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PAGE 1 1 HYPERBOLIC TANGENT YIELD FUNCTION OF FLORIDA CITRUS By LAN CHENG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Lan Cheng PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS First, I would like to express my deepest a ppreciation to all my committee, Dr. Charles Moss, Dr. Thomas Spreen, and Dr. Mark Brown. I thank my committee chair, Dr. Moss, for his understanding, guidance and support throughout my masters study at the University of Florida, and my other committee members for their advice, valuable ideas and kindness. I also thank my fellow graduate student in the Department of Food and Resource Economics for their support, suggestions and all the unforgettable memories. Finally, the greatest thank you goes to my parents for thei r loving encouragement and support, which motivated me to complete my study. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ..............9 CHAPTER 1 INTRODUCTON...................................................................................................................10 Overview....................................................................................................................... ..........10 Random Events Impacting Citrus Yield.................................................................................11 Economic Response to Risks..................................................................................................12 Problem Statement.............................................................................................................. ....15 Objectives..................................................................................................................... ..........16 Hypothesis..................................................................................................................... .........17 2 LITERATURE REVIEW.......................................................................................................18 Discussion on Crop Insurance................................................................................................18 Yield Function for Perennial Crop.........................................................................................21 3 MODELS AND PROCEDURES...........................................................................................25 Optimal Control Setup.......................................................................................................... ..25 Yield Function of Florida Citrus.............................................................................................27 Spatial Autoregressive Model.................................................................................................28 Data Sources................................................................................................................... ........31 Estimation..................................................................................................................... ..........33 4 EMPIRICAL RESULTS AND DISCUSSION......................................................................35 Estimated Yield Function.......................................................................................................35 Deviation of Yield............................................................................................................. .....36 Test of Normality.............................................................................................................. ......37 Comparison with Results without Considering Spatial Correlation.......................................38 PAGE 6 6 5 SUMMARY AND CONCLUSTION.....................................................................................46 Summary........................................................................................................................ .........46 Implications................................................................................................................... .........47 Limitations of the Study and Sugge stions for Further Research............................................48 APPENDIX A SIMULATION MODELINGPROGRAMMING BY GUAUSS...........................................49 B COMPARISON OF SPATIAL CORRELATION BY GUAUSS..........................................56 C SPATIAL WEIGHT MATRIX W........................................................................................62 LIST OF REFERENCES............................................................................................................. ..64 BIOGRAPHICAL SKETCH.........................................................................................................67 PAGE 7 7 LIST OF TABLES Table page 41 Parameter estimates and relative statistics.........................................................................39 42 Final estimation........................................................................................................... .......39 43 Estimated boxes of fruit per tree by age group from USDA.............................................40 44 Skewness and kurtosis of disturbance term.......................................................................40 45 Parameter estimates and relative stat istics without spat ial correlation..............................40 PAGE 8 8 LIST OF FIGURES Figure page 41 Average yield curve of early and midseason oranges........................................................41 42 Average yield curve of valencia oranges...........................................................................41 43 Average yield curve of white seedless grapefruit..............................................................42 44 Average yield curve of colored seedless grapefruit...........................................................42 45 Deviation of total yield of ear ly and midseason oranges by counties................................43 46 Average yield curve without spatial co rrelation of early and midseason oranges.............44 47 Average yield curve without spatia l correlation of valencia oranges................................44 48 Average yield curve without spatial correlation of white seedless grapefruit...................45 49 Average yield curve without spatial co rrelation of colored seedless grapefruit................45 PAGE 9 9 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science HYPERBOLIC TANGENT YIELD FUNCTION OF FLORIDA CITRUS By Lan Cheng August 2007 Chair: Charles B. Moss Major: Food and Resource Economics This study models Florida citrus production as a function of th e age profile of a given tree stock. The age relationship is estimated using a modified hyperbolic ta ngent function and the parameters are solved by Spatial Process Models and Maximum Li kelihood approach. The estimation is based on the production data of four citrus varietie s in 25 regions of Florida from 1992 to 2005. The results show smooth Sshaped yi eld curves of Florida citrus. This analysis offers yield function of citrus which provides useful informa tion for the design of citrus insurance program and the valuation of citrus trees. PAGE 10 10 CHAPTER 1 INTRODUCTION Overview Florida is an important supplie r of citrus products to both U. S. and world market. Florida provides most of the worlds fresh and processed grapefruit products. In addition, Florida is not only the largest citrusproducing st ate in the United States; it is also the second largest citrusproducing region in the world followi ng the state of San Paulo, Brazil. However, the Floridas citrus industry faces major challenges fr om citrus diseases such as citrus canker and citrus greening and other natura l calamities such as hurricanes and freezes. As a result, there was a significant decrease in Florida citrus production from 292 million boxes in 20032004 to 169 million boxes in 20042005, a 42.1% decline. To determine losses to diseases and other natural calamities, for example for insurance purposes, this study analyses citrus production and tree yields by formulating an Ssh aped treeyield function by age of tree and incorporating the spatial dimension of citrus yields across growing areas. In order to determine producti on losses to diseases and weat her events, this study estimates a yield function for individual citr us trees focusing on the age prof ile of the yield function. Citrus trees (like most trees) tend to yi eld different quantities of fruit as the tree ages. Initially, the yield of the tree is fairly small as the bearing volume of the tree (the area of the tree canopy where fruit sets) is small. As the tree starts to age, the yi eld increases at an increa sing rate as proportion of the bearing volume of the tree incr eases. However, as the tree appr oaches maturity, the growth in the trees yield slows down. In some cases, the bearing area of the tree may reach a sustainable maximum, but in other cases the yield may actuall y decline. This growth pattern is sometimes referred to as an Sshaped or sigmoid yield function. Further, this Sshaped functions is important when modeling the impact of di seases and weather on citrus yields. PAGE 11 11 Random Events Impacting Citrus Yield Citrus diseases such as c itrus canker and citrus greeni ng are significant concerns for Florida citrus industry. Citrus ca nker is a bacterial citrus diseas e that causes premature leaf and fruit drop. Remaining fruit can be unmarketable or much less valuable. C itrus canker is highly contagious and has several means of transmissi on. It is mainly spread by human contact and wind driven rain. These human and weather processes may possess a spatial nature of transmission. The disease can spread from 