner (73) and Fenton (74) used micro data to estimate demand for housing equations.- In general all these past studies use a least squares technique and estimate t..'o housing demand equations -- one for owners and another for renters. One purpose of this thesis is to sho'.. that under certain conditions these least squares estimates will be biased. This bias becomes apparent once the owner and renter equations are v.iew.ed as limited dependent .ariables that are determined simultaneously ..ith a rent v.ersus ow..'n choice function. It is this simultaneity which causes the OLS estimates to be biased. Therefore, other unbiased estima- tion procedures have to be used. One such method is a "t.wo stage" technique proposed by Lee (76). Another is the t..wo step maximum likelihood procedure described in Amemiya (74). Both procedures are used in this thesis and the results are compared to the usual OLS approach. I limited dependent .'ariable is a dependent variable that has a lower or upper limit, and takes on the limiting value for a substantial number of respondents. Models of this type have attracted great attention recently. For examples, see Cragg (71), Maddala and Nelson (74) and Heckman (74). A second purpose of thi: thesis is to derive a model of housing demand with interdependent choices about owning or renting. To .accomplish this task housing is viewed as a bundle of several goods such as location in a given area, type of structure, size'and quality of neighbors. In general any given bundle will have one price if it is owned and another price if rented. Following this line of thought, some families will be better off renting while others will prefer to buy their own home. The model derived in this thesis assumes that this rent versus own choice function and the expenditure equations depend on certain independent variables as well as stochastic disturbances that are not necessarily iidepen-dent. Thc-se '.ariables include a measure of permanent income; personal characteristics of the family head such as race a r and se:: family background 'ariables such as mobility and family size: various price indices; and regional variables to capture the effect of varying land val'ues.in different parts of the U.S. In Chapter II past housinQ studies are critiqued. Chapter III presents the formal. model. In Chapter IV con- sistent estimates of the rent and Cown expenditure equations are ob:tainedr by the t'.-o stage method. The choice function is estimatedl by Frobit analysis. Chapter '.' estimates the m,.,iiIel '.-.ith .a tl.'o step maximum likelihood procedure and tests [or simultancitY '.-.'ith a maximum likelihood ratio test. Cliapi-tr ''I coniailns the conclusions. Finally, the appendices contain a discussion of the data and an extension of the model tc the multi-period case. CHAPTER II A CRITIQUE OF PREVIOUS STUDIES In the past fifteen years numerous studies of the house- hold's demand and choice of housing have been presented in the literature. The earlier papers studied macro data such as aggregate U.S. time series data or the average income and housing expenses of given SMSA's and cities. The main con- cern of these studies was to obtain estimates of the income elasticity demand for housing. In general these elasticity estimates '.'ere greater than one. For example, see Muth (60), Reid (62) or Winger (68). luth's (60j study uses time series data and, in a separate analysis, he also uses cross-sectional data. He concludes that the income elasticity for desired housing stock is at least 1.0. In a review. article, de Leeiu (71) makes adjustments in the Muth (60' reid (62).and Winger (68) estimates. lie concludes that the income elasticity of rental housing in the United States is between .8 and 1.0. A second conclusion is that the income elasticity for ow'.ner- occupied housing is probably higher than that for rental housing. llowever, even de Leeu'.:'s results are quite high '..hen compared to the estimates obtained with micro data. For example, using micro data Carliner's (73; estimates .were ]These "adjustments" stem from the fact that past studies did not use the same data base or the same definition of variables when estimating the elasticities. For example, some did not include imputed rent in the definition of income for homeowners. 3 .631 for owners and .520 for renters. Also using microdata, Fenton (74) estimated income elasticities of .41, .46 and .46 for all renter households, and high and low income renter households, respectively. Polinsky (75) concludes that these past macro results are biased upward and the micro results biased downward. 2 He then states that the true income elas- ticity is around .75 and the true price elasticity is approx- imately -.75. While Polinsky's criticisms deal with the specification of the price variable, the critique in this clha ipter is i me1d it: the technique used to estimate these pist model]p. Since i typical T-i-mple is comprised of families '.:ho either o'.:.n their home or rent, these models cin be divided into or e oF thi -re .roups. ;, nai -e approach is to di-ide-: the simple data into o,-..ner s .nnmd rente-rs and estimate t each subg rroup sep-r tely. Thi-- method rei'-iults in t'.:o unique e~:pe.-' di t:t e equations:3 oniE for homeo'.:ners, estimated withoutu t .usin the information in the renter subsample: e ncother for renters, estimated ..itliout ,using the in form tion cont-in ed in the o.-'ner sub- sa-mple. For e:-:amples see Lee (68), Oh]s (C-), C Irliner (73) or Fenton (741 . l'o l 1. insi ;y L'.-:u s tI-iC t Lhoe omission or misspecif ication of the price term Lbas'.? the u.1n rol ups- (i.e. micro) estimates do'.-.'nwa.rd and: the rocups (i.e. micrc) estimates up'.:ard. P' !tlher thl-n cli'.'ie the i-riiple into o'..-ners *and renterFs one c-n assume 1that all or 1,part of the parameters in the t'.:o expendi- tuLr eCquations are the sime for o..wn.ers and renters. A single equationb can then be estimated] from the entire s-mple. Dunmmy *-rir .bles are uscd on the subset of para meters assumed differ- zent for o..:wners -jnd renters. The attractiveness of tie above approach is that an ordinary least squares estimation technique (OLS) can be used. A second alternative, the Tobit model, requires maximum likelihood estimation. This method, first des- cribed Iby Tobin (58), results in two unique demand equa- tions. It differs from the first approach in that each equation is estimated using all the sample information. This is done by assigning a zero value to the renters ownersr) w-hen estimating the demand equation for owners (renters). In his paper Tobin examines buyers and non- buyers of a durable good in any-given year. He shows that OLS on the subset of buyers is both biased and in- efficient. He then suggests a maximum likelihood techni- que (Tobit) that is both unbiased and efficient. In these. first t'.o approaches either housing demand or e::penditure equations are estimated. The dependent variable is either the homogeneous commodity called "housing services" discussed in Olsen (69) or various Lancastrian "housing characteristics" described in King (76). A third approach is to only model the choice of whether to buy or rent. This can be accomplished with a linear pro- bability model, a Probit model or a Logit model. With this method one does not obtain demand equations but rather an equation to predict the probability that a given family will own their ow.n home. Examples of the Probit model can be found in Ohls (71) and Poirier (74). Quiqley (76), in a similar analysis, uses the Logit model to estimate the probability that a given family will choose among 18 types of residential housing. The coefficients obtained by these models are estimates of the true coefficients in the choice equation divided by a common scale factor. The coeffici- ents can only be estimated up to a scale factor because the dependent variable is never observed and therefore the variance of the disturbance term cannot be estimated. For example, if the dependent variable is the utility obtain- a-ble '.when a family o'w.ns less the utility obtainable w..hen thit s-me family' rents, this difference of utility is never obscr',,,d. Rat her, ..'e only obse'rv.e a one (o'.'n.. inqg) when this difference is positive and a zero (renting) %'..'hen this differ- ence is negat'.'ie. The Probit and Log it modIels use a ma: likelihood technique to estimate the cor fficients. These crefficients are then used to assign each family '..'ith a numb-er bet.-'een ;ero and one. This number can be interpreted as the probab ]lit tllh t a gi'.'en family '.-.'1ll o..'n their own home. For Probit Analysis this probability is derived by first multiplying the independent variables by their respecti'.e coefficients. The scalar obtained from this multiplication is a standard normal '.ariate. The probability of owning is simply the Standlard Normal Cumulati'.e function evaluated at this scalic. While numerous e:-:amples of the abo'.'e alternatives can be cited in the literature, no one has studied models where the decision on how much to spend is determined simultaneously '.*-:ih the rent-o.'-:n decision. An approach designed to fill this gap in the literature is deriv'.ed in the following chapter. CHAPTER III A MODCL OF IiTEIRDEPEIiDEINT CHOICES 1. An Interdeplendent Choice Model Based on Utility Mlax: imiza t.Lon The major focal point of this section is to sort out the factors thnt determine '.hy some families ow..n their ow.n home and others rent. While many specific reasons--such as investment criteria or simple Tobit analysis --for buying i home can be i-.'en, the choice mechanism presented here is based on the simple economic principles of utility analysis. The hope is to h-.'e a general and '.-idely applicable model. The model will ':ie.-.: the decision to procure shelter in an imperfect world. First, the choice of a family to rent or buy in a particular neighborhood is limited to an "either- or" situation. In w'.ords, a family must choose bet..,een a number of discrete alternatives available. That is, given price and nonprice constraints some alternatives may not be feasible; for example, renting an apartment '..:ith a large q rden. Second, individual families face a number of con- straints such as budget and time constraints. Financial IIn Tobin's; (58) pionocrin i piper oil limited dependent vari- ables he presents an estiim-tion technique for models .w.hen the dependent '.',riable often takes on a lower or upper limit. His method is now know-.'n as Tobit analysis and has many appli- cations in economTics. For e:-:ample, 3 family's e::penditure on a major durable such as house may be zero until the house- hold income exceeds a certain level. 7 institutions constrain families to loans of limited size by requiring, for example, that housing expenditures not exceed 20-25 percent of income. Third, families purchase both durables and nondurables, consuming durables over time. Fourth, housing is not a homogeneous commodity. Rather, what we call housing is really a bundle of several goods: location, house size, yard size, type of structure, central air-conditioning and'even type of neighbors. To get around this nonhomogeneity problem Olsen (69) defines a homogeneous commodity which he calls'"housing services." Olsen argues that whenever the housing market is out of equilib: rium t:he e 1:istinq housing stock .'ouldl filter (i.e. and increase or decrease in the qluantity' of housing serv.icosi up or do.