SPATIOTEMPORAL MODELING OF ASIAN CITRUS