12 to 3474 meters in a period of 30 days. Most commercial citrus varieties in Fl orida are susceptible to citrus canker, especially lime and grapefruit. Citrus canker has been a periodic problem affecting Fl orida citrus since the early 20th century. The most recent out break in residential citrus trees originated in 1995, and since then citrus canker had spread into 24 Florida counties by 2006. Citrus greening (also known as huangblongbing or yellow dragon dis ease) is a bacterial disease that reduces citrus production. This disease is spread by a leaf feeding insect known as the Asia n citrus psyllid. Since psyllids appear to prefer new growth for feeding, citrus greening will likely have a larger affect on young trees. Previous experience with citrus greening in Asia suggests that this is a potentially devastating disease for citrus. Citrus greening was found near Ho mestead of Florida in 2005 and then was detected in a commercial grove of DeSoto County in 2005. Many of the commercialproduction areas of Florida have since reported this disease. Further, while most crops are susceptible to weather events, hurricanes produced a huge effect on the citrus industry in Florida in 2004. Sp ecifically, Floridas citrus groves were directly affected by three hurricanes in 2004 (Charley, Fran cis, and Gene) (Figure 1). Of these storms Charley and Francis had the most severe implicat ions for the citrus indus try. In the late summer of 2004 hurricane Charley struck the main citrusp roducing region of the state for the first time PAGE 12 12 since 1962. This storm along with th e other two storms following it spread citrus canker into a large portion of the commercial ci trusproducing area of the state. Earlier, another notable challenge was the seri es of freezes that struck the industry in the 1980s. Both bearing tree numb ers and production declined by 40% between 1975 and 1986 as freezes destroyed a large portion of the industry in Lake, Orange and Pasco counties. The two most serious Florida freezes during this pe riod occurred in December 1983 and January 1985. Together, they are estimated to have killed a pproximately one third of the state's commercial citrus trees. Freezes are still poten tial threats to Florida citrus but migration of the industry southward has reduced the probability of a devastating freeze to some degree. Before January 10, 2006 FDACS implemented an eradication program which destroyed trees within 1,900 feet of a tree infected with citrus. At th e same time the USDA provided compensation to all eligible citrus growers in Fl orida for losses resulting from the citrus canker eradication program. However, the hurricanes in 2004 and 2005 spr ead citrus canker in Florida so extensively that it was determined on Ja nuary 10, 2006 that eradication was no longer a scientifically feasible option and the practice of destroying tr ees was stopped. Similarly, in 2005 APHISUSDA announced it would not pursue an er adication program for citrus greening. The strategies to deal with fre ezing include moving south, adopting microsprinkler irrigation, and planting new varieties and rootstocks. Economic Response to Risks While several agronomic and mechanical adva nces have been suggested for protecting Floridas citrus industry against natural risks, they have not pr oved effective on the large scale. As a result, several contractual mechanisms have been developed to manage risk associated with diseases and weather, including insurance programs and disaster aid programs that are designed to reduce risks due to losses of citrus production and citrus trees. An overview of programs that PAGE 13 13 provide protection due to yield lo ss is provided below. Programs also exist that provide income replacement when prices are low. Initially, the USDA offered a lowyield di saster assistance program (DIS) to crop producers. This program provides assistance fo r losses that result from drought, flood, fire, freeze, tornadoes, pest infestation, and other calamities. Under disaster assistance program producers are eligible for disaster aid if harvested yield is less than or equal to a yield guarantee level of the program yield which is set by th e Agricultural Stabili zation and Conservation Service (ASCS) or of the expected area yield wh ich equal to an area's historical mean yield. Another source of disaster aid is emerge ncy loans offered by USDA s Farm Service Adiministration (FSA) to help producers recover from producti on and physical losses due to drought, flooding, other natural disast ers or quarantine. Florida citr us growers were eligible to receive disaster aids for fruit or tree loss due to hurricane in 2004 and 2005 and citrus disease such as citrus canker. For example, the FSA 2005 Hurricane Citrus Program (CP) provided financial assistance to Florida producers w ho suffered citrus crop production losses and associated fruitbearing tree damage that result ed from Hurricanes Ophelia, Katrina, Rita and Wilma in 2005 (FSA, 2006). However, the maximu m combined benefits producers can receive from the CP is only $80,000, which is relative a sm all amount compared to the value of a citrus grove. In 1980, the principal form of crop loss assi stance in the United States was provided through the Federal Crop Insuranc e Program. Congress continued to provide intermittent disaster assistance to farmers via direct payments and em ergency loans. One of cr op insurance policies is an Actual Production History (APH) policy. Under this policy, crop insura nce yield is based on expected yield which is estimated by the APH for each producer. APH is a simple average of PAGE 14 14 from four to ten consecutive years of actual yi elds based on farmers production records. Once the insurance yield is establishe d, the yield guarantee is calculated as product of insurance yield and yield guarantee level. Then indemnity paym ent is calculated as the difference between the yield guarantee and farmers actual yield multiplied by the indemnity price that the insured choose from several elections. Premium rates ar e based on historical yields and the production and loss history of the county in which the farm is located. In 1993, Congress authorized area yield insura nce as a pilot program and FCIC began offering a new insurance program, the Group Risk Plan (GRP). Unlike APH policy, GRP is based on an average areayield loss (insured and uninsured farm ers) with no individual loss calculation. The probability of collecting an inde mnity payment is the same for all insured farmers in the area, which means indemnity pa yment is determined by the difference between average historical arealevel da ta and the actual arealevel yield. Individual farmers cannot influence the indemnity payment they receive by altering production and/ or harvest practices. Accurate farmlevel yield data are not needed to actuarially determine insurance premiums. For Florida citrus, insurance policies should consider the characteristics of life cycle production of perennial crops. C itrus is a perennial crop whos e production activities involve planting, removal, yield and time dimensions not similarly encountered in annual crops. Perennial crop production is dis tinguished from the production of annual crops such as corn by (1) the long gestation peri od between initial i nput and first output, (2 ) th e trees of perennial fruit crops are not uniformly productive over its be aring years, and (3) eventually a gradual deterioration of the productive capac ity of the plants is realized (French and Matthews, 1971). In the case of citrus, the trees could remain produc tive well beyond 50 years and do not bear fruit for the first two to three years af ter they are set in a grove. Generally by the fourth year there is PAGE 15 15 some harvestable production. Yield per tree grad ually increases until some threshold when potential production reaches a maximum, ultim ately declining if the tree is not removed. Furthermore, and weather and disease events may affect output across several production periods. For example, the freeze events Florida experienced in the 1980s resulted in not only crop damage (i.e. oranges), but al so tree damage. This tree damage implied a significant loss in production due to the Sshaped yield function discussed earlier. Based on citrus trees special production pattern, the USDA provi des citrus fruit insurance policy and citrus trees insuran ce policy which share certain co mmon characteristics with above policies but keep their uniqueness. These two ci trus insurance plans are independent from each other. Florida citrus fruit policy adopts Dollar Amount of Insurance (DOL ) plan that estimates yield guarantee based on arealev el historical data and deri ves indemnity by the difference between yield guarantee and harv ested individuallevel yield. Tree Based Dollar Amount of Insurance (TDO) policy is provided to insure Flor ida citrus trees. This policy set tree values which are different by citrus type and age of the tree to calculate yield guarantee. So far, tree value is based on three stages: 5 years, 68 year s and 9 years and above. However, the details on the process of the tree valuation could not been found from previous literature and government documents. Problem Statement For government risk management programs including the widelyused APH and GRP together with DIS, DOL and TDO currently Florid a citrus rely on, the measurement of expected yield and tree value is a vital issue which determines yield guar antee, indemnity and premium. Either APH or GRPs yield guarantee is simply an average value of historical records, just different in individual level or area level. In DIS, DOL and TDO exp ected yields are also assigned based