-.:n and net.' dwelling units '.*.ould be con- structed L'isequillbr ium is said to e:.:ist whenever r price is greater tlhan long run average cost. This filtering and construction continues until there 3re no profits to be rad3e on bundles of housing of -an size. This zero profit equilibrium ,r's iti-on requi rre the price per unit of housingq services for bundles of all sizes to: be the same. Follo'..'- ing this line o:f thou iht, if t..'o families in an equilibrium market speijr- different amounts on' housing, the implication is that both families are colnsumiing different quantities 'f the same good--" lousincT setr'.ices." Tl is quantity com- parison is possible because of tw..o crucial assumptions. First, it assumes that the equilibrium price per unit of "Ihousing services" has been reached. Second, it assumes that families do not have preferences for particular types of "hoIusi n servicCes" such a li'.'ing in do.wn to'...'n few' York City versus living in flew Y'cI' I: City suburbs. A Lancastrian approach to the nonhomogeneity problem is taken b:y j:inq (76'). He 'postulates that housing is really made up of several hon'ogeneous characteristics. An individual has separate demands for each of these characteristics but not a demand for*housing per se. SIn order toc distinguish bet'.-:een the demand for renting-- ..which .was primarily the demand for ap--irtments in 1971--and the demand for hou]Ises, a slightly diff rent ,approach to the ronhomrrogcnei Lt issue is taken here. For simplicity this approach is presented as a one period maximization problem. Ho..-e'ver, the model is general enough to be easily extended to the nmulti-period case. For example, one could assume the fainily mr.xiimi zes ;:pected utility summed over the entire occupancy of a dwelling. A multi-period model taking 'this approach is presented in the Appendix. Assumi'e for the moment that housing merely amounts to the size of the dw..:elling. The larger the size, the greater is the quantity of Ihousinq consumed. In this simple case an owned home and a rented apartment are perfect substitutes. Indeed, they are the same commodity and a family's decision to o'.-.'n or renLt '.-:i]1 l e based on price alone. From consump- tion theory w..:e kno'..' that in equilibrium the family will equate the -r:atjos of marginal utility to the ratios of marginal cost for all qoods. That is, if the family owns IlUm (( 0) fIU (I) o will hold, where MU is marginal utility, MCo is a marginal cost function for owning, H is the size of the owned home, X represents all other goods and P is the price of all ^x other goods. Similarly, if-the family rents H units of housing SICr (H) MU(II) Cr( MU(X) P x will hold, where MC is a marginal cost function for renting. r If MC and MCr are different functions', then given prices, preferences and income, the family will rent or own depend- ing upon which is cheaper.2 For example, consider the partial equilibrium ain.-alysis dej:,icted in Figuire 1. In this case '.-'e wo-uld observe family 1 renting H1 and family 2 owning H2. 11- igenera 1 the shape of these marr,:ainal cost functions would de- pend -on the location cf the structure. A family in t[ew York City with higi h land values s would therefore face a different mi:rgiqnal cos L pattern than a family in rural Alabama with relatively, io.- land '.alues. ,!liil, in qgneral the choice between t'o perfect substitutess result in an cithli-er'-or situation there is one ec:-eption. If the ma.rgninal 1-ite of sul:.stitution and: the p.:ice ratio ha'.'e iden t ica slopes, then the consumer is indifferent between ani con: bina t ion of the t,.-.o commodities. Unless the individual could rent Lhhrce troo'ms of a five room house and o..-.n the other tw.:o, I-1iis '-'LouIl not be the case foi housing. For most con- sumiersG the implicit time cost of consuming t ,.-.io separated hous- int units constrains them to an cithcr-ori choice. Of course thrre may I:b situations hereee a familIy would simultaneouslly o( -. ii reGnt. C'n o.al'e is thle businessman \- o o-ns a house in tl!e suburLs but also rents an aparttrient near w'.or For him thl opportunity cost of commuting ever,' day outw.'eiqhs the cost of renting an apartment. lo-.' consider housing--or if you like, housing services-- as? being made up of several commodities. Let this set H be represenEted by a vector of n different housing service com- modities such that II = [hi i=l . n, .-here h. is one element of the sectorr 1. lie:-:t, let the set of prices of these n cominodities be sector r functions of h.. Call this set P for renting and 0 for o.-'ning such that P(11) = (h ], i = 1 . n 0D(1) = [o (h.)], i = 1 . n R' (il) = (r (h i = 1 . n i i (11) = [o '(h .)], i 1 . n where P' (ll) and O' (H) are sectorss of marginal cost functions. At any point in time a family can consume all the h.'s or only a subset of them. Utility theory states that in equili- brium the family '-.'ill equate the ratios of marginal utility to marginal cost for all goods. Utility theory also tells us that each family :.ill ha''e demand equations for all the hi goods. As shown.n in the partial equilibrium diagrams of Figure 1, wvhethcr they o:.'n or rent these qoods '.-'ill depend on the marginal cost and marginal utility functions. But at least four problems arise when it comes to estimating these separate demand functions. First, a family generally does not o\-.'n housing commodity h. and rent housing commodity i. For e:.aminlc it v.'ould be very difficult cost w-ise for a family to rent a lot with a nice location and then build a dre'aml hlcuse :.'hichl they plan on taking with them when they move. Second, most or all of the h. 's are either unobserv- able or difficult to measure. Indeed, many of the hi 's MC (H)' R AC (H) / R / / MC (H) -I / I / / - I rIi (U it -II House Size (lI) Fiu.-jre 1.PaLr al equilibrium housing demand 4 may be subjective qualities in ique to e.'ery family. Third, because housing for any family may only. contain a subset of the h.'s, ..e have a limited dependent variable problem. That is, for any h. we willconly observe positive quanti- ties bcing consumed by some of the: families. For the rest of the families w.e ..ill observe some limited quantity being consumed. In general this limited quantity is zero. Fourth, the price functions o. (h. a'- d r. (h a are hard if not im- 1 1 1 1 possible to observe. Ideally of cours.:e w.,e should estimate demand equations foi: all the h. 's, but qiv'en the four problems just discussed, other approaches must be substituted. One method is to reduce housing to a fewr. reasonable and measurable Lancas- train characteristics and estimate demand equations for each of these char icter istics. Tiis is the approach taken by i.ing (76). Another alterna tive is to simply define a homo- geneous good called "housinci services." As discussed in Olsen (69. the problem no'..' reduces to the estimation of a single demand equation. Uhile both of these approaches have merit, they give little insight into the rent-o'.'n decision. The analysis to follow combines these tw'o ap- :)roaches. The result is a model where the expenditure decision is determined simultaneously '..'ith the rent-ov.n decision. Kecpingj in mind the fact that families face time, financial, and budget constraints, the model can be stated as follow.s. The family can choose to purchase or rent some amount of the homogeneous housing commodity H.*. Here 11.* is simply one of the many possible subsets of H such that 11.* = F.(I) j=l . m. Here F.(-) is some function. More precisely, ;.* should be viewed as a homo- geneous subset of H with the remaining m-j commodities held constant at some level.3 This fixed level can be zero or any positive number. The idea here is to break housing up into several types much like one'can break fruit up into apples, oranges, and pears. For example, you often hear a family say they are looking for a "big house.in the city," or "a ranch house in the suburbs," or "a split level just outside the cit, '.-.ith a pool," or "a two'-. bedroom apartment near '.-.ork," or perhaps "just a one bedroom unit near a shopping center." .After pricing these specific types of rhousi ri the fla- il rlI.:i\' or may eo t settle on their first choice. They will of course settle for some unique com- bin action of the set H. The simplifying argument made in thi' [ -1.'p" r is that '.-:E can .'ie'.. this decision. as a choice oL one t,'ye of housing from a set of several la 1ternati'.'es. The per unit cost of buying H is represented by O.*H. ) suchI that C0 (I i ) f Cf OlH) = 1 . m, '..lh re f, is soni' function. Similarly, the per unit cost of rental. I. 1 Ls hi'.'en bL such that i . I4ll ) = f [ (II)] i = 1 m , ...'-herc f is some function. If '..e ignore for the moment "If I i = II, then this model reduces to the one discussed in C'lIen (69). the possible ta:-: reductions From o'.-ning, then if -the family were to purchase housing the maximum utility obtained would be derived from (i1) la:-: (4 *ix, H ) Subject to [" X-, H. ) 1 = 1 . m, .'where : is a bundle of all other goods including sa'.'ing, F f (-, HI. ) is the "ow'. inci consumption possibility frontier." In this simple case Fr (:, 11. ) -is ni''en b-: F :' r ;, ) = '-P( ) -0. (H )H , 1.I 1 k H:; -H j . v.'here 0 ((ll includes ani' financial payments or opport- unity costs and IH only includes that part of the dur- able qood bought for consumption purposes. That part of 11 + purchased as an investment is included in X. Y is income in period t. The first order conditions to (1) are U Fo (2) i F J 3 .here U Fi a nd F 1 are partial deri'vati"es. rl, U F. and 1r are nothing more than the marginal 1 1 utility of X, the marginal utility of H.*, the marginal cost of X and the marginal cost of buying 1i respectively. When' (2) liolds let the utility obtained be represented by U (i ,*), where 1 = 1 . m. Assume that when ow.-ned the Kth subset of H produces the highest le'.'el of utility. Let this utility level be represented by .U o* (11 ). A similar anial.'sis holds for renting. If the family rents the subset II. the maximum utility would be derived froi (3) (3) Max U(X, Hi*) Subject to: F (X, H.*), where F (X, II.*) is the "renting consumption possibility frontier" and is given by FR(X, H.*) = Y-P(X)X-R.*(H *)II*. j ( J J J The first order conditions to'(3) are U FR x X (4) F H .* F ] .* RJ where F is the marginal cost of renting Hj*. When (4) holds let the utility obtained by represented, by TF (nI *) \where j = m. Assume that the Lth subs.'Eh cf H pro- duces che highe.-st le'.eel of utility by renting. Let this Ltilit'y le'.'l. be represented by I-I* ( ) Figure 2 sh'..o s R LL Ht e eC]qu 1 iibr' i ,m po, c :ition ii'.'enll by" b I H *) and -*0H ). The f mily' '-'ill buy, tlhe su:,set of housing ser'.'ices H * if lU li(H.* is larger than UI ) Con.ersel the family '...'ll r nL the subset cf housing ser'.'ices 1 L' if U l(HL i greater than LU Itl *I ) If one includes the ta:.: ad'.'antaqes from owning, then the "ou'ninli consumption p.ss ibilitt frontier" constraint reduces to r" IX, [ *) = C'-rM.1) il -t)-PI :-:) .'-O .*(H .*)ll.*, ] j .i j .-.'here r is the interest rate orc the mortqgae, I1 is the amount of the mortgaLq"e ou stand inq and t is the tax: rate. In general lhore o'.-ninq '.'ill no'.