on past records of yield. These methods could not reflect changes which happen PAGE 16 16 in that insurance year like tree growing and pl ant pattern altering. Especi ally for citrus trees, neither of policy reflects the cha nges of age profile of citrus, si nce the ability of production is increasing and the value of trees alters as citrus is growing. In addition, the rough classification of tree ages and setting of tree values in TDP could also be doubted. However, a more accurate and reliable estimation of tree value can be re alized using yield function under age profile. Therefore, how to simulate the yield function of perennial crop appropr iately is the focus of this research and should also be of interest for policy makers of in surance program and other government assistance program. Objectives The first objective of the analysis is to provi de a descriptive framework for analyzing the factors that impact the yield of Florida citrus and produce appropriate yield curve that match with observationbased assumptions. In particular the focus of the work is to describe the response of the average yield of Florida citrus pe r tree to age of trees, one of these factors. The second objective is to examine the spatia l effect produced by factors from contiguous counties on citrus yield, given ra ndom events special na ture of transmission. In order to meet this objective, this research uses a spatial stru cture which combines with information on citrus yield. Third, the study plans to generate useful information for identifying the probability distribution of yield of Florida citrus. Specifically, the objective is to estimate mean of the distribution, assuming that average yield follows normal distribution. The last one is to identify other factors that could be added into the model in future study to refine the function. PAGE 17 17 Hypothesis 1H: The form of the ageyield relationship follows an S shape. This assumption is made based on the characte ristics of perennial crop production. Casual observation in the case of citrus suggests that: during the early bear ing years, growth is relatively low, and changes from one year to the next are small; at some point, however, changes in growth increase but level off at some maximum yield; growth remains stable for a long period of time until at some age growth begins to decline (Z anzig, Moss and Schmitz 1998). Therefore, yield function with an Sshaped curve matches with the actual life cycle of citrus production. 2H : Spatial autocorrelation is sign ificant enough to affect estimation. We assume that there is a certain relati onship between the yields of two contiguous counties since productions in the same region tend to increase or decrease simultaneously. This relationship is estimated by spatial coefficien t that is defined between 0.1 and 1 although mathematically the spatial correl ation could be negative. When spatial coefficient is below 0.1, the estimated parameters could be inaccurate und er spatial autoregressive model (Livanis, Moss et al. 2006). Besides, spatial Autocorrelation only exists be tween two contiguous counties. Because the closer the locations of two counties are, the more signifi cant the spatial effect is, we make this assumption in order to estimate the major spatial effects and simplify the structure. At last, spatial autocorrelation is a ssumed to only exist in disturban ce term but not in the dependent variables. This is because the spatial effect in dependent variables will be included in spatial coefficient of disturbance term. This assumption made the structure easier to analyze as well; 3H : The proportional relationship between the to tal bearing trees and that at each age is constant for two successive years. PAGE 18 18 CHAPTER 2 LITERATURE REVIEW Discussion on Crop Insurance Economic research on crop insurance can be trace d at least as far back as Valgrens 1922 study of the private insurance market. The amount of research on crop in surance has increased dramatically over the past ten years, paralleling th e growth in the program itself. Most of articles recognized that current crop insurance has actuar ial problems which might be mitigated by more accurate and reliable measurement of insurable yi eld. However, there are no articles specifically discussing problems embedded in insurance progra ms for perennial crops that this study can contribute to. The General Accounting Office criticized the poor actuarial performance on expansion of coverage into new areas without having adequate data to rate ri sks which contributed to adverse selection problems and the difficulty in monito ring producer behavior which contributed to moral hazard issues. Adverse selection arises because producers who recognize that their expected indemnities exceed their premiums are more likely to purchase coverage than those whose premiums are actuarially high. As a resu lt, the insurers expected indemnity outlays exceed total premium income, and, in the l ong run, the insurance operation loses money (Miranda, 1991). Moral hazard occurs when an insu red producer can increase his or her expected indemnity by actions taken after buying insurance (Glauber and Collins, 2002). Knight and Coble (1997), and Skees and Reed (1986) establ ished that adverse sele ction problems arises because of the use of aggregated (typically coun ty) measures to estimate individual yields and rates as farmers with lossrisk s above the area averages compri sed an everincreasing proportion of the insured pool. Miranda (1991) notes that e fforts by the insurer to avoid these losses by raising premiums only result in a smaller and mo re adversely selected pool of participants; PAGE 19 19 besides, areayield crop insurance would cover only systemic individual yield risk but not nonsystemic risk. However, although based on farml evel data, APH also received criticism for moral hazard and high administrative cost. Good win (1993) concluded th at APH rating methods were flawed because they assumed the same rate for farms within a county that had the same mean yields. Skees and Reed (1986) summarized that the manner of de veloping premium rates and APH yields in the FCI (APH) program have the potential to cause ad verse selection. Except expected yields, Skees, Barnett & Hartell (2005) put forth another type of basis risk resulting from the estimate of realized yield. The errors in measuring realized yield can also result in underand overpayments. The most common prescription for mitiga ting above problems is by providing more accurate classification of insured into homoge nous risk such as. Goodwin (1993) proposed refinements in premium rate set ting techniques; changes in the t echniques used to calculate yield losses are recommended by Miranda (1991), Carriker et al.(1991), and W illiams et al.(1993); Skees and Reed (1986) analyze the relation betwee n the mean and standard deviation of farmlevel yield and suggest that levels of protec tion ideally should be tie d to some measure of variability (e.g., standard deviation). Perennial crop insurance including Florida ci trus insurance suffered from the same problems as other crops but also from special difficulties due to its unique perennial life cycle of production. As a result some m odification had to be made to better protect perennial crop growers. For example, USDA s Risk Manageme nt Agency (RMA) changed its procedures to enable perennial farmers to request a yield de termination from their crop insurance agent in instances of decreased production because of le gitimate adverse weather conditions. This is RMAs response to request of New Yorks a pple and grape growers to reconsider the PAGE 20 20 methodology for calculating APH for perennial crops. And RMA also noticed that, the cyclical nature of perennial crop producti on, where weatherrelated impacts can have negative production effects for several years, needs to be better recognized within the crop insurance program. The RMA also revised the insurance premium rate structure for Georgia (GA) and South Carolina (SC) peaches to reduce the incidence of adverse select based on the study of Miller, Kahl and Rathwell (2000) who suggested that the premium rate of GA and SC peaches should vary with the individual growers expect ed yield. They estimated the actuarially fair premium rates by simulating farm yield. The farm yield is explai ned by parameters of the statelevel yield, the difference between the yield of county and the st atelevel yield and the difference between the yield of farm and county. Unfortunately, this issue has not attracted mu ch attention in academic and literature on citrus insurance could not be f ound. Since citrus trees potential yields changes over years, the estimation of tree yield, tree value, the distribution and va riability of production are not as simple as that for annual crop. For annual crops the in surable yield is simply approximated by an average yield of past several years. But fo r perennial crops, this method would always underestimate the expected yield because perennial trees like citrus will usually produce more in next year than this year. Another important issu e is how to estimate trees value for perennial crop insurance. Usually, the olde r the trees, the more fruit and therefore more value produced. Once trees are damaged due to adverse weather or disease, trees at different age needs to be evaluated differently. Besides, we have to cons ider not only how much fruit they could produce in that year but also how much fruit each tree could produce in the future. Skee and Reed (1986) recommended that the levels of insurance pr otection should be tied to some measure of variability like standard deviation. The distribution of pere nnial trees yield is also very useful to PAGE 21 21 measure variability for the design of insurance policy. And the distribution could be set upon the mean given by yield function. In summary, the yield function of perennial crop such as citrus can be used for the estimation of yield, tree value and the distribution of pr oduction resulting in an more reliable and accurate insurance design for perennial crop. Yield Function for Perennial Crop The determinants of perennial crop yield func tions are discussed in the economic literature dealing with perennial, crop supply response func tions. In general terms, supply at any given point in time is determined by the stock of trees Much of the literature has focused on modeling the decision (both voluntary and involuntary) to remove trees and to plan new trees and the yieldacreage relationship. The inventory of trees is identified and it is substituted into some type of yield equation to generate an overall supply response. Prior to 1960 there were almost no attempts to estimate supply models for perennial crops. The earliest work could be tracked back to Fr ench and Bressler s s upply response model of lemons in 1992. They focused on the change in bearing acreage wh ich was given by the difference between the acreage coming into bear ing category and the acreage removed, under the assumption that the yield per acre is a constant average value. A lthough they also noticed that the acreage of new plantings was influenced by the age distribution of exis ting trees, their model only contained a ratio of bear ing trees at a given age. Batemen (1965) recognized the natural life cycle of perennial crops when examining the supply function of Ghanaian cocoa. He modeled the yield age relationship as k i i t i tX b Q ) (* (2.1) where tQ is the potential yield of coca in year t, i tX is acreage of coca planted in year ti, ib is PAGE 22 22 the potential yield per acre in year t of coca plan ted in year ti, and k is the age at which coca trees first begin to bear. This e quation described the pattern of yi elds over the life cycle of the perennial crop. Bateman also modified this general equation by groupi ng tree acres into two distinct growing phases as ) ( ) (2 1 1 s i i t s k i i t tX b X b Q (2.2) where s is the year in which the second distinct increase in yield occurs, under the assumption that the transition from one yield level to th e next is instantaneous so that in year 1 sthe crop yield per acre is 1b and in the following year it immediately changes to 2b. Similar work was done by French and Ma tthews (1971) who provided a more fully specified yield function as: it it H k i it i tv T b A a Y (2.3) where itAis the acreage in the ith age category in year t, k is the initial bearing age, H is a reasonable maximum age of the plant, T is time and v is a disturbance term. However due to a lack of data and a large number of variables involved in the e quation, French and Matthews also grouped individual ages into what se em to be arbitrary age groups. Baritelle and Price (1974) pr oposed a yield function similar to 2.3 but based on number of trees. Total quantity produced is equal to number of trees multiplied by yield per tree: i t i t i tA Y Q, (2.4) where Q is total production in year t, t iY, is yield per tree for trees in age group i in year t, and t iA, is the number of trees in age group iin year t. This equation allows differing yields PAGE 23 23 according to tree age. They also specified yiel d as a function of current and past weather condition, i.e., ) ,..., (1 n t t t itW W W F Y (2.5) where W denotes weather conditions in fluencing yield. However, it is difficult and complex to specify the relationship be tween yield and weather. Wickens and Greenfields (1973) specified a model that included a yield function similar to equation 2.3 assuming yield is a function only of the number of productiv e trees. But they did not directly estimate this function instead they substituted it into a reducedform supply response equation. The final supply response equation wa s estimated in a polynomial form that was criticized by Akiyama and Trived i on the ground that th e polynomial fails to reflect a realistic yieldage relationship. Akiyama and Trivedi (1987) noticed that the age distribution of trees and the total stock of trees were important in determining feasible levels of production. They modeled the supply function of a perennial crop from a twofactor production function. Total output ) ( t Q is defined by )] ( ), ( [ ) ( ) ( ) ( v t L v t K F v t q v t q t Qv (2.6) where ) (v t qdenotes production at time t using inputs of vintage v in the vintage production function F[.], ) (v t K denotes capital of vintage v used at time t and ) (v t L denotes the labor combined with K Capital primarily refers to the tree stock at time t of a particular age or vintagev. The variable labor means all noncapital inputs which are used in fixed proportion to labor. Total output for this model can be written as: PAGE 24 24 vv t K v t t Q ) ( ) ( ) ( (2.7) where ) (v t is the average productivity or yield per unit of capital, give n by q(t,v)/K(t,v) Following Akiyama and Trivedi, Kalaitzandonakes and Shonkwiler applied their model to Florida citrus for the first time in 1990. A common denominator of the liter atures listed above is that the yield age relationship is essential to determine the supply of a perennial crop in any given year; however, due to a lack of data, more elaborate modeling of this relations hip was not done. This problem continued until Zanzig, Moss and Schmitz (1998) developed a specific functional form for modeling Florida citrus yield with a rich data set of the age pr ofile and yield for a rela tively long time period. The simulated results shows an Sshap ed yield curve consistent with expectation, but two out of six parameters are not statistically significant whic h may imply some problems with the process of estimation. Their functional form is adopt ed by this study but process is modified. Another difference from past research is that the presented studys yield function is developed based on optimal control theory. Opti mal control theory is a mathematical field concerned with control policies that can be de duced using optimization algorithms. Kamien and Schwartz (1991) discuss various optimal control problems. In their book, variables are divided into two classes, state variables and control va riable, and the movement of state variables is governed by first order differential equations. PAGE 25 25 CHAPTER 3 MODELS AND PROCEDURES The main objective of this study is to estim ate yield function for various varieties of Florida citrus based on yieldage relationships. To achieve this object, we use a spatial model to measure the spatial nature of citrus yields and reduce the impact of spat ial autocorrelation on the estimates. The detailed data on Florida citrus prod uction and the tree age profile are used in the analysis. The maximum likelihood a pproach is adopted to estimate the parameters due to the desirable properties of ML estimators under spatial autoregressive model. The first part of this chapter discusses optimal control theory from which the yield function is derived. The second part describes the struct ure of the functional form we use to estimation citrus yields. The third secti on introduces the model used for the estimation, the spatial autoregressive model. Lastly, data sources and es timation procedures are di scussed in parts 4 and 5 respectively. Optimal Control Setup The simplest control problem is one of se lecting a piecewise con tinuous control function u ( t ), to 0 0 1 0 ) () ( )), ( ), ( ( ) ( )) ( ), ( ( max1 0x t x t t t u t x t g t x t s dt t u t x t ft t t u (3.1) where f and g are assumed to be continuously differ entiable functions of three independent arguments, one of which is a derivative; the cont rol variable u(t) must be a continuous function of time; the state variable x(t)changes over time according to the differential equation governing its movement. PAGE 26 26 We assume that the various decisions a gr ove manager makes follow the basic optimal control formulation. In this formulation, a ma nager determines the level of input usage that maximizes the expected value of profit through time. In the vernacular of optimal control, fertilizer and other variable inputs are the control variables which are varied to control the level of state variables thr ough time (Kamien and Schwartz 1991). In our formulation, producers optimize 0 0 0 ) () ( )) ( ), ( ( ) ( )) ( ), ( ( max k t k t u t k t g t k t s dt t u t k t FT t u (3.2) where )) ( ), ( (t u t k t F is the discounted profit function, ) (t k is the state function (the bearing area of the citrus tree), ut is the control variable (such as fertilizer), and )) ( ), ( (t u t k t g is the equation of motion which depicts the growth in bearing area over time Previous study has discussed a lot about grove managers decision making of planting. However, this study is interested in the consequences of the decisionsbearing area ) (t k. From equation (3.1), bearing area ) (t k can be derived as tds s u s k s g t k0)) ( ), ( ( ) ( (3.3) Further complicating our formulation is the fact that the bearing area of each citrus tree cannot be observed. Given this c onsideration, we start by focusing on the citrus yield that is a function of bearing area as PAGE 27 27 ) ( )) ( ), ( (t k t u t k t y where is a constant parameter. As a further simplification, we assume that the bearing area is largely a function of tree age ) ( ) ( v h k v t kv (3.4) where v is the age of tree cohort. Replacing k with h in)) ( ), ( (t u t k t y, the yield function y becomes a