-.' be more attracti'.'e the frontier for o'..'ninq '...'ill slope, more to the right in Figure 2), but the analysis w.iill bte the same as before. t could, of course, be made a function of income. S :: l ) U 4 4, 1- St A ( ,,l, I 4 1 = I. ll ) T -B R L FIGUR.C 2. Consumption possibility curves Up till now I have proceeded as if the quantities ,Hk* and II can be observed and measured. Indeed, I have pro- L ceeded as if the whole set of housing types H.* can be ob- served and measured. In pr.~ctice, both observation and measurement are difficult. To get around this difficulty, housing can be measured in units of dollars. These dollar units can then be deflated by regional price indices. The dependent variable for both-owners And renters now reduces to deflated dollar expenditures. This view of housing is easily reconciled with consumption theory. The consumer :..ill ima:-:imi utility by e.quating the mariaginal utility of the lSt ldolla.r spent on housiln to the m.-ar-jinal utility of the last dcll.ar spent on all other gcds. More speci- fically, the model de'eloped: in this chapter can be written 03S =5) 'In n' 1 + L n 15) In n + 2n *71 I =' ,' 1i 1 hereee 'ln is LOal annual e:-xpenditures on housing if the family owns, is r' al anni.ial expenditures on housing if the family rents, I is an unobser\ vablle inde:-:, : n: ::-, nd Zn are "ectors of incaeperenten v.ar tables -and l Un ', U ) are triv.-r iate In _n 1 normally d t s tribl-.u I-d Also, the model assumes Yn and n .rie rILItL.1ly l :I cusi'.'e and cannot be observed simultaneously for any one indi.'jdual. .One either observes the family' own- 111I ajnd sFpndinr-' Y if I 0 or renting and spending 'IY IInIl orall this is a s intc i e ssi if I 1). Mrire formally, this is a 's%.itching regression model '.with sample separation and can be writtenn as (8) lIn = ln. 'l. + tn (9) '2n = :2. + U n 2n but w.e only obser"e Y where 'Y = IY ,iff 1,, U i n In Z n T n (10) In 2n i n Znn *U This s'..itching regression model is nothing more than an extension of Tobin's (58) paper. In Tobin's paper the dependent variable is zero or some constant if a family does not have expenditures on a durable .ood. In this paper the dependent variablee is rent ex:penditures--a random "ariable-- for families w..ho do not ow.n. In this section I ha"e described a --.'itchinq repression model with sample separation where the expenditure equations and choice function are simultaneously determined. Although the model itself is simple, its estimation is not straight- forward. The estimation problems stem from the fact that U is assumed to be ccL-related with U1 and Un . In n 2. SpecificaLion of the Nodel The model was estimated on a sample of 3,028 families from A Panel Study of Income Dynamics, "olume II (72), (hencefor th tie "Panel") These data include voluminous q]uesti.onnairec data for a sample of 3,4152 American families over a period of fi.'e years. A detailed explanation of this data base is presented in the Appendix. For empirical purposes, a logarithmic functional form 20 was used in the demand equations and the choice equation. This makes the estimated demand equations compatible with previous studies. The dependent variable .was housing expenditures-- described as housing cost in the Panel--divided by a price of housing index. This variable includes utility payments, amount saved on additions and repairs.when the work was done by a family member,5 property t'xes for homeowners, 6% of house value for homeowners and annual rent payments for families who rent. That is, the department variable is 1'in its of housing as measured in real annual dollar e :-: nd i tur c s. The p[:rimary ind.1eperndent -ariable '-.'as a measure of prrmanent income. The variable e used: '.-:as a fi'.e ye.ar a'.:C'rag of "fmrimily noi-ne,'"--as described in the Panel--plus inp,:ute.d rental income for .homeo'-.'ners (i.e. 6l. of net equity), all di '.1id1: by a general price inde>:. This '..-r able in- cludes labor income of head and/:or w'.ife: asset inbo me from fa-m or business; rental, interest and di-idend income; arid transfer pri, ments such as Aid to Dependeint Children. Other iiid.iepc dcrt "ariables included city population and distance from the center of the nearest city of 50,000 .r mount saved ion additions and repairs \-'as included to make hou sing c:*: pcrn.d it L es fo.-r renl tle s andr o.-.lners comparable. In _inCrCL ii 'L'c-, c.: icnterls dc thicir c'.'n main tcnance '.'crl:. Tiiis is bec-iusc t;hieir r,:-nt]tal p:'aymen ts--.and therefore housing e::pend i tures--i nclu.de a3n implicit maintenance cost. 6 population or more. These two variables '.'ere included to capture the effect of higher land values as land becomes relatively scarce in densely populated areas. Also, to capture demographic differences, family size '.*.as included as an independent variable. This variable is defined as a number of people (children plus adults) living in the family unit. Dummy varliables for .age, sex:, and race of the family head were included, as '..as a dummy variable if the family moved more than once between 1963 and 1972. The reason for including this ls-t variable in the expenditure equa- tions '.:as to capture the different- search costs as '.ell as different demographic factors of the mobile families. It was included in the choice equation for other reasons. The moving family, because of the transactions cost in- volved in buying a home, is more likely to rent than own. Finally, six price indices '..ere included. Each house- hold has associated with it a relative (to all other goods) price of renting, a relative (to all other goods) price of owning, a price of renting, a price of owning, a price for all goods other than housing if the family, rents and a price for all goods other than housing if the family ow.ns. These price indices were based on data taken from the In the Panel these two variables were coded in groups. For c-:amIpl1e, c ty size W.'as cla-ss ficd as beinti in one of six groups: r-c-atcr thD1n 500,000; 100,000 to 199,999; etc. To make the variable cont iluous, a real iumber-- the midpoint, when appropr iate--',..as used for each gr [)oup. That is, for the Z 'wo above examples the family was assigned a value of 1,000,000 or 300,000, respectively. Bureau of Labor Statistics (DLS) Handbook of Labor Statistics, 1972. The three "renting" price indices were constructed from Table 136 in the BLS "Annual Budgets at a Lower Level of Living for a 4-person Family, Autumn 1971." This table breaks the budget into nine categories for forty-four cities and nonmetropolitan areas in the United States. The housing expenditure data assumed all families were renters and was used to construct the rent index. The price index for all other goods was constructed by subtracting the housing expen- diturres fromL tLhe total bl:.udget. Similarly, the three "ow'.ning" price indices ..'ere constructed from Table 138 in the BLS - "Annlual Pudgets at a Higher Lc'.'el of Liv.inq for a 4-person Family', Autumn 19'71." In this table- the housing expendi- ture data v'ere ..'eiqhted by the following proportions: 15 percent for rental cost and 85 percent for homeow.ners cost. Using these t:.o BLS tables, it '..as th-refore possible to assign one price imnde:-. for renting and another for ow'n- ing to e-ach of the forty-foour 3reas. Depending on where they li'.'ed, each family in the Panel could then be qi.'en three price indices for o.'wning and three for renting. For e::ample, if a family li'.'ed in the Boston SMSA the price indices for Boston .-.ere used. If a family li'.ed in the ilor'th -:ist andj in a city of less than 50 thousand popula- tion, then tlhe .price indices for nonmetropol itan areas in the lortheas:t .-ere used. A similar procedure '..'as followed for other parts of the country. i.]thouigh the abov'.e price indices made careful use of all available data, at least five cautions should be made. First, section one of this chalipter argues that housing is really a bundle of several goods. One price index is only an approximation for several price indices and therefore its coefficient is only a surrogate for several price elasticities. Second, Polinsky (75) notes that even if housing is a homocieneous good, using metro housing price indices ignores price variations among observations from a given metropolitan area. This can' cause do',.'nward biased estimates of income elasticity and up'w.'ard (toward zero) biased estimates of price elasticity. -Third, the price inrd:.: Fnr n.wninn waq hansr-d on 1071 data. IUnless rPlativ'e prices across cities remain constant over time, the rice index for o.-.'ners is only valid for families '.ho bought their homes in 1971. Fourth, since the rent index '..'as based on lower income families, it may not reflect the true price of renting for higher income families. Simi- larlv, the owniing prrice index may not reflect the true price of ow-ninq for lo'..'er income families. Fifth, families may believe that the selling price of houses will rise faster than the g-eneral price index. This may encourage them to buy, a house larger than their consumption needs. Their dollar expenditures on housing would now..' reflect investment as .,well as consumption decisions. CIIAPTER IV TWO STAGE ESTIMATION 1. Estimation Problems Because of the assumption of correlated disturbances, estimation of equations (5) and (6) in Chapter III with ordinary least'squares is biased and inconsistent. As in the Tobit model, this bias and inconsistency is due to the presence of a disturbLaince term \w-hose e:-xpectation is nonzero and not constant for -all obser'.'ations. In the econometric litert--Ljure the e::pectatioin of this disturbance tern- is o fteri referred to .as a "Hissincg Variable." lore specifi- cally, it can be sho-.wn that E ( |I ) 0 E (U ) 0 In n1 n 2n n n .aind -lso tllat the error sterns U 1 will be correlated with S1n' 2 n their regCressCors. Thus, the direct least squares method can- not Ib:, applied and other consistent estimation methods must be used.C While one possible procedure_ is nm.aximum. likelihood estimation, Lee (76) hais proposed t'..o much simpler computa- tionl-l "t'-.'o staqe" methods. These procedures can be applied to eqij'ations 5-7 of Chapter III as follows. Let the obser'.'ed sample separation be denoted by a dichotomous '"'ri.alle I That is, (1) I = 1 if0 f U ]n n n If '..e assume U is normally distributed the above can be w.'rittoen as the following model. (2) I = - n n n but we only observe" I where: n (3) 1 = 1 iff I 0 n n - n = 0 iff I 0. Finally, the Probit model y'- ilds the relationship (4 ) I = F ( ) + E - n n n where F is the cumulative distribution of a standard normal variate. The first procedure proposed by Lee (1976) uses the com- bined ow-ner and rcnter sample and estimates B1 and 82 together. To derive this procedure let i' be the observed sample of the th endogenous variable for the nt family. Combining the two separate equatioiLs into one singIle equation, .'e have S + (Il I 1 '' -n 11 n I L - (5 = + (1 I : + I U + (1-1 )U (5) = 1 ]lit + n n 212 n in n 2 "n .Al1thloug:j (5) looks like a single regression equation, ordinary least squares cannot be applied directly to it. First, since U of equation (2) is correlated '..'ith UI and U,2 the error terms do not have zero mean. In fact, Iwe have (6) E(I Ui ) = Ol U f(U )dU n In 1 1 n n n Z 'y (7) C(1-1)U2 = ~2. n U f( )dU where n", 2' 1 L0, denote the standard deviations of Uln, U and the correlation coefficients between U and U and 2n In n between U2n and Ul respectively. Also, f(U ) is the standard normal density function._ Thus, the mean error term of (5) is (8) E[I U + (1 I )U ] = (P l-p 2) U f(U )dU Oo Furthermore, the error terms of the equation are also correlated with the regressors -since (9) E[InUnI Xln] = EmUln]Xln = POlXn/ Unf(U)d f 0, --OO and (10) E[(1-I )U (1-I )X ] = E[(I-I )U ]X n 2n n 2n n 2n 2n Z Y = -p OX21 U f (J )dU 0 P2 Z X2/ n Unf n)dn n 0 --O In order to estimate (5) we must first adjust the mean of the error terms to zero and take care of the correlation between Lth- error t-rms a1nd the rec ressor We no'.' ha'.e (11) Y = F(, ,)::ln + (1-F C n ));:2n 2 n n In 1 /n + (I'1 -r'n .' i n U (LI Id + W ...'hL,-e U = I iU -i (1-I U.. ( .i. -n, ,.. 