function of bearing age of trees as )) ( ), ( ( ) ( v t u v t k v t y v y yv (3.5) Yield Function of Florida Citrus Zanzig, Moss, and Schmitz (1998) recognized that perennial cr ops including citrus demonstrate the following production characteristic s: during the early bearing years, growth is relatively low, and changes from one year to the next are small; at some point, however, changes in growth increase but level off at some maximum yield; growth remains stable for a long period of time until at some age growth begins to decline. Based on this observation, we assume that the form of the ageyield relationship follows an S shape. The hyperbolic tangent function provides an ideal Sshaped functional form for modeling perennial crop production, and can be approximated with a transformed hyperbolic tangent function proposed by Zanzig, Moss, and Sc hmitz (1998). The formal representation of average yield is max max0101(,,,)(1tanh()) 2 y vv (3.6) where max ,0 and 1 are parameters to be estimated, max01(,,,) y v is the yield of each citrus tree, and vis the tree age. The transformed hyperbolic tangent function yields an Sshaped curve. The range of the hyperbolic tangent function is (1, 1). Thus, in our formulation, the range PAGE 28 28 of citrus yields ismax0, yv. 1 adjusts the relative slope and growth rate of the yield curve. Ideally, 1 should be a small value less than 0.5; otherwise yield function would reach max very fast so that the curve appears to be flat rather in creasing gradually. 0 shifts the sigmoid shaped graph. In its original formulation under which00 the hyperbolic tangent has an inflection point at0v. Hence, when00 the yield curve is convex (0 v); when00 the yield function depicts first a concave and then a convex curve while it increases at an increasing rate until 1 0 v and then increases, at a decreasing rate. Since, casual observation suggests that the growth rate for citrus yields, first increases and then decreases, the latter matches with expectation well. Unfortunately we do not have treelevel yield da ta. Instead, we aggregate the yield of each citrus variety over each county based on this formulation as max01 ,,,z ititv vayyvT (3.7) where it y is the estimated c ounty level production and itvT is the number trees of age v in county i at time t In this formulation, max01(,,,) y v is the expected yield for a particular age cohort. Spatial Autoregressive Model Given the yield age relationship and the spatial nature of se veral random events effecting citrus production such as freezes, hurricanes, an d disease outbreaks (inc luding both citrus canker and greening), the paramete rs of Equation 3.7 are estimated us ing spatial autoregressive models and the maximum likelihood approach. PAGE 29 29 The model begins with the nonlinear regr ession model with spatial autoregressive disturbances. We assume that spatial autocorrelat ion only exists in disturbance term but not in dependent variables. The spat ial structure is written as (,) yfx W (3.8) where y is the countylevel production fo r a given citrus variety, ) ( x f is the nonlinear model of citrus yields presented in Equation 3.5, are the parameters to be estimated, x is a vector of exogenous factors including the tree age v or the numbers of tr ees in each age cohort, is the spatial autoregressive coefficient, and is an identically and inde pendently distributed error term (2~0, N ) (Anselin 1988, Livanis et al. 2006). The W matrix is the spatial weight matrix that is determined by the specific locati on of counties. If two counties are contiguous the corresponding cell for th e two counties in the W matrix is set 1; otherwise, the cells are set to 0. In this formulation we usually assume that 0,1 with 0 representing the standard ordinary least squares model. It is mathematically possible for to be less than zero, it raises some empirical questions. Further, following the intuition from generalized least squares, we note that the true spatial formulation is always at least as efficient as ordinary least squares. However, estimated generalized least squares is not guaranteed to more efficient than ordinary least squares since estimating the heteroscedastic ity process introduces some error. Hence, if 0.10 we are more confident that adjusting for sp atial autocorrelation improves the efficiency of estimation (Livanis and Moss 2006). Based on Equation 3.6, we derive the error term wh ich is a function of production, spatial coefficient, independent variables and s as PAGE 30 30 1(,)(1) ()((,)) yfxW IWyfx (3.9) The likelihood function for the specificati on in Equation 3.7 can be expressed as 1ln(2)lnln 22 1 (1)((,))(1)((,)) 2 NN LIW WyfxWyfx (3.10) Maximization of Equati on 3.8 with respect to for a given yields ((,))()()((,)) ,MLEyfxIWIWyfx N (3.11) Substituting this result back into the loglikelihood function Equation 3.10 we get the concentrated likelihood f unction with respect to : 1 ln(2)lnln, 222CMLENN LIWN (3.12) Given that the eigenvalues of W matrix can be written as iw and that 1lnln()N i i I WIw (3.13) the final expression for the concentrat ed loglikelihood function is given by: 1ln(2)ln() 2 1(,)(1)((,)) ln 22N ci iN LIw WyfxWyfx NN N (3.14) Finally, we estimate this likelihood function for the same set of counties over several years so that the likelihood function becomes PAGE 31 31 1 1ln2ln 2 1(,)(1)((,)) ln 22N ci i R tttt tNR LRIw WyfxWyfx NRN N (3.15) where R represents years, and N is number of observations (counties) for each year. Data Sources The detailed record of Flor ida citrus production and tree ag e profile provides convenience for yield analysis. The data collected in this study came from two major sources: Commercials Citrus Inventory and Citrus Summary. These two documents present the resu lts of Florida citrus census survey conducted by USDA since 1966. We abstracted following types of data from above two documents: 1) citrus production by eac h variety, county and year; 2) citrus tree number by each age, variety, county and year However, since citrus census survey was conducted every two years, the tree numbers in odd years are not available. This research estimated the tree number in odd years based on Citrus Inventory of previous year and corresponding Citrus Summary. All of the data are used to estimat e parameters in yield function discussed in last chapter. The Florida Agricultural Statistical Servic eUSDA conducted the states complete citrus tree census survey as of January every two y ears since 1966. Ground crews update the survey by identifying all the states groves by fruit type, ro w spacing, and year. The design of the inventory survey allows for quick and ec onomical updating. Sin ce 1966, the citrus belt has resurveyed each second winter to determine changes. The resu lts of the census survey are presented in the Commercial Citrus Inventor y and the Citrus Summary. The Commercial Citrus Invent ory is published every two year s. The data in this report relate to commercial groves, those containi ng a minimum of 50 trees from which fruit is PAGE 32 32 generally sold. This report mainly contains th e number of acres and the number of trees by variety, county and year. This study uses the information on quantity of bearing trees by each age, county and variety from 19922004. The Citrus Summary is produced every year, reporting production by county and variety. As for the same season as commercial citrus i nventory, the data was given by complete citrus census survey; for the other seasons, citrus pr oduction was estimated by counties using objective survey data obtained from the citrus crop estimates program. Production for Florida has been distributed to counties based on th e biennial citrus tree census, lim b count survey data adjusted for drop page to end of season, and size data at maturity. We abstracted the total production by county and variety from Citr us Summary from 19922005. However, because the Commercial Citrus Inve ntory is published every two years while the Citrus Summary is reported every year, data on number of trees in odd years had to be estimated to match with data on production. Assuming prop ortion relationship between the total bearing trees and that at each age is constant for two su ccessive years, the quantit ies of trees in odd years can be estimated by Citrus Inventory of previ ous year and corresponding Citrus Summary. For example, for estimating tree number in 1993, first we mark trees as age from 0 and added total bearing tree number with the tree number at age 2 in 1992s C itrus Inventory, resulting in estimated total bearing tree numbe rs for 1993. Second, calculate th e ratio of total bearing trees which equals to total bearing tree number from 1993 citrus summary di vided by estimated total bearing tree number resu lted from last step. Third, the first 23 categories of bearing trees in modified 1992 citrus inventory were remarked as age from 1 to 23, and the rest after age 23 was aggregated as one group at age 24. The next st ep is multiplying ratio of bearing trees with number of bearing trees older th an 2 years which resulted from step 3, resulting in estimated PAGE 33 33 bearing tree number data for 1993 from age 3 to 24. Finally bear ing tree number are transformed into percentage of trees divi ding by total bearing tree number, as the same as production, for emphasizing the weight of trees at in dividual age among total bearing trees. The gestation period before the tr ee bears fruit is set from tree age 0 to age 2, which means that citrus trees start to bear fr uit from age 3. The range of yieldage profile is from age 3 to 24. After combining the production of Early oranges a nd Midseason oranges, finally we select four citrus varieties complete data in 25 Florid a counties from 1992 to 2005: Early and Midseason Oranges, Valencia Oranges, White Seedless Grapefruit, and Colored Seedless Grapefruit. The final data set does not represent all the counties growing citrus. This is because: first, citrus summary and citrus inventory ignored the coun ties with relative sma ll production and acreages; second, some counties had tree age profile da ta but production records missing and some had production but missed tree numbers. The share of s eedy grapefruit is relatively small so it is eliminated from data set. Usually the period of a data set has to be at least ten years to reflect certain problems or tendency, and the period co uld not be too long eith er since production is impacted a lot by technology which is quite differe nt today from 10 years ago. Therefore, this study collects data for past 14 years so that there are enough observa tions to find out the relationship and some influences of i rrelevant factors could be avoided. Estimation Given the share of treesitvTr, the estimated production was calculated by Equation 3.4 and 3.5 while unknown parameters s were assigned initial values for the first iteration. Under spatial process model we use maximum lik elihood rather than ordinary least square to estimate the model. This is because parameter estimates from OLS are inefficient and not consistent due to the nondiagonal structure of the disturbance varian ce matrix (Anselin, 1988) PAGE 34 34 but ML estimator achieves the de sirable properties of consistenc y, asymptotic efficiency and asymptotic normality (Andrews, 1986). The spatial weight matrix Wand its eigenvalue are determined by the specific location of counties. In this study, we assume that sp atial autocorrelation only exists between two contiguous counties because the cl oser the locations of two coun ties the more significant the spatial effect. Thus, the W matrix used in this study becomes (3.14) where the cells of two co unties are set to be 1 if the two counties are contiguous; otherwise the cells are set into 0. Plug estimated production it y and its observation vector as we ll as W matrix into Equation 3.13. The maximization of this equa tion generates estimated parameters s and We also set constraints on the seconddegree derivative of yield function with resp ective to bearing age v which is negative at tree age 24 and positive at 3. These settings make sure that growth of average yield increases fast at early beari ng years and slows down after certain point. 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 W PAGE 35 35 CHAPTER 4 EMPIRICAL RESULTS AND DISCUSSION The first part of this chapter presents the es timated parameters and yield curves of four citrus varieties considered in th is study. The deviations betwee n actual yield and estimated yield are then calculated for Early and Midseason oranges by county. Next, the normality of disturbance term is tested and a method is proposed to correct the nonnormality problem. Finally, the last section compares the result with the assumption that spatial correlation does not exist. Estimated Yield Function The empirical results show that citrus in Florida has an Sshap ed yield function and significant spatial coefficient The parameter estimates and related statistics for the yield function are presented in Table 41. White seedless grapefruits exhibit the larges t maximum average yield per tree which is 4.47 boxes and Valencia the smallest, 3.11 boxes. 0 s are all negative wh ich means that the minimum yield starts from value less than half of max Negative 0 also indicates that horizontal ordinate of the inflection points is a positive age and the yield curve starts with a concave. Small positive1 s reflect that yield curve is in creasing gradually. The spatial coefficients range from 0.48 to 0. 64 and all are statistically different from zero at the 95% level, which verify that the spatial correlation has si gnificant impacts on the estimation. Given that all the estimated parameters pass t test except 0 of white seedless grapefruits, the final estimated parameters are shown in Table 42. The shapes of yield function for each variety are presented in Figur e 41 through 44 which also shows the estimated boxes of fruit per tree by USDA. USDA estimated yields based on PAGE 36 36 official endofseason production estimates and the number of bearing trees indicated by the citrus tree inventory surveys. The estimati on done by USDA does not have enough points to show the shape of yield curve (Table 43). Hence we smoothe d their estimation and graph it together with yield curves obt ained by this study. Figures belo w show that the trend and bounds of two curves are almost the same and several critical points, such as minimum and maximum yield as well as inflexion point, coincide very well. The estimated yield functions exhibit an S shaped curve which increases with a positive second derivative during the early bearing years and once reaching a certain age the growth rate decreases rapidly and the yield levels off. For example, Fi gure 41 shows that early and midseason oranges estimated yiel d goes up rapidly with a concave curve until age 10. After that the curve becomes convex and then flat, endi ng with a maximum average yield of around 4.5 boxes. Other varieties have the similar charact eristics of yield curve except white seedless grapefruit. The reason for white seedless grapefru it nonS shape may be relative to the fact that few young trees of white seedle ss grapefruit were planted in re cent years so that the trees reflected by data are mature trees with high level of production. Deviation of Yield Although we assume that all the counties have the same yield function for the same citrus variety, deviations vary widely among coun ties. The average deviation for each county i were calculated as 14 1 22 1 1 0 max1 ) (it i it it it it itv itT y y T v y y (4.1) PAGE 37 37 Theoretically, i s should converge to zero, but Figu re 45 shows that some counties present deviations close to zero such as Hardee, Palm Beach and Seminole while others possess a large deviation like Hendry or one with an opposite direction like Hills borough and Glades. The deviations of average yield range from Hi ghlands 2,337.1 thousand boxes to Hillsboroughs negative 816.3 thousand boxes. The reason for this huge difference is various. One of the reasons may be the impact of tree density, because tree density is different among counties and it impacts average yield per tree. Other reasons might be re lative to the different locations of counties. The difference in deviation also indicates th at productive capacity of the same citrus variety varies among counties. Therefore, insu rance program requires a procedure which could fully consider the difference in productiveness. However, as for Volusia, Seminole, Palm Beach and other counties with very sma ll deviations, tree values and expected yield could be estimated directly by yield function derive d by this study. For the counties with large deviations such as Hendry and Highland, this yield f unction could be used as a base estimation which needs to be adjusted by other countyl evel information. Since 16 out of 25 counties have deviations less than +1000 boxes, this yield function is very useful for most counties In addition, almost all the deviations are posi tive, which means that yield function overestimates yields. One possible reason is th at hurricanes and citrus canker that reduced production from 2004 to 2005 are not included in the function. To improve this yield function, future work may involve adding weather a nd disease variables into the simulation. Test of Normality The research assumes that disturbance term follows normal distribution with zero mean. Plugging estimated parameters into equation (3.7) is calculated and it s skewness and kurtosis PAGE 38 38 are presented in Table 44. For testing normality of we use JarqueBera test proposed by Jarque and Bera (1980). The formula of the test is ) 4 ) 3 ( ( 62 2 K S n JB (4.2) where Sis the sample skewness, K is the sample kurtosis and n is the number of oberservation. Since JBis asymptotically chisquare distribut ed with two degrees of freedom and JBis of all varieties exhibit extremely high value (Table 44), it is rejected that follows normal distribution. The high JB values can be explained by the heavy right tails. Comparison with Results without Considering Spatial Correlation Spatial correlation is an im portant and necessary consider ation during th e process of estimation in this research. Specifically, assuming that spatial correlation does not exist by setting spatial coefficient to be zero, the estimators of yield function fail to pass ttest or their significance decrease (Table 45). max And 1 for early and midseason oranges as well as 1 for valencia oranges are all insi gnificant different from zero at the 95% confidence level. More important, Figures below show that estimated curves of yield without considering spatial correlation obviously diverge from USDA estimation a nd depict flat lines rather than Sshaped curve. Further residuals under =0 are much more skewed and kurtotic than that under0 All of these verify the significan ce of spatial structures for simu lating citrus yield further. In case of white seedless and colored grapefruit tvalues of estimators and the yield curves almost keep same as considering spatial correlati on which implies that these two varieties have much less spatial correlation. Oranges are pl anted throughout the south east of the state and concentrating in from Polk county down to Collie r county area, so that the planting regions have a lot of connections with each other. Grapefruit, however, ar