1 L fW( )dUL I'I I III I 0. I IIl "1 n . '11 4 1 ^ ,n n' It can be c sil'- s-l-..n that Ef I = 0 .nd W .is unco-rrela ted l' rn '.' l hI tI1, r.;i r-es or s. Ho.-.'e"r, erjuation (11) is non-linear in the parameters 1 id :2. But since s,-imple separation information is a'.'-i lable, a simple technique to estimate the unkno'.-.'n co- *ffici .ntl s 1- 1 : d i. -., is possible. FiLst, .]uation i) crn l e estimated b.y th.: usual Probit nal-1sis. This is the first staqe of the estimation procedure. il-ir- scon.i stage is to estimate the fol l'-.'inl equation by oud. inary 1' I nse t souar- , (12) 1 r ( "):.:: 1 + (1-F (Z ') :.: ". II In 1 n S7i n ), + ( i"' -0,0,) .1 U f (IU Id U + W z1 I n1 n' n n Since is a consistent estimate of y, U will have an asymptotically zero mean and be asymptotically unrelated with the rcqressors. Even though W does not have the same variancec e for each n, the estimated ~1 and 8-, by ordinary least squares w.'ill be consistent. To see ho..' P 1 and F '2 are identified, equation 12 can be rewritten as (12a) = :' + X2 + ,- n -+ .- , (n in 1 2n 2 n n w\.herIe X = F'. = F(-Z ) 2 a ( P F0 ) in 12 I11 1 2 1 and 3 = i U f )d" When all the independent .variables 3 n n n n , in X: and are different, OLS on (12a) yields estimates of ., ', './ar (,) and '.'-c r H( 2 directly. fiHowev.'er, if = "" 2"2 -2n, OLS on (12a) will only gi' estimates of ,, ) '.'ar- (2) and '.ar ( 2,). This is because (12a) now reduces to (12b) = :: 2 ( [ ) + + n' 4. ^ 1 2 1)1 e = = and : = F Z ,) Estimates of ..'ar e n In "2n 1n n n1 (B6) can be obtai ined by using OLS on the follo'.-'inq (12c) : ;= + >; (, ) + a:e; + , S1 2 2 I 3n where r = F(-Z x) x. Since the three sets of independent variables in equations 12b and 12c ..'will be highly correlated, estimation of 1 and -, \.'ill ha"e multicollinearity problems. In order to av.oid these multicollinearity problems a second two stage technique is available. This method esti- mates F1 using only the o,..ner data and 2 using only the renter data. That is, consistent estimates of 81 are 1 The standard errors will be slightly biased. This is because unbiased estimates of 01 and a, require a somewhat different forimulation than the one used in OLS. obtained by using OLS on -f(Z y) (13) Y X lnB1 + T ]( + V n Similarly, consistent estimates of B2 can be obtained by using OLS on f(Z y) (14) Y = X 2 + o [ J + V . 2n X2n 2 Zu (-Z) + 2n In the two above expressions Cu and a 2uare the covari- A lu 2u ances between Uln and Un; and U2n and Un respectively. Vln and V2n are stochastic disturbance. Equation (13) is der1.'ed by t,~kinq the expectation of iln conditional on I = 1 arid adjusting the conditional mean of the error term to zcro. That is, (15) EC(Y In = 1) Xl n + E(U- 1n In 1) n + I In d = n1 I + 1" Ulng (In n = l)dU n, ..'here 1l(U n I = 1) denotes the conditional density of Uln conduit ioncd on In = 1. Before proceeding, rev..'rite q(U r, I = 1) as the joint density divided by the marginal density, q (U In = 1) (1;J (( l II = 1) = I rI In n qC(I = 1) n n f(U IU )cU in' n n f (1-1 ) d n r 77 1 .rA f (II. F(7 F)ln 171 n n 1 .n .I f (U ) f (U )dU F(Z ) In n n n =, 1 n (f ( lU U ) f (U )dUn n --, ..'lher-: f (U .n',I n and f('ln IUn ) are the joint and conditional I.i'.ariate normal density functions, respecti'.'ely. By substituting (16) into (15 it follow'.s that (171 ) l i n I 1) = v I .. in n n + n J (" U f(U ILl ) dL ) f (U ) dUl F( Z1 ] 1 1 n In n n 1By substitutinI in i or f (U I U ) '.-'hen i 2 = 1 and i nte r a t i ri n '.'.e r e t (19) CFl'inI = 1) = : n 1 nr r. U f (ndu 1n in 1 1 n' n U f L )dLI *I n I n n n n 1 lu F (( ) Ln n in 1"I Finally to use OLS on Chapter III's equation (8) directly, '..e need to adjust the conditional mean of the error term to zcro. ['oinci this '.:c cet in = in 1 lu F (Z ) n 7 + n ilu L C.Fn) S[ -E(Z1 1 S"1'nl lu F(Z ) n' .where .' = tL ,- and C' l I = 1) = 0. Similarly, Equation (11) is deri'.'ed Ly taking the c::pectation of Y2, conditioned on I = 0 and adiustinq the conditional mean of the error term to zero. As mentioned earlier, this two stage approach avoids the multicollinearity problems that arise when Xln =2n Since nine of the ten variables in Xn and X2n are the same for the model specified in Chapter III, 81 and B2 were estimated from equations 13 and 14 respectively. 2. Discussion of the Results 'The model was estimated.,usinc 1971 data. The Probit estimates of the choice equation are reported in Table 1. The OLS and two stage estimates are reported and compared in Tables 2 ard 3. A discussion of the results found in these tables ollo'..'s. Table 1 contains the Probit results w...ith el.een in- dependent .ariables. Both age dummies are significant and inJndcate that families .'ith a head o'.'er G6 are the most likely to o..wn their ow..n home and as expected, families headed Lby an individual under 36 are the least likely to own. Thie coef[icient for The Black durmmy is highly signi- ficant and neqativ'e, indicating that Blacks are more likely to be renters than owners. This could be due to either price discrimination or merely a stronger preference for those housing goods that are cheaper to consume by renting. A similar interpretation applies to the ngati'.e sign for the female dummy. 'ihe ineg t0i'.'e coefficient for the mover variable sug- gests that transactions costs of buying and selling a house effectively raise the price of a home for the family that frequ,-'-nt 1y mov.es. These families are therefore more likely to rent. TA BL 1. Probit CoefficienL:s for Owners' Intercept Age < 35 Age 36-64 13 lack Fem ale flo'.'er Log (City Size) Log (Distance from Center of Cit'y) Log (Family Sizel Log (Ile.al Permnainent Income) Log (Pelati.-ve Price of Penting Housing) Log ( lciati.ve Price of Owning lorme) Lumber of Owners Number of Renters 'Standard errors in parentheses -4.6586 . 5350) -1.2001 (.109 4) -. 6567 (.099 1) -. 3256 (.0728) (. 0632) -.9591 (.0721) -.1060 (.0199) .1389 . 0276) S1328 (.0498) .7586 (.0535) 2.1735 (.4015) -.3684 (.3793) 1803 1225 TABLE 2 Comparison of Two Stage and OLS Estimates for Renters* Intercept Age < 35 Age 36-64 B a c: F em,:- I e Lo;i (['lct.-incc froni Cen ter of Cily) Log (Fariu1l; Siz,-) Lo (. (F.1e PcF-i:ranent Income) Two State Estimates 2.0515 -.1994 (.099.0) -.1320 (.0730) OLS 2.0965 -.1774 (.0580) -.1190 (.0558) -.1823 -.1776 .04941 (.04641 .1061 .1107 (.0329) (.0282) .12 6- 1407 (.0593) (.0286) .056 4 .05 81 (. 100) ( 007 6 . 0 n005 -.0017 (.0310 (.0299) .i 0 2 .0863 S0215 (.02041 .50231 .490: (.0506) (.0236) Loi (Pe la t "i r'Lice of PFenting Ilousin ) -1.2:372 (.22 65) -.0329 (. 1192) -1.3216 (. 1 3 5) [ iussini '.', i -,:a b 1 e n2 [lumber of Obscr-'a t ions *St.andard errors in parent theses 1225 .4 2r 9 1225 TABLI: ] Coniparison ofL Two' Stage and OLS Estimates for Ow'ners* T',iwo SLta -- Estimates 2.0392 Intercept Ace -: 35 Ag le 3-6641 Blacl: Female lover OLS 2.0909 -.1-54 -.1343 (.0693) (.04071 -.1563 -.1505 (.0440) (.0330) -.2534 -.2502 (.0585) (.0561) .0772 .0006 (.0352) (.0308) -.1018 (.0660) -.0909 (.0367) Log (City Size) Log (Distance from the Center of City) Loci (Family Size) Logr (Real Permanent Income) Log (Relative Price of rOwning Home) flissinci Variables .0282 .0295 (.0088) (.0060) .0193 .0182 (.0363) (.0357) .029P .0285 (.0234) (.0224) .5712 .5644 (.0396) (.0206) .1784 .1768 (.1914) (.1912) -.0195 - (.0979) .441 77 Number of Observ'.ations 1803 .4490 1803 *Standard errors in parentheses. The next two variables, city size and distance from the center of the city, indicate that in rural areas the family is more likely to own their own home. This could be caused by higher land values in thi more densely populated areas. In populated areas where land is scarce and therefore expensive relative to land in rural areas one would expect to find more people living in land economizing apartments. Conversely, in rural areas one would expect to find a rela- tively limited number of people living in apartments. For S:-;amp le, few fami lies could d af ford to Gjuy land and build single: family lhous es in :do-.'nto:'n He'-. York City. Instead of houses :.'e see high risc apartments' that economize on the scarce inpuit land. Ia sed on mcre availahi] ity then, it foillo.'s that city' size should have a negative sign and distance- from the city a positive sign in the choice equation. ThI- coefficient for family size is positi'.e. This i.lndicat.cs that larger families prefer the more spacious living coni-i it ions pro'.'jded by home ow-.'nership. The positive coefficiE-Crit for the income variable suqqests that lo\-.'er income families are unable to obtain a mortgage for the size of house th'v desire and therefore rent rather than o'.'n. Finally, the t.wo relati'.'e price 'ariables ha'.'e reason- -tile sigq ns. ,fs the price of renting goes up or the price of owning i J goes .io'..;n, a fariily is more likely to own their o'.'n home. Tables 2 and 3 contain the t,..'o stage estimates of the demand equations (equations 13 and 141). For comparison pur- poses these tables also contain OLS estimates of the renter and owner demand equations. These OLS estimates were ob- tained by splitting the sample into two groups, renters and owners. Demand equations were separately estimated for each group. A discussionj of the two stage estimates follows. The coefficients for the-age dummies are negative and significant for both owners and renters. For owners the coefficients give a U-shaped pattern of housing ex- penditures versus age of head at given income levels. For renters, expenditures on housing increase as age in- creases. The coefficients for owners are consistent with a theoretical result derived in Muth. (74). Using what he calls the "income effect, price effect and length of stay effect," AMuth presents a justification for this U-shaped patt rn. The black and female dummies indicate that blacks spend less on housing than nonblacks, while females spend more on housing than males. While this may suggest the existence of some discrimination against blacks in 1971, it does not prove it. In general the outcome for both blacks and females largely depends on their preferences relative to others. Ceteris paribus, a stronger (weaker) preference for housing relative to all other goods would result in more (less) housing expenditures. The coefficient for the mover variable in the renter group is positive. Therefore, moving renters spend more than nonmovers. Mobile families may spend more for rental units because oC search costs. That is, since they only 36 spend a relatively short time in each location, the marginal benefit from an additional search in terms of a lower price - does not outweigh the marginal costs. For the nonmoving renter the optimal searching rule in terms of the number of searches or stopping price probably results in a rela- tively lower expenditure. For a review of search theory see McCall and Lippman (75). In the owner group the mover co- efficient '.'as negqati.v This ma; be due to three factors. First, transaction costs are lo..wer for less e:-:pensi:.'e houses rincc reactor fees and the like are a percentage of the cost. Hoi.ile families can economize on these costs by buying less e:-:Lensiv.e Ji..wellings. Second, it may be easier to resell a lo'w.,er priced hocusc. Third, mobile families ha:.'e little incenti.'e to "buiy hl-at dream house" and maintain it. Rather, they aLre more liLely to settle for a simple place to li.'e. Thie ; positi.'C coefficients for the city size -.ariables indicaLc thlt: bctlh osv'ners and renters spend m,-r-e on housing Ias city size increases. Since the dependent variablee is real dollar e::penditure the expected a prior result depends on the '.arlious price elasticities of housing goods. Since the demand for "housing scrv.'ices" are generally .'iewed as price inrelastic, then the c:-:pected effect on expenditures would be p:ositiv-.e. That is, a larger city implies higher land values .which in turn simply higher housing prices. Given inelastic demarid, hiihe r prices would lead to more expenditures on housing. 