e growing intensively in tree PAGE 39 39 separated regions, Polk, Indian River and Saint Lu cie, and Hendry. Therefore, spatial correlation does not have much impact on the estimation for grapefruit but does have that for oranges. Table 41. Parameter estimates and relative statistics Parameters Estimation Sta ndard deviation Tvalue Eearly & midseason max 4.472 0.247 18.142 0 1.778 0.653 2.723 1 0.235 0.102 2.300 0.640 0.048 13.255 Valencia oranges max 3.111 0.133 23.469 0 1.041 0.328 3.177 1 0.177 0.049 3.618 0.576 0.052 11.140 White seedless grapefruit max 5.476 0.229 23.960 0 0.463 0.370 1.251 1 0.139 0.064 2.162 0.556 0.056 9.920 Colored seedless grapefruit max 4.961 0.216 22.997 0 0.984 0.225 4.370 1 0.175 0.038 4.582 0.482 0.060 8.064 Table 42. Final estimation Parameters Early & midseason Valencia White seedless Colored seedless max 4.472 3.111 5.476 4.961 0 1.778 1.041 0.000 0.984 1 0.235 0.177 0.139 0.175 0.640 0.576 0.556 0.482 PAGE 40 40 Table 43. Estimated boxes of fruit per tree by age group from USDA Age of trees Parameters 35 years 68 years 913 years 1423 years 24 years and older Early and midseason 1.238 2.408 3.415 4.292 4.885 Valencia 1.115 1.931 2.400 3.046 4.046 White seedless 1.900 3.200 3.969 5.700 5.508 Colored seedless 1.923 3.100 4.000 4.554 4.962 Table 44. Skewness and kur tosis of disturbance term Early & midseason Valencia White seedless Colored seedless Skewness 11.481 8.628 9.842 14.241 Kurtosis 186.574 110.102 119.555 242.969 JB value 499138.9 171625.5 195615.4 851613.4 Table 45. Parameter estimates and relativ e statistics without spatial correlation Parameters Estimation Standard deviation Tvalue Eearly & midseason max 6.996 4.094 1.709 0 0.726 0.269 2.702 1 0.045 0.025 1.812 Valencia oranges max 3.333 0.835 3.992 0 0.126 0.136 0.921 1 0.052 0.047 1.091 White seedless grapefruit max 5.498 0.203 27.073 0 0.343 0.372 0.921 1 0.133 0.065 2.047 Colored seedless grapefruit max 4.843 0.213 22.755 0 0.792 0.183 4.334 1 0.157 0.035 4.536 PAGE 41 41 0 1 2 3 4 5 63456789101112131415161718192021222324Bearing AgeBoxes Estimated average yield Average of estimated yield by USDA Figure 41. Average yield curve of early and midseason oranges. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 3456789101112131415161718192021222324 Bearing AgeBoxes Estimated average yield Average of estimated yield by USDA Figure 42. Average yield cu rve of valencia oranges. PAGE 42 42 0 1 2 3 4 5 6 3456789101112131415161718192021222324 Bearing ageBoxes Estimated average yield Average of estimated yield by USDA Figure 43. Average yield curv e of white seedless grapefruit. 0 1 2 3 4 5 6 3456789101112131415161718192021222324 Bearing yearBoxes Estimated average yield Average of estimated yield by USDA Figure 44. Average yield curve of colored seedless grapefruit. PAGE 43 43 1000 500 0 500 1000 1500 2000 2500 3000 3500 4000Brevard Charlotte Collier DeSoto Glades Hardee Hendry Highlands Hillsborough Indian River Lake Lee Manatee Martin Okeechobee Orange Osceola Other counties Palm Beach Pasco Polk Sarasota Seminole St. Lucie VolusiaCounty1000 boxes Figure 45. Deviation of total yield of early and midseason oranges by counties. PAGE 44 44 0 1 2 3 4 5 63456789101112131415161718192021222324Bearing AgeBoxes Estimated average yield Average of estimated yield by USDA Figure 46. Average yield curv e without spatial correlation of early and midseason oranges. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 3456789101112131415161718192021222324Bearing AgeBoxes Estimated average yield Average of estimated yield by USDA Figure 47. Average yield curve without spatia l correlation of valencia oranges. PAGE 45 45 0 1 2 3 4 5 6 3456789101112131415161718192021222324 Bearing ageBoxes Estimated average yield Average of estimated yield by USDA Figure 48. Average yield cu rve without spatial correlation of white seedless grapefruit. 0 1 2 3 4 5 6 3456789101112131415161718192021222324 Bearing yearBoxes Estimated average yield Average of estimated yield by USDA Figure 49. Average yield curv e without spatial correlation of colored seedless grapefruit. PAGE 46 46 CHAPTER 5 SUMMARY AND CONCLUSTION Summary It is complicated to describe the production decisions for perennial crops due to several factors including the timespecific nature of yields. There is a proportional re lationship between changes over time in the yields of perennial crops and th e size of the trees in the case of citrus. This research provides a descri ptive framework for analyzing the yield age relationship for Florida citrus. In particular, the focus of the wo rk describes the response of the average yield of Florida citrus per tree to the age of trees. The yield and age relationship was derived from optimal control theory and desc ribed by transformed hyperbolic tangent function which depicts an Sshape curve. Further, sp atial effect produced by factors from contiguous counties on yield was simulated under spatial autoregressive m odel and maximum likelihood estimation providing clear advantages than OLS. Commercials Citrus Inventory and Citrus Summary of Florida citrus provided data base for estimation. The results indicate that yield curve depict s a clear S shape for Early and Midseason orange, Valencia Orange and Co lored Seedless grapefruit. The si mulated yield curves of these varieties coincide very well with approximated true yield curve. The possible explanation for the convex shape of White Seedless grapefruit may be involve even more significant influence produced by hurricane and citrus canker on its yiel d. Estimated parameters were tested by ttest resulting in being significantly di fferent from zero. From the result s, it is also indicated that spatial effects could not be ignored in the process of simulation. A ll estimated spatial coefficients are significant and larger than 0.4 within 1, whic h could include spatial effects of factors such as weather, temperature, catastrophic events, and human mobility. And the results were compared PAGE 47 47 with those without considering sp atial correlation, which further verify the importance of spatial structure in this study. Implications This study provides the basic stru cture of citrus yiel d function that would be helpful for analyzing impacts of random events such as fr eezing, hurricanes and di sease outbreak including citrus canker and citrus greening. Take citrus canker for example. Citrus canker mainly was not a severe problem that impacted citrus production un til 2004. However, because infected citrus will not be eradicated any more, citrus canker can be added into model as a f actor impacting betas in form like(1) Z whereis canker coefficient and Z is index of citrus canker. Another factor impacting citrus yield is tree dens ity. Brown and Spreen (2000) not ed that historic boxes per tree may be higher than future tree yields due to increasing tree de nsities, so that the projections based on tree yields may ove rstate future production. The unique perennial life cycle of production is very important for Florida citrus insurance program designing. The yield func tion could describe the characteristics of the life cycle and reflect the change of age profile and plant pattern. Hen ce, the results would help to accurately estimate expect yield and tree value which are cr itical variables in insu rance program design. The yield function is also useful to generate the conditional probability density function of Florida citrus average yield. The mean of the di stribution has been estimated by this research and its variance may be of inte rest for future study. The difference in deviation i ndicates that productive capacity of the same citrus variety varies among counties. Therefore, insurance program requires a procedure which could fully consider the difference in productiveness. As for Volusia, Seminole, Palm Beach and other counties with very small deviations, tree values and expected yield could be estimated directly PAGE 48 48 by yield function derived by this study. For the counties with large deviations such as Hendry and Highland, this yield function could be used as a base estimation which needs to be adjusted by other countylevel information. Since 16 out of 25 counties have devi ations less than +1000 boxes, this yield function is very useful for most counties. Finally, the estimated sp atial coefficients can be used in other studies which involve the factors with spatial nature su ch as weather, temperature, catastrophic events, and human mobility. Limitations of the Study and Sug gestions for Further Research Although our research findings have important implications for m odeling perennial crop yields and insurance program design, ma ny important research issues remain. The first is the difference of productive cap acity among counties whic h could not be shown by this yield function. Therefore, future study may change the assumpti on that all the counties have the same yield function for the same variet y by adding variables of factors such as tree density, location, weather and dise ase variables that infect yiel d and varies by county. Another problem is skewness and kurtosis of the distur bance term which is assumed to follow normal distribution. Future work coul d correct this problem by the procedures proposed by Ramirez, Moss and Boggess who used a modified invers e hyperbolic sine function to transform nonnormal random variables. PAGE 49 49 APPENDIX A SIMULATION MODELING PROGRAMMING BY GUAUSS library co; coset; load data_early[350,23]=e:\rese arch\data\earlymidseasondata.dta; load data_valencia[350,23]=e:\res earch\data\vale nciadata.dta; load data_white[336,23]=e:\resear ch\data\whiteseed lessdata.dta; load data_colored[350,23]=e:\res earch\data\colore dseedless.dta; load data_specialty[276,23]=e:\res earch\data\speci altiesdata.dta; load ww1[25,25]=e:\research\data\w1.dta; load ww2[23,23]=e:\research\data\w2.dta; load ww3[24,24]=e:\research\data\w3.dta; w1=ww1./sumc(ww1)'; w2=ww2./sumc(ww2)'; w3=ww3./sumc(ww3)'; v=seqa(3,1,22); {va1,ve1}=eigrs2(w1); {va2,ve2}=eigrs2(w2); {va3,ve3}=eigrs2(w3); /*early and midseason oranges*/ x_early=data_early[.,1:22]; y_early=data_early[.,23]; proc m1(b_early); local err,j,x,y,z,s; s=0; for i(25,350,25); j=i24; x=x_early[j:i,.]; y=y_early[j:i,.]; z=rows(x); PAGE 50 50 err=(eye(z)b_early[4]*w1)*(yx*( b_early[1]/2*(1+tanh(b_early[2]+b_early[3]*v)))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(1b_early[4]*va1))+s); endp; /*set second derivative at v=22 <0; at v=1>0*/ proc ineqp1(b_early); local a; a=b_early[3]^2*b_early[1]/cosh(b_early[2]+b_ early[3]*3)^2*tanh(b_early[2]+b_early[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_early); local b; b=b_early[3]^2*b_early[1]/cosh(b_early[2]+b_ early[3]*24)^2*tanh(b_early[2]+b_early[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 10 10, 0 1}; b_early0=(50.50.50.3); {b_early,f,g,ret}=co(&m1,b_early0); /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_early~stdb~b_early./stdb; /*valencia oranges*/ PAGE 51 51 x_valencia=data_valencia[.,1:22]; y_valencia=data_valencia[.,23]; proc m2(b_valencia); local err,j,x,y,z,s,dens,n; s=0; for i(25,350,25); j=i24; x=x_valencia[j:i,.]; y=y_valencia[j:i,.]; z=rows(x); n=i/25; dens=D[n,1]; err=(eye(z)b_valencia[4]*w1)*(yx*b_valencia[1]/2*(1+tanh(b_valencia[2]+b_valencia[3]*v))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(1b_valencia[4]*va1))+s); endp; proc ineqp1(b_valencia); local a; a=b_valencia[3]^2*b_valencia[1]/cosh(b_valencia[2 ]+b_valencia[3]*3)^2*tanh(b_valencia[2]+b_v alencia[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_valencia); local b; b=b_valencia[3]^2*b_valencia[1]/cosh(b_valencia[2 ]+b_valencia[3]*24)^2*tanh(b_valencia[2]+b_ valencia[3]*24); retp(b1e10); PAGE 52 52 endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 10 10, 0 1}; b_valencia0=(40.50.50.3); {b_valenciahat,f,g,ret}=co(&m2,b_valencia0); print b_valenciahat; /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_valenciahat~stdb~b_valenciahat./stdb /*white seedless oranges*/ x_white=data_white[.,1:22]; y_white=data_white[.,23]; proc m3(b_white); local err,j,x,y,z,s; s=0; for i(24,336,24); j=i23; x=x_white[j:i,.]; y=y_white[j:i,.]; z=rows(x); err=(eye(z)b_white[4]*w3)*(yx*(b_white[1]/2*(1+tanh(b_white[2]+b_white[3]*v)))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(1b_white[4]*va3))+s); endp; proc ineqp1(b_white); local a; PAGE 53 53 a=b_white[3]^2*b_white[1]/cosh(b_white[2]+b_wh ite[3]*3)^2*tanh(b_white[2]+b_white[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_white); local b; b=b_white[3]^2*b_white[1]/cosh(b_white[2]+b_wh ite[3]*24)^2*tanh(b_white[2]+b_white[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 0 10, 0 1}; b_white0=(40.50.50.5); {b_whitehat,f,g,ret}=co(&m3,b_white0); /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_whitehat~stdb~b_whitehat./stdb /*colored seedless oranges*/ x_colored=data_co lored[.,1:22]; y_colored=data_colored[.,23]; proc m4(b_colored); local err,j,x,y,z,s; s=0; for i(25,350,25); j=i24; PAGE 54 54 x=x_colored[j:i,.]; y=y_colored[j:i,.]; z=rows(x); err=(eye(z)b_colored[4]*w1)*(yx*(b_colored[1]/2*(1+tanh(b_col ored[2]+b_colored[3]*v)))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(1b_colored[4]*va1))+s); endp; proc ineqp1(b_colored); local a; a=b_colored[3]^2*b_colored[1]/cosh (b_colored[2]+b_colored[3]*3) ^2*tanh(b_colored[2]+b_color ed[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_colored); local b; b=b_colored[3]^2*b_colored[1]/cosh (b_colored[2]+b_colored[3]*24) ^2*tanh(b_colored[2]+b_colo red[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 0 10, 0 1}; b_colored0=(40.50.50.5); {b_coloredhat,f,g,ret}=co(&m4,b_colored0); PAGE 55 55 /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_coloredhat~stdb ~b_coloredhat./stdb; PAGE 56 56 APPENDIX B COMPARISON OF SPATIAL CORRELATION BY GUAUSS library co; coset; load data_early[350,23]=e:\rese arch\data\earlymidseasondata.dta; load data_valencia[350,23]=e:\res earch\data\vale nciadata.dta; load data_white[336,23]=e:\resear ch\data\whiteseed lessdata.dta; load data_colored[350,23]=e:\res earch\data\colore dseedless.dta; load data_specialty[276,23]=e:\res earch\data\speci altiesdata.dta; load ww1[25,25]=e:\research\data\w1.dta; load ww2[23,23]=e:\research\data\w2.dta; load ww3[24,24]=e:\research\data\w3.dta; w1=ww1./sumc(ww1)'; w2=ww2./sumc(ww2)'; w3=ww3./sumc(ww3)'; v=seqa(3,1,22); {va1,ve1}=eigrs2(w1); {va2,ve2}=eigrs2(w2); {va3,ve3}=eigrs2(w3); /*early and midseason oranges*/ x_early=data_early[.,1:22]; y_early=data_early[.,23]; proc m1(b_early); local err,j,x,y,z,s; s=0; for i(25,350,25); j=i24; x=x_early[j:i,.]; y=y_early[j:i,.]; z=rows(x); PAGE 57 57 err=(eye(z)0*w1)*(yx*(b_early[1]/2*(1+tanh(b_early[2]+b_early[3]*v)))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(10*va1))+s); endp; /*set second derivative at v=22 <0; at v=1>0*/ proc ineqp1(b_early); local a; a=b_early[3]^2*b_early[1]/cosh(b_early[2]+b_ early[3]*3)^2*tanh(b_early[2]+b_early[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_early); local b; b=b_early[3]^2*b_early[1]/cosh(b_early[2]+b_ early[3]*24)^2*tanh(b_early[2]+b_early[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 10 10}; b_early0=(50.50.5); {b_early,f,g,ret}=co(&m1,b_early0); /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_early~stdb~b_early./stdb; /*valencia oranges*/ x_valencia=data_valencia[.,1:22]; PAGE 58 58 y_valencia=data_valencia[.,23]; proc m2(b_valencia); local err,j,x,y,z,s,dens,n; s=0; for i(25,350,25); j=i24; x=x_valencia[j:i,.]; y=y_valencia[j:i,.]; z=rows(x); n=i/25; dens=D[n,1]; err=(eye(z)0*w1)*(yx*b_valencia [1]/2*(1+tanh(b_valencia[ 2]+b_valencia[3]*v))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(10*va1))+s); endp; proc ineqp1(b_valencia); local a; a=b_valencia[3]^2*b_valencia[1]/cosh(b_valencia[2 ]+b_valencia[3]*3)^2*tanh(b_valencia[2]+b_v alencia[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_valencia); local b; b=b_valencia[3]^2*b_valencia[1]/cosh(b_valencia[2 ]+b_valencia[3]*24)^2*tanh(b_valencia[2]+b_ valencia[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; PAGE 59 59 _co_Bounds={10 10, 10 10, 10 10}; b_valencia0=(40.50.5); {b_valenciahat,f,g,ret}=co(&m2,b_valencia0); print b_valenciahat; /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_valenciahat~stdb~b_valenciahat./stdb; /*white seedless oranges*/ x_white=data_white[.,1:22]; y_white=data_white[.,23]; proc m3(b_white); local err,j,x,y,z,s; s=0; for i(24,336,24); j=i23; x=x_white[j:i,.]; y=y_white[j:i,.]; z=rows(x); err=(eye(z)0*w3)*(yx*(b_white[1]/2*(1+tanh(b_white[2]+b_white[3]*v)))); s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(10*va3))+s); endp; proc ineqp1(b_white); local a; a=b_white[3]^2*b_white[1]/cosh(b_white[2]+b_wh ite[3]*3)^2*tanh(b_white[2]+b_white[3]*3); retp(a1e10); PAGE 60 60 endp; _co_IneqProc=&ineqp1; proc ineqp2(b_white); local b; b=b_white[3]^2*b_white[1]/cosh(b_white[2]+b_wh ite[3]*24)^2*tanh(b_white[2]+b_white[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 0 10}; b_white0=(40.50.5); {b_whitehat,f,g,ret}=co(&m3,b_white0); /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_whitehat~stdb~b_whitehat./stdb; /*colored seedless oranges*/ x_colored=data_co lored[.,1:22]; y_colored=data_colored[.,23]; proc m4(b_colored); local err,j,x,y,z,s; s=0; for i(25,350,25); j=i24; x=x_colored[j:i,.]; y=y_colored[j:i,.]; z=rows(x); err=(eye(z)0*w1)*(yx*(b_colored [1]/2*(1+tanh(b_colored[2]+b_colored[3]*v)))); PAGE 61 61 s=s+rows(err)/2*ln(det(err'err/rows(err))); endfor; retp(14*sumc(ln(10*va1))+s); endp; proc ineqp1(b_colored); local a; a=b_colored[3]^2*b_colored[1]/cosh (b_colored[2]+b_colored[3]*3) ^2*tanh(b_colored[2]+b_color ed[3]*3); retp(a1e10); endp; _co_IneqProc=&ineqp1; proc ineqp2(b_colored); local b; b=b_colored[3]^2*b_colored[1]/cosh (b_colored[2]+b_colored[3]*24) ^2*tanh(b_colored[2]+b_colo red[3]*24); retp(b1e10); endp; _co_IneqProc=&ineqp2; _co_Bounds={10 10, 10 10, 0 10}; b_colored0=(40.50.5); {b_coloredhat,f,g,ret}=co(&m4,b_colored0); /*ttest*/ stdb=sqrt(diag(inv( _co_finalhess))); print b_coloredhat~stdb ~b_coloredhat./stdb; PAGE 62 62 APPENDIX C SPATIAL WEIGHT MATRIX W EARLY MIDSEASON ORANGES, VALENCIA ORANGES, COLORED SEEDLESS GRAPEFRUIT 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 00 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 00 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 00 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 00 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 00 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 PAGE 63 63 White Seedless Grapefruit 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 PAGE 64 64 LIST OF REFERENCES Akeyama, T. 1987. 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PAGE 67 67 BIOGRAPHICAL SKETCH Lan Cheng was born in 1982, in the Province of Inner Mongolia, China. After graduating from high school, she enrolled in th e Shanghai Fisheries Un iversity, majoring in international economics and trade, Shanghai, Ch ina, in 2001 and graduate d with a bachelors degree in economics in June 2005. From August 2005 to August 2007, Miss Cheng studi ed in the Master of Science program of the Food and Resource Economics Department (F RED) at the University of Florida. After obtaining her masters degree, she will enter the Department of Agricultural and Applied Economics at the University of WisconsinMadis on for the Ph.D. degree and will continue her specialization in agricultural finance and crop insurance. 