7ilso, if w.e assume that all families require some nILnimuni size house, and therefore some minimum size lot, another explanation is possible. .Once this minimum size is reached families can no longc-r economize on lot size by buying smaller lots. Higher land values s therefore imply higher expenditures once this. lot size constraint is -ffectliie. The two stage estimates of the distance co- efficients are positive for both o''ners and renters but are never significant. The coefficients for family size show that as family size increases renters spend'more oi housing. For owners hov.'e'.er, family size has no significant effect on housing expenditures. As expected, if income increases both o',ners and renters spend more on housing. These coefficients are well below one on similar to those found in Lee (68), Carliner (73) and Fenton (71). The coefficient on the relati'.'e price '.ariable is negative for renters but positive for owners. Gi.ven all the problems of norhomogeneity and cautions about the price indicies discussed in Chapter III, these results are hard to interpret. However, the -1.2872 result for .renters is similar to Fenton's (-1.27, -1.35 and -1.09) for all renter households, high and lower renter households, respectively). Also, the p-ositi'.e coefficient for owners is not significant. The coefficients for the "nfissin '.'ariable" in the renter and owner equations are estimates of the covariances clu and o2u, respectively. The negative sign, although in- significant, indicates the renters spend less as renters than they would spend if they oi.ned. Similarly, oi.'ners spend less as owners than they would spend if they were forced to rent. Finally, although the two stage estimates are theore- tically better than the OLS.-estimates, both sets of results are similar. The reason for this similarity is the in- significance of the-Missing Variables. This implies that olu and o2u are not. significantly different from zero. Without using the two stage.iethod however, this zero correlation conclusion could not have been drawn. CIIAPTJP '. V MAXIf-lUHl LIVELIHlOOD ESTIf-U.TIOf[ Iii trod uc tion In Chapters III and I'V' 7 housing expenditure model with interdependent choices betw..een o'...ning or renting was studied. Because of th-is interdependence the binary o'.r'n or rent choice functions is determined simultaneous ly with the housing expen- diture equations. It is the simultaneity '.w.hich males that model an extension of past studies on housing. While others such as Poirier (74j and Quigley (76) have presented models of consumer choice, they did not include an analysis of corn- sumer e:penciture equations. Similarly Lee (E), Carliner (73), Fenton (74j and Polinsky (75) ha'e studied and esti- mated housing expenditure equations without considering the implied choice equation. To the casual observer these over- sights are minor. A closer examination, ho..'ever, showed that estimation of the expenditure equations is not straight- forward. In general simple ordinary least squares (OLS) on the expenditure equations w..'ill lead to biased estimated. Other unbiased estimation techniques have to be used. One such procedure is a two stage method proposed by Lee (76) This is the procedure used in Chapter IV to estimate the model. Ilow.ev'.er, the estimates of the correlation between the disturbances in the expenditure equations and the dis- turbance in the decision function were not significantly different from zero. This is an important result since it implies that simultaneity does not exist and therefore OLS estimates of the expenditure equations are unbiased. In this chapter a further test of the simultaneity issue is made. To do this I start with the two stage estimates as initial consistent estimates and then apply a two step maximum likelihood procedure (2SML) to. give the final estimates. Since the 2SML approach does not ignore the simultaneous effects and the usual OLS approach does ignore these effects, a maximum likelihood ratio test'is employed to compare the two methods. If simultaneity does exist thnii this test should show. the 2S;1L estimates to be signify ic-itly Lbtter than the OLS estimates. 1. '.iim'miyia' s T',o Sterp l:.in.ium Likelihood Procedu're lThe iiimodel presenrtc here is the same as the one in Chapter. III except for tl, he specificatiron of the prices. In Clhapter III t';.o sets of indices .e.'ere used, one for o'.wning a'd one for renting. This clhnpter only uses cne set Cof Cg'ij):re,.ate price it-ddices -for bo th i-wners and renters. A different specific:i tiocn '.-as emnplo,'/ed because the positi'.'e coef ficient f-or the .price :.'arL i-ble in the owner eqluatin obtained in- Chapterl I" is lIard to, justify. Each h household h1ia associ'atdc with it ] housing price inde:-: and a price inIle:-: for all gcqoods. The purpose of these indices is to stanJardize housing e:-:penditures and incoine for different cost of housiingr and cost of living across the U.S., res- pecti.ely. Gi.'en this price specification, the relative price of housing services w'.ould not enter into the rent-ovwn decision function. It o.'ould, of course, influence the allocation of the budget between housing and all other goods. iore specifically', the model studied has the follow- ing specification (1) crn in + r n (2) C2,1 2n L 2n (3) = , n I n ,.'here Cn .ire annual expenditures on housing if the family owns, C2 are annual expenditures on housing if the family rents ard (r 'c are tria'.'riate normally distributed H1 (0, ) Here Q is a 3 x 3 co"ariance matrix. Without loss of generality the .variance of E is assumed to be one. I is an unobLserv'.able index. Also, assume C and C, are n In 2n mutual l e'c;lusiv' e and cannot be observed simultaneously for any one ind '.'iual. What is observed are oxogenous .ariables , the binary inde:: I and e:-:penditures Iln 1 n n L on housing, C such that (4) C = C ; I1 = 1 iff I 0, (5) C C, ; I = 0 other..ise. n 2.n n The price indices were constructed from table 128 in the Bureau of Labor Statistics (BLS) Handbook of Labor Stattistics 1972 "Consumer Price Inde::, 23 Cities or Standard metropolitann Statistical Areas, All Items and liajor Groups, 1974-1971." This table gives six price indices including ai housing index with a base of 1967 = 100. T'-.-enty-one of these SMCSAS (Honolulu and Washington, D.C. ..'ere excluded) were grouped into one of four regions: Northeast, North Central, South and West. Two average price indices and a relative price i-ndex were then cal- culated for each of the four regions and assigned to the appropriate family. The rast of the data are from A Panel Study of Income Dynamics (72) on a sample of 3,028 families. In the expenditure equations the dependent variable is logarit}hm of housing e-:pendiltures, di" ided by' the regional price of housing inde:-:. Th'" same t-n explanatory '.ariables are used in thll t.wo eX-.pend iture equations. They include the personal characteristics of the family' head (age, race, se:-:) ; fa;mi 1'. bac.: round (mover, logarithm of family size); loqgaithm of family "tcrmanent income"; regional variables (logarithm of city size and logarithm of distance from the c-onter of the city) and a relative price index of housing. .'*11 these e;.-'pl nnatory '.'ar able e.-.cept the relati'.'e price of housing are include inue n the decision function. For a more detailed description of these 'variables and their e:-.pected effects see Chapter III. The model can be re..'ritten as the following switching repression model *I6) C = "'" E. + iff :' n IIn In n I 7) C = + 4-C ifF Z y F n n an 11 1 - .A\ssulime that : L, and e are tri'.'ariate normally dis- tLibuLed wi th zero mean and co'.'arlance matrix ? such that I 2 '01 12 2Ic ^ l a2 2 o7 1 L - As noted by. Lee (76), all thi, parameters in the above model are identifiable except n12 cov'c I' 2). The maximum like- lihood procedure described in Amemiya (74) is used to esti- mate this model. Followinrg'Amemiya 's proof it can be easily shown that those estimates are asymptotically efficient. Let f and f, be the jointly normal distributions of (51 ,:) and (-, ) respecti.',lv. Dropping the n subscripts on C, i ], 2, Z and I, tile likelihood function for this model is (S) L( IB 2 'O 1 ', 2 ') 0 T Z', T 2 = II [. l C- 1 )d?] f 2 (Cj122' c)d ] 12I S1 (pc-:.; I )q1 (c-'Y I d. ]I- , n=n=l1 n 1 1 i (- 1 1 f C C- J ( -X. )d 1-I S- '-" i\2 27 2 2 2 i 1 (C-X11) /f1 1C-x1)JcII[g2(C-x22) f~2(c C-Y22)1-I i = -1'.'. " 1- 1 (C-: ) = e::p (C-: )2 TI I 1..n 2 2 !-, 11'1 fl(c C-::1 1) 1 .1 11 cxp (- ---- l (C-XlB )]2}, 2(1-Pl ) 1- 44 f2(EIC-X2) = 1 2 /2 /- 2 1-pq- 2 exp 2 [E P2 C-X( 2 2(1-pq ) 2 -% 2 where'p1 and p2 are the correlation coefficients of (cEE) and (E2,c) respectively. The integrals in equation 8 can be simplified by trans- forminc the variables. Following Freund (62),. if "y F(Zy) = / f(E)dE, -- then given a new variable t such that " -1 t = h(E); E = h (t), where h(E) is increasing, one can transform variables and obtain h(Zy) -1 dh (t) G[h(E)] = / f [H( t)] dt. dt Ey selectinrq li( ) as t [> (C-:'. ,l I 1L and sol'.'inq for G.[ () ], thL first integral in equaLtion 8 simplifies to 2 S I - L (dt. S2ii A sinlil ar analysis holcs for the second integral in equation P. Ilnial. ly afctern. making these t.-.'o transformations and taking .loqarithms one gets ( ) L = In L(e ',l,2 ,, l1, 02 , 2P , T I ln(q] (C-:; 1. )) + t=l in 'I (t) dt) (1-I) []Jn(q2 (C- .' C' ) + In ( .f.(t)dt).]} wherc P'( t) is the standard iiormal density function, 1 . r S2 01 ('2 L2," '2'2o) __- [ ,- (C-- 2e2) ]) S2 2 In ordLe to calculate the maximum likelihood estimates. a numerical optimization algorithm must be used. Since the likelihood func lion dcri'.'-d above is nonlinear these com- putations will be :quite complicated. However, a simplified procedure discussed in A;memiya (74) is available if con- sistent estimates of ,L,':2,,',o l,,' 1 and 2 can be obtained. Let (D0 be these consistent estimates. Amemiya's "t'.wo step maximum likelihood estimates" (2S1iL), 0 M, are calculated from InL.(11) I. nL ( ) (10) ', = )[ L l U O ,: ,- " The first s-tep of A.'memiya's procedure is obtaining the con- sistent estimates 0. These can be found using a two stage method described in Lee (76). Lee's procedure uses probit analysis in the first stage and OLS in the second. The details of this procedure are found in Chapters 2 and 3 r of Lee (76). The second step of Amemiya's procedure is substituting 0 into.(10) and solving for 0M. 2. A Comparison of OLS and Maximum Likelihood Estimates and a Test for Simultaneity Tables 1 and 2 present the probit and two step maximum likelihood (2SML) estimates of the -decision function res- pectively. Both estimation'procedules yield similar results, but the 2SML estimates have slightly smaller standard errors for all the estimated. parameters. All-the coefficients are significantt and ha'.'e the e:-:pected signs discussed in Chapter III. This implies that all the included variables have a significant impact on the- buy .'ersus rent decision. Table 3 presents the t'.'o stage and- 2SHL estimates of the e::l:pernd:it ure equat ions tor ,'.-.'ners. Table 4 contains similar estimates for renters. is e:-:pected the asympotically efficient 2Silj. estimates generally .have much smaller stand- ard error- s tha t the t.'o stage estimates. Hlo'..'ever, the two sets _of estimates are quite similar with three exceptions. First, thie m .'ver coefficient in the o:-.'ner equation changes from the tw'o staqe estimate ,of -.04407 to the 2Sill estimate of- -.1 61.99" and becomes significant Second, the city size coeFficient in the ''l ne-r e]ua tion decreases from the t'.-'o tage estim-te -of .065C2 to, the 2SnL estimate of .04237. In terms of the 2SIML standard errors, these represent more than a three standard error change. Third, the distance arablee in the renter equation changes from a positive TABLI 1 Probit Estimates of th., Decision Function \' I.. L E S CSTI ATTS STArIDARD ERROP Constan t Aqe 35 36 .: aqge : 6-; Black, Female In (City Sizo) In ( Dist-nce from center of city) in (Family Size) 1n lati"e permanent income) -3.2r6292 -1.17Q45 -0.6 .3 96 -0.30715 -0. 32102 -0. 9.1359 -0. 14969 0. 12779 0. 13-134 0.69437 0.47282 0.1075,1 0.09768 0. 07131 0.06729 0.07093 0.01763 0.02721 0.04929 0.05168 TABLE 2 Two Step Maximum Likelihood Estimates of the Decisicn Function VARIABLES ESTIMATES STANDARD ERROR Cons tant Age < 35 36 < age < 64 Black remale -- rri, m I e flo'.'e I.n CI ty Size7 1n (Lr i t .-rin c e ) o m center', of citLy) In i[aniily Size) in P i co r im-nc n>ti L income) -3. 24123 -1.16230 -0.63151 -0.30971 -0. 31895 -0. 94527 -0. 14991 0. 12 864 . 0.13381 0. 6 3159 0.47183 0.10681 0.09711 0.07076 0.06625 0.07053 0.01755 0.02700 0. 0-1904 0.05155 C -1* L 4' *:o0 -- r I -- r -'i (- I '--4 '. "4 '1 r"- r ', rC rC . '-4 0 0 0 0 '0 '0 0 0 .0 0 ,. CO r-4 r-I r'] ,--4 r0 0 0 -7 -7 r-, CO r- oI r- rI, *- 0 -I-1 Ll F-" H-- r L. ' SLl .Q4 1, 0 [ * C_ '-1 : _. 1-, .-i r 1 L 0 [- r'- iLr 'i Li (r . I- C --- C -C) -7 -r- -7 3 ,', ,- r, 7 C)0 D.D '- -O'. 0 '- '- ',D r'l r-; r'| ,-4 ,-4 c'. u-r i) .r 'j'. r-l 0 0 01 0 0 r- r', r'l r r rJ r-1 r, r- r' I -I L.' r- ,-4 r ,- C) I I I I r- r]i ,-4 r--i r- u-i r'1 rC 'D r i n n '0 0 '-4 r-, C) C) 0 0 I L'r 47 '2' C i 1 4J 0 I ., 3 D .--4 r'l -: 0C r' c r'I 0 Si in -4 r rCi -" i G) r- Or *r', -4 n | u-I u.I r- 0 0, ", -' '.*D r-1 rl - L", rj -i4 r,' --4 D0 ,- 0 -4 D C' I I I I C 0 ,i) -I U [ S (1) o - 0 0 G '0 C 0 I C: 3 ,-4 4 .*4 0 0 } ,i In () _1) O - -4 q 0 C *-1 V ,4.J QJ 3 4* ) ( * :2:, 1 rC rL4o- 1 Li i r' , 71 S II 4 r,-4 0 .0 Li --1 '-D r .] i .. I 31) I .0 - -I j 1 0 JI I'4 3 U 0 "0 14 CO I' (0 ,-I U 4 --1 0 ,- ru-i! . O 1 -< C IO *. c LO 0 *-I Li :*: ) L4 C" LI 1) 1 E *' er -.l - Si-. n C L4- (II m' II L) " ,n i11 u < 0 <0. -. CO ri 'D ri 0 r- u1, un U-, u-, fl CO a-I E U. C -4 J-' c '2 -4 CL, --I '--1 14 u1 E 0 0 - i fl) L I S 4- 4 N 0. *' N *^ L ) 1_} -4 U > II L C "., -4 ') --4 44 _ (l :,, 4J -4 4 PT- Ln L' :-, E ,- *T' 70 0 .1 -4 -44 -) 0l C || ,-4 Qi rj "' [- -4 *V U "' > 3- C. 'o 'l 0 C C 1-i C C ' ri r -- -,3 O C CO O E-i L, H U) U) Q 4 ,0 WM 0 E-i L[i ,l I H 41- r-) LJ Q 4:* J . C ,r r '.CI i PI -- 1 r- *. - .-., ,r- r"- -' --4 --4 0' 0C. C. I I I I -. rq .-C r- ,-I , C, C r-, i C. C Co C o. o. . :.: C C. "1 '. "- .H -,--4- ,- , S .,C C C, C, I Ir -J I o I r r ^ ^ r- I r^ ^-i ^-i O- r-i r- (N in N rn (N C) rCl C o o o o co N < m m o r-1 C - C) r-4 to . I, I IC l-1 C- C o --l ,- r-, r i o C- o 0 '. t , icT, 0 0 .U "1) L L7 I-7 C i, 0 4J _J = I Ii C-' '4 -I C ,-- C) ,,--I 0 q- S50 , N ---i4 CO !. C rl o o o oN oo S N i- r- o o i Co r- in oo r o 0 ri in in 1 n "tN N oC cl C) CD C0 CD 0 C) ,'U a - IJi E - ,---I D_ .' II ,' C : u i '.) 4 il '24 C : *-I U -13 .1i ul 1i1 :s J .-- *-1 4 Sf, 1J E 1 :c c C .' -- ,'I 0 i C '- C CO , : ,"- 1' L ,D--CO -- -4 .C 0a -r-1 0 .I 41 ji (U '0 4.- Sr w 3 ,4 1 - Lo r I IJ ,1) E -- r .L 0 c"0 I' L ') n C) 'l G! ,E O E II 'i 41 0 -1 q> .1-J I.) C 0'- '2J 4 1 u'4 4- X I-4 1J r j nilll S 4-1 0 C E 4c (a 0-1 40 ' 0 ui 'J .0 n IC, - E 0 c -- J ,. il r . U) *-f C' 0 L., C C Ln '*2 c.- O 01 --A r] cU CJ E r--- si nll to t i"e. sia.n nd ':...comes s iqni ficant. CGi'.'-en Chapter. III's is us ioi.i on l .:rid -.al, es .a-id cl ca t .ion this neigati .'a sn scenes more appropsri, te. i itli the ex::eption of price in the renter jequrtion,', all the .riL -c e fficier, ts ar,. siq r i if ri nt. I s th, : a ve h .: te- s s i s - .cuissecl in Chaptr III . ithl u.: li i [ :st [.:.a -.L-E suci aus .c-h i':, Carliru 3 , Frni to n i 4i} ani rolinsk. 7. sp d iC si ii t] di fcr- ,nt .-xI iCl iditur: e-uations F f ..' c Mlpa sons are pc'ssib'l '. ITe er- ,riii t inc,,m,- ci 'L ic i t L i rites. *f t.':" ? fo r o'..'I ..rS ]and .lOJi. for r n t-r arc .:u re C lose t' Ca ir l ir r' s 1 Lst nat-s of 31 .:r o..rneirs .:inc:] '. 0 moi : ,EVncr. S. rcn to'n estlimrated perranenrit Ii u e plastic ities *f -1i, 16 arrid 16 [cIc all re, tu househlcds and high -an i. isc,"TI r1ent er hlciusehol'ds, r [c--ti'. Le' i ",.'.: r ?['-'L '-1 se'c.'iL l s of .income- el sticity c l;.ri atus. His *.-:= ir.at s of p rPf nrr i,-'- i.n.:-,oinr1 e l a:s ti- ci L t r oC o C-.n:. I rt s ,nrigqc f iom ...0:. to '.2 '. L reni--cIrs hi [-.-rma 13 1,2.it income elasticity: estimates ranged from 387 to 6 2. Pol insk; c nci'lud s tliat tI2- true irnc . c-lastici t,' is appr :: irna t l-,' 5 f -.c b'tl- r n teirs a i-id C'...'rier s . Finally', the co efficient ts for the re lativ.'e r:.rice :ariab es are .n2qaLi'.'e. l'] -se coeffici'en s :re riot reliable e.tinrates of trad i io ai 'ri,- c lastLicit --l'rC r ntl:age clhaiin je in c1uL nti tv [)er pe rcen ta c n qe ii [ '' : This is tri.r": for ti.'c ru s sr s . C r 'rline"r U.1 S tie sanec data h.base a.= 'ii C]c. ilis s tc Limate s of . 631 arnd 51'0 aire based oin a re- ;riess '1in u-siing fIour year '.'eraiCi .i.nico'nie, re' la iv 'e p1r ice o:f Iou s i rn. :nId di i r s f, 'r Llack, f:iiale ind ,-a e is c:-:planato'y '.' r i-i ] r l o.- l'i- cocff icic' ts 1 for the- BlacIJ :, female and anci cl-u-iiic ::, are si. r, la r L o, I'l' e s ti lna .c s . First, the dependent variable is deflated dollar expendi- tures on housing. The price coefficients are therefore percentage change in deflated dollar expenditures per percentage change in price. Second, the poor quality of the available price data makes any estimate of price elas- ticity equally poor. This would be the case even if the dependent variable were measured in units of quantity. The negative signs of the two price coefficients do indi- cate that families 1. ing in those regions .. here hoi-_is in is relati'.elv i.to al 1 other g ods) more e.:-pensive tend to spend less o:n housincq. It '...ould be useful to compare the S'lfL estimates of the e>:[pe id iture equations with O- LS estimates. The OLS estimates .ere- ol-tained b.' di.i ng the sample into t.wo' grouf'.s ow..rers and renters and appli..'ng OLS to each grCoup separate-ly. These results are presented in Tables 5 a'ni 6-. cc:mpar iscn of Tables 3 and 5 shco.'s the 2SI1L and OLS estimates to le: sligihtl '.' different but co nm a tile. One e::ception to this cormpr:- tibility. is the distance V ari- able coefficient. The '.alue obtained w.;ith 2SI-1, is five L . times greater than the OLS estimrlate. Also, the 2 S-1, estimate is -igi-ifican t a=id the OLS estimriate is insignifi- cart. compar-isorn of Tables 4 and C aqain shows a difference in the distance coeff icierit. Thle 2Sl1L ,estimate is negative and significant. The OLS estimate is .ositiv.e and inrsinificant All the other coefficients lc nearly the same. IWhile these casual comparisons do reveal some differ- ences in the t'.wo approaches, a more detailed investigation on w.'hetlier or not simultanra.t occurs is possible. This investigation in'.ol'.'es the use of a maximum likelihood ratio test. Consider the null hypothesis that there is no correla- tio! b t.-.'een the disturbances in the.housin.g expenditure equations and the decision function. That is, the null hypothesis contains rl, and 2 to zero. The maximum likelihood estimates of the expenditure equations no,., reduce to simple OLS estimates, '...hile the maximum likelihood esti- mates of the decision function are profit estimates. Denote these estimates as 0 Tne alternative hypothesis does not constrain 1, and nL to zero. The maximum likelihood esti- mat-es are no'...' the 2Si!L estimates presented in Tables 2-4. Denote these estimates as Given *' and C.I the ratio -2 In h< as a ,. distribution asymptotically. For this model the ratio is 6.771 and significant at the 0.05 level. This implies that simultaneity does occur, albeit the e"i- dence is weak. It then follows that OLS yields biased esti- mates of the e:.xpenditure equations, e-ren if this bias is not very strong. Therefore estimation procedures such as the two staqe and 2SrIL approaches, v.hich account for the simul- tarn eituy, are more appropriate. TABLE 5 Housing Expenditure Equation of the Owners COS VARIABLES ESTIMATES STANDARD ERROR Constant 1.49477 /1qe 35 -0.15 45 0. 0413 3n .ae 64 -0 i 520 0 .0359 Black 260 0 i: 057 : 9 F.i .J e 0 I. i: 3 2 0312 9 ro'.'er -0.09916 0.03732 SIn (C it S-eze) 0.05.922 0. 00.572 In i c)istance roi n cen er of cit ') -0.:001 563 0.0 3646 In (Fainir,' Size) 0. 03752' 0.02284 In ( Pel l : .' pi.-ice of -Lousini) -2.62209 1.085 In r, n.' .. p trm ian ieni t income) 0.5c399 0.02079 TABLE 6 Housing E:-penditure Equiition of the Renters '.'/.IABLES EST I PATES STAllDAPD FPPOR Constant I: 1.77267 Ag 5 -0. 13423 0. 05946 36 C : age 64 -0. 12172 05634 Black -0.19701 0.0465 Female 0.11751 0.02936 l.o'.'e 0. 13378 0. 02877 In (City Size) 0.06964 0.00778 In I [Distance from center of city) 0.00698 0.02998 In (Family Size) 0.08309) 0.02050 I.n i Re 1 In t 1'.' price of housin.:j) -1.81576 1. 36963 In ,-n1 pc.rian.ment income) 0.50212 0.02368 Estimated slandalrd error of disturbance -, = 0.4348 CHAPTER VI CONCLUSIONS The.model derived in chapter three of this thesis begins with the premise that "housing services" are not a homogeneous good. Father, hoIsing is ,'iew'.'d .As a bundle of several commod- jties such as location, size of house, tLpe of structure, neighbors ard other humanities. This "ie..' of housing is some- .-hat different t han the Olsen (69) approach '.here homogenitL, is assumed and the Fing (7i6 approach '.-hich estimated demand equations for Lancastrian characteristics. It is then aLcqued that some bundles are cheaper to con- sume if the family buys their ov.n home. Conversely, other bundles may' be cheaper to consume if rented. A family is .assumed to. consume that bundle which maximizes utility. Follo-.in. this line of thought a general model of consumer choice bet'.-.een o'ninq and renting and ho.' much to spend - is developed. The model is a s-itching regression model '..'ith a choice function that is not assumed to be independent of the expenditure equations. This differs from past studies of housing '.-'hCLec independence is tacitly assumed and simple OLS is applied directly on the o''ner and renter e:xpenditur-e equations. Gi'cn that some simultaneity is present, these OLS estimates will be biascd. Other consistent estimation procedures must be used. In this thesis consistent estimates of the model are found wi ith F' "two stage" technique and a two stop maximum li ke li hood procedure. Both the "twoo stage" and "t..wo step" procedures perform quite well in terms of standard errors and e::plained 'ari- ation. For example, all but one of the coefficients esti- ma-ted Li the tw'o step procedure] e are significant at the 57 level. Only the estimate of price elasticity in the renter equation is insignificant. .'Aso, w~lth the e:.:ception of the price elasticity estimates the empirical results are com- patible :-.'ith economic theory and other recent studies. The two step estimates of permanent income elasticity of .60678 for owners and .50046 'or renters are significantly less one. These estimates support the results found in Polinsky (75), Fe:nton (74i Carliner 173) and Lee (69i but differ with nutlh's ('30' results. ILuth f( 60 uses a l re-j ate time series data and concludes that the income elasticity for desired housing stock is at least 1.0. The two steps estimates for owners indicate that families living in rural areas, families w.ho frequently mo1''e, and families headed by a black tend to consume smaller units of housing. Con'ersely, large families and families hea-ded by either a female or by someone over 64 tend to consume lari2er units of housing. The results are sim- ilar- for renters with one except ion. Families who frequently nmo.ve tend to consume larger units of housing. The two step estimates of the decision function indicate that families living in rural areas, families- who frequently mo,'e, and families headed by a black, female or someone- under 36 are the least likely to own:. Conversely, the larger the family size or the higher the permanent income, the higher is the probability of owning. The only discouraging results were the estimates of price elasticity. Two different price specifications were tried and both yielded unsatisfactory results. The specifi- cation used in Chapter IV resulted in a positive estimate cf price elasticity focr ow,,ners. The. specification u sed in chapter f iv'.e resulted in a nlega ive' but elastic (-2. 59 for ow.ners and' -2.11 for renters) estimate of price elasticity. This ela.stic estimate is not supported b recent studies. Fo r e.-.am ple, Pol ins.y .75) concludes that housing demand is price inelastic. These poor price, elasticity estimates sugglcest that price data should be collected by the Sur'.'ey Rese-arch Center's ne-:t "Panel Stud'.." ito test the hypothesis that simultaneity bet'.-'een the e::penditure and choice equations does e:-:ist, the two step ma:-:imum likeli- hood estimates are statistically compared to the u.-uail COLS estimates. By using a maximum likelihood ratio test to make this comparison, evidence ..,as found that simul- taneity doe.- e_:ist. Therefore, the procedures used in this thesis to e.=stimate the e::penditure equations are more appro- priate than 'the usual OLS approach. Finally, a few suqgge. t io-ns for future research are '.-:orth making. Firstly, a housing demand model using the entire f'.-e year ?ur'.'ey Pesearch Center sample could be deri'.'ed and estimated. Thi s '..could not be a simple chore since the Lrquired estimation technique needs to be workede d cout. In general, the procedures used in this thesis would not be appropriate to estimatl:e a model based on pooled -cross-section and time series data. Secondly, a model that described li'.'h families move '..'ithin a qi'.'en area could be foL-mulated and estimated. 'The underpinnings for such a model are presented in the Appendi:x of this thesis. Lastl.y, tlie computer proq ram used to obtain the t'...'o step estimates has many applications. For example,'it could be used to estimate a model that explai ns electric utilities' demand for oil and coal and '-'hy these utilities "switch" from coal to oil and '.ice v.orsa. APPENDIX 1 DISCUSSION OF THE DATA The majority of the data were constructed from the Survey Research Center (SRC) .sample. The only exception was the construction of the sundry price indicies. These indicies were constructed from the Bureau of Labor Statis- t1cs' IlI lnd-: o.Il ofl L.ibor Statistics 1972. For an e:plana- tion of lhi'.' th,?se indicies '.*'ere formulated see Chapter III, cctiron 2 Dnd Chl.pter '.', Section 2. lThe ori.- i1~-l SPC s-imple cacm from t.-o sources. Abo.t 10 percentt of the f:mrilies first inter'.ie'.-'ed in 1969 had been intcr'.'io*.'cd pF e'.'iously in 1966 and 1967 Iby the Bureau of tllh Ccunsus .is pirtt of the S'.ilrey of Econcmic Opportunity iSE 'l lhe repal.in in 60% con listed or a cro-ss-section smr'pi[plE rof '-]'E IlinigCs of the coterrmiinouI United States. The 19).69 tr, 1972 sacrimples consisted of all panel members living: in tamI ilies f.h.at '.*.ere inter'".'i:'.-'ed the pre'.,ious ye-ir and nc'.-,ly foi-mcd famijilie containing ..ny adult panel mcibbeLr '.'hoc had mn'.'crd since 1E968 f rom a sample fmnlily. The res- pondcnt '.-.'.is usual]'- thc head of the family. ,'11 of tlhe o-riglinal SECO families selected in 1968 had income s in 1966 equal to cr belo'.-.' twice the. federal poverty line at that time. The selection formula w.'as $2000 + r (51000) '...h1ere I is the number of indi'.'iiduial s in the family. E:-:cluded were families where the head .-'as o'.ver 60. Also, except in the South the original SEO subsample only included families w'ho live'cd in SlISA's. In summary, the 1968 interviews were taken with 4802 families by'SPC, 1872 from the SEC selection frame and: 2930 from its cross-section samples. Li.en though the SPC sample seems hiased tow..ard poor families in 1968, by 1970 the data 'w.'re a fairly represent- ative cuoszs-sectionh of families. This can be seen in Tables 1 and 2, w.-lich were taken from The Panel Study of Income Dynamics (1972 '.'olume 1. These tables compare the SPC data for 1970 to i corresponding national sample taken at approx- imately the same time, the 1970 Current Population Survey. Variables used for the comparison were family money income, race, sex of head-, size of family, number of children under ei llteren in family aqe of family head, and size of place of residence. For e:-ample, both distributtions of family income shio'. about the same percent below and above S5,000, though SFC does have a slight underrepresentation of very poor families (Income under 51,000). The comparisons according to the demographic variables are also close. There are a few notable exceptions. For black families SPC does have a slight loss among two person families and those with no children under aic eighteen. For white families there .is a slight undcr-epresentation of those with heads of families forty-five or older. Finally, this thesis only uses the 1971 data. The sample consisted of 3,452 families. E:-:cluded from this TABLE . Comparison of Census and SRC Data: Total Family Income by Race, for all Families family Census S income White Black White $1000 3.0, 6.9' 1.5 -1999 6.4 13.0 5.5 -2999 5.9 9.9 6.1 -3999 5.8 10.4 5. 5 -4999 5.4 9.0 5.9 32.0 33.2 24.0 17.4 16.5 15.4 23. 4 20.1 12.1 5.8 $5000-7499 $7500-9999 $10,000-14,999 $15,000 or more 99.9 100.3 Total money Under $1000 $2000 $3000 $4000 RC Black S2.7 13.0 14.5 10.0 9.0 18.1 12.4 14.9 5.4 99.9 100.0 TABLF .? Comparison of Cenr u.s and SP.C Data: Pace by .'Various Demoqgraphic a.'ariables, for Families with Tw.o or More Members C Th IN u SRI Sexc: of family head White Black White Black MIale Fema.i i Size of family 89.7 10. 3 90.0 71. 7 9. 1 28.3 00 .0 100. 0 69. 1 3 0. 9 100.0 100.0 2 [personsS 3 persons 4 persons 5 persons 6 person s 7 persons or more l.imber of children One Three Fouilr or more- [lot ascertained .'. of f aimi y hI e, C ad under 25 35. 2 20.9 19. 7 12 .6 6.5 5. 2 28.7 19.9 16.1 11.7 14. 1 100. 1 100.0 142. 9 18.2 10. : 9.1 30. 5 10 .8 17.1 11. 1 21. 99.9 100.1 100. 1 99.9 Under 25 25-2 c 30-341 35- 1 3I5 5 .1 *15- 5.1 55-61 , 65- 7.1 75 and over 99.9 100. 1 Si.ae of place of r es i dence Son-metropolitan [ion-met ropol i tan 63.7 36 .3 99.9 99.9 63.3 7.5.9 36.6 24.1 99.9 100.0 72. . 27.4 100.0 100.0 35. 3 20. 3 19. 5 12.0 6.7 6.3 22.4 21.3 16. G 1 4. 7 : .3 16.6 29 . 19. 3 19. 5 10.6 10. 7 0. 2 100. 1 2 -1 5 1 9. 6 12.3 1.6 99. 9 6. 6 10. 5 .9 21.2 21. 3 16. 3 9 4 4 7 9.3 11. 6 11. 21.9 19.6 1 C 15.3 i. 3.E 7.7 11.0 9.o 2 23.2 20. 2 15. 5 9 .4 3.7 7.7 10. 2 12.0 23.2 21.2 12. 0 9. 9 3. 7 sample were families who at any time in the five year inter- view period lived outside the continental U.S. Also excluded were those families who at any ,time during the study either both owned and rented or di- not own or rent. This second group included families who received housing free from a friend or relative, or received housing as part of their wage. 'The final sample used for estimation consisted of 3,028 families, 1,225 of these rented in 1971 and 1,803 owned their o'...rn homes. APPECDBIN: 2 L:;E:TL'SIOI OF TIL' fICDEL TO T1CIE HULTI-PERIOD CASE In this secti.on I will assume there are only two differ- ent bundles of housing goods. One of the bundles can be owned, the otler rented. Ii .ight of chapter three, this assumption rmay' seem quite restrictive. It is not. The re- sults derived in this section apply with equal force to the more genectal case of several housing bundles. The f-l.lowing notation will be used. U = U(:-:,Ih is the household's utility function, where h is the .ow.' of housing services received per unit of time when the house is owned and < is dollars of e:.:pendituire .per unit of time on all other goods. U = U(::,hr) is the same as abo.-e, except the housing is rented. l, = h l. (t) is the flow of housing services per unit of time by an owned dw.'elling of initial size IB. The depreciation function B(t) has the following pro- perties: (0) = 1 6 (t)-0 as t",, and < 0. hl = h .(t) is the same as the above, but the initial size d,.,wellinlc, hI is rented. pB(t,ll) = per unit price of owned housing.at time L, p (t, IU) '..,here --- -- = PB (') 0. icLr-e B n PR(t,hp) = per unit price of rented housing at time t, where p' () 0. P. > Y(t) = family income receipts at time t. CB(hB) = transaction costs of buying a home of initial size h These costs do not include the time and money costs of moving. DCB(hB) _h - Ct(h ) > 0. Th B B B mb(lhB [t]) = transaction costs of selling a home in period t. Again, mb(') does not include the time and money costs of moving. 3m (h 6 [t]) h 6t = 6B(t)m (hb [t). > 0. k = time and money costs of moving. For any family, k is the same for all types of moves, That is, it makes no difference if the family; mo'.es from a rented house into an o(..ned house, or vice versa. f(t) = a probability density function of li'.ing in a gi'.en area for e:-:actly t years. Ef t) is defined o'er tlhe int:erv.al 0 : t .: t w.'here t is the m m household's time to retirement. Ech family is assul-Icd to possess such a densi t:y function. For s ir ii ci ty, I first assLue that the family can occupy only one d'.-.elling during their stay in any given area. This as sumpt ion w'.ill be relaxed later. Follo'..'inq lith (71) if %'e assume no time preference o'n the part of tie household, its utility aggregated for t years over the occupancy of an o.'ned d'ellitn of initial size h is 13S nnL-) = L1 :.:(.v) 1,h ] 1: B B Apart from its initial '-.ealth, the present discounted '.alue of the stream of wealth up to time t for the family who buys h1 i . i i :(t) y( h BPBI. ,h)6 B(')e~ 'd. C (hl -It mb(h 13B1 [t]"e where i is the household's discount rate. In selecting an owned dw.'llin characterized by initial size hp, assume the family maximizes (1 C (I) = i'm U (t) f (t)dt. The maximization is subject to the constraint that expected v.'ealth when the family moves from the area is equal to 1:, or, t (2) C('..') = I m ...t)f(t)dt = Y h X T = k, 0 D D where: t t -1 y = m I y ,., e- d.]f (t).dt, 0o o BP = l'm [ tpB V 'hB )"B( ie "' "]f(tdt, X = f m [/ xt:('7)e de f (t).dt, 0 0 T = C (h ) fm [m (hBC (t))e -i lf(tdt. Let L = EfU) + E (W) where is an undetermined multiplier. The household selects the function x(t) and h so as to maximize L. The function x(t) must satisfy f(t) t U L.'))d.' nf(t) t e df' = 0 o A 0' and upon differentitiatinq with respect to t -it (3) U (t) = ne :\ Differentiating L with respect to hg yields t (4) / [f 0 Uh B (v)dv]f(t)dt (4)m [ft (pB(Vl1) 6B(V) 0 0 i dfc b + hgp (v,h )(p (v))e dv]f(t)dt t-it {C (hB [r (i) + B6 (t)) 6B(t)e ]f(t)dt} = 0 0 B I B Dividing by p we get Ii , m [.i t b 0 0 .: ('') ]f (t)d t / m It (p ('.',h ) (.) + h p (vh )6 ('.'))e d'.']f(t)dt t -it C' C h ) + m (mr n h (t) ) (t)e- ]f t)dt, or, substi uting u =- and combining terms, ..t -h %. - P,B (5) .-m [-- - IC (.) ] ('.v)e 'dv )f (t)dt L t -It S ,-, U _t L. = C (h ) + 1 (hp [m (t)), (t)e ]f(t)dt, ',here I1C ('.') = P ('' h ) + h P- ('.',h ), the marginal cost at time '.' of buying a larger house of initial size h. The cerni insideL braces in (5) is the '..'eighted average ('weighted by ('.'.e.. ) )of the difference between the maL-inal rate -P - of cubstitution of housing for other consumption and the relati'.'e (to other goods) marginal cost of ov.ned housing ser'.'ices. The right hand side of (5) is the e::pected pre- sent discounted marginal transaction cost of buying (and selling) h -. The higher the probability that a family will ino'.'e in the near future, the higher are these trans- action costs. These transaction costs do nothing more than shift to the left Chapter IIl's "consumption possibilities frontier for o'.ning." When (3) and (5) hold, the family maximi :es (1). Let the optimal solution to (3) and (5) be E(U*) where t m ( ) ) = (I*) I'* (t) f t)dt. b n b If the family chooses to rent, they max:-imize (1) subject to (7) E U ) = Y P = S m t -i pR = [..t p '. (v I .")e ]f(t)dt n r, P. Again, the function :(t) must satisfy (3). The optimal hR must satisfy t h (8) / I [-- I C (_.'I 1. ('.'e d'.'d f(t)dt = 0, 0 0 U R R where 1CR(') = Pp('',hp) + h pN ( ',hR ), the marginal cost at time :. of renting a larger apartment of initial size hp. Let the optimal solution to (3) and (8) be C(l~') where r (9) CI(U* = i m U* (t)f(t)dt. r ,1 r *The farnil y '..'ll choose to own if (6) is larger than (9) Simi larly, thle' will rent if (9) is larger than (6). In qcneoral this choice will depend on U' (x,hb) U(x,h ) 6B(t), 6R(t) P (-), Pp(-), y(tl, C (hi ), m (h (t) ), f(t) and the families discount rate i. Ceteris paribus, a higher T or a higher f(t) in early pcriods will slant a family's choice to,.ard renting. This is consistent with the "mover" variable discussion in Chapter IV. Prices and preferences enter the rent-own decision in a manner similar to Figure 1 in Chapter III. Up to now I have assumed that the family must occupy the initial dwelling until they move from the area or retire. More generally, there will be some optimal number of moves. That is, it may be optimal f6r the family to first occupy a rented dwelling of initial size hR(0), and at some time, .tl, move to ah owned home of initial size h (1). Conversely, they may want to own first and rent second. If we assume that there are only a finite nuLmblr cf moves possible, then the family must choose mcv.-in periods t t, . t (t t m and h so as to rnma.;inmize Li t L. t I 1 t, t t, + I. U (:(v),h('.'v j_.' f(t .dt + . . t t -t.t 1 1 2 SI [r lm L (v ) h 0' Jdv I] f ( t)dt t t .'t.t In nl m .'herl e l(vi, is either h, or h .r The nmax;imi nation is subject to the cons'-taint that expected w.'ealth when the family moves from the area is again a constant I], or (11; [EU = :. (h(1)P(1 + . h(n P(n)] TTC = K t t t _ hl(llP( 1 = r p (v,h ( ) )6 (vie d'. If t )dt, t -. t t , (2 PI = i p(v,h(2 (v)e dv f (t dt, etc. tl tl 2 TTC = total transaction costs of all moves. These costs depend on w.lhen the moves take place and the type of move. Also, hi, p, 6 are either h'- p 3, 6 or hp, pp or ( i. )Whi le the solution to (10) and (11) is not straightforward, in general the- optimal strategy will depend on prices, pre- ferences and the density function f(t). For example, if the optiimal strategy is t tm (and therefore, L,. t2, S. all equal zero), then the solution reduces to (3) and (5) or (3) and (8). In this section I have presented a multi-period model of housing demand. It extends :;uth's model in four important ways. First, I hnve allowed for t.w.'o homogeneous bundles of housing services rather than cone. Second, iluth assumes the price of housing is constant. In my model price is a func- tion of quantity. Third, I e::plicitly distinguish bet..ween mov incr costs for owners and renters. :1uth does not. Fourth, I allo'-' the fara.ily to plan mo'.'es while still liv.inq in a gCiven area. .luth treats moves "within the area" the same as moves "away from the area." Therefore, in his paper each time the household mo'.'es prior to its retirement, it repeats the decision process under the conditions then prevailing. In other %-.orcs, flulth does not allow for the optimal moving strategy implied in (10) and (11). REFERENCES Amemiya, T., "Multivariate Regression and Simultaneous Equation Models When the Dependent Variables are Truncated Normal," Econometrica, 42, (1974). Carliner, G., "Income Elasti'city ofHousing Demand," The Review of Economics and Statistics, LV ( .o.' 1973). Ciragci, J., "Some Statistical fl1oels for Limite d Dependent '.'.ar1 i i].es .-.ith Appl ication to the Demand for Durable Goods," Lccncmetrica, 39 (Sept. 1971). de Leeu',.', Frail:, "The Dcrimand for Housinq: A Pe'.ie'.-.' of C-oss_-Sect icn E idence," The PF'e.'e'.' of Economics and Statistics, LIII (Feb. 1971). Fentor', C. "Vhc I'erimInent Income l'ypr.o hesis Source of Incrme and the Demand for Pental liousinq," in iiL iC,.nt .er for r1.rban Studi s, ,',nal'.-sis co Selccectd Cenisus Wleirre Pr.'e o i api Data to Dcterriiinie I laticn of iiouselr_'l d Characteristics, iiousinQ lirlLet Characteristics, and Adiiii.nistra- Li'.'c t l.'c fa Policies to a Di.eCCt 1lousinq Assist- ance [-rog.an. (Draft -- Final Peport, mimeo-raphed, Jul' 31, 1974) . Freund, J., Nlatherniatical Statistics, 2nd Edition, Prentice- ila11, Inc., Engle..'ood Cliff, [I.J., 1962. Ileclian, J. l '..' Pr ices, flar. et Wa.Tes, and Labor Supply," Lcocnometrica, 42 (July, 1974). King, A. T., "The Domand for Housing: A Lancastrian Approach," Southern Economic Journal, '.ol. 43, October 1976. Lee, Lung-Fr.i, "Cstimation of Limited Dependent Variable rrldels by' T'-.o Sta.ig e Ilcthlods," U npubl islied Ph.D. .Dissertation, Un .'iersity of Rochester, September, 197 . Lee, T. i. "liousinq and Poermanent Income: Tests Based on an [hree-Y-ear Peinterv'ie..' Sur.'ey," The Revie'.. of Ecoinomics and Statistics, L (1968). rad:dala, G. S. and Ielson, [-. "Maximum Likelihood Methods for the Estimation of Models-of Markets in Disequilibrium," Fconometrica, 42 (Nov. 1974) McCall, J. ,1. and Lippman, S. \A.,-"The Ecbnomics of Job Sea rch: A Si.irr ." 'Discussion paper 55, UCLA, Department of Economics, February 1975. Mouth, R., "The Deninnds for [Honfarm Housing," in Harberger (Ed.) 'i'l. Demand for -Durable Goods, Chicago: Uni-ersity of Chica.o Press, 1960. MIuth, R. "Mo'.ing Cocts and Ilousing'Expenditures," Journal of Urban Economics,%Vol. 1, January 1974. OhlI, J., "A Cross Section Study of Demand Functions for Housing an:] Policy Implications of -the ,Results," lIniv1c-rsity of Pennsyl'ania 'Ph.D. Dissertation, 1971. Olsen, C. 0. "... Comrpetiti'.'e Theory of the Housing Market," ,Timeric-jn Economic Pe'ie'..', September 1'969. Poiricr, D., "The Determinants of Home Buying in the New .Jerse.y Graduated Work Incentive Experiment," r.aciilty Working Paper, University of Illinois at Urbana Champaign, December 1974. F'olinsky, A'., "The Demand for Housing: A Study in S[,;pcifical-ion and Grouping," Discussion Paper rHumbI-cr 432 Hiir'.'ard Institute of Economic Research, HIar'.'aLrd Uni'.'ersity, Cambridge, Massachusetts, Septemuer 1975. Quigley, J., "Housing Dem.nd: in the Short Run: An Analysis of Folytomous Choice," E:-:plorations in Economic Research, pp. 76-102; '.'ol. 3,-No. 1, Winter 1976. Reid, M., liousing and Income, Chicago: University of Clucago Press, 1962. Sur'.'ey Rc-scrch Center, A Pancl Study of Income Dynamics, '.Vol. ues I and II, Institute for Social Research, University of Michigan, 1972. Tobin, J., "EC. tijn l-ion of RelationshipS for .Limited Dependent r.',riables," 26 Econometrica, (1958). Winner, 1., "llou.isinq and Income," Western .Economic Journal, June 1968. BIOGRAPHICAL SKETCH .Robert P. Trost was born on June 7, 1946, in Rochester, S'..' York. le g. rld uated fro:nm .'.uinas Institute Hi. h School in June, 19,I' lie received a B-chelor degree in Mlechianical iEni ineer inii fi-rom the UniveLrsity of Detroit in May, 1969. . ipon ..ra iatingq, le ser-"ed in the United States Marine Corp i :,s f[ i co n June, 19'9 t o A ril, I1971. In M-arL-. ch, 972, Mr. Trost enrolled in the GradJuate Scliool c:,! tlie LUni"ersity' of Flori.ida to pursue doctoral stud ies in the Department of Economics. He has held research and teachiniii assistantshi ips in the Depar tment of Economics from Tune, 1973 to March, 1977. I certiFy thlit I h.a"e r:1 l this study .and that in Imy opinion it conforms to accep'Jable standards' of scholar 'r , resen-l. nation and is fully a rd.quate, in scone and] quality, as a dissertation for the rdeiree of Doctor of Philosophy. G.-S. S l dda la a, Clha irma n Professor of Economics- I c: r t i f that I Iha.'" t.eir this s ttrlyd an-l thiat in m,' opinion it conforms to acct.ptable s tandrarlds of scholar) ly uree. n tat ion and is full'," .-,:'adequ1at, in scope- and quality, as a r di.sert.ation for the degree of Doctor of Philosohy. Pobert D. Emerson Assistant Professor of FooCd and Resource Ec onomics, IFAS I ccirt if. thlit: I h1 .'-1 re-ad thiis st ud-', arid that in my opinion it confo'rmr to arceptiblc r standards of scholarly presentation and is fully. adequate, in scope and quality, as a di sertation for the de.q--ee of Doctor of Philosophy. S'.,., 1. '. I i i. l .1ii,,_ Je'iome WI. Ilill.iman Professor of Economics I certify thLnt I ha'.'e read this cturl, and that in my oniniron it confcrnm to acceptable standards of -cholarly ni-esentation an-d is fully adequate, in sccoe and qualityt, as a diss-rta.tion For the degree of Doctor of Phil os;nrhy. Sa','mnd B. Roberts Associate ProfessorL ocf Economics This dissertation was submitted to the Graduate Faculty of the Department of Economics in the Collecge of Business Administration and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. March, 1977 Dean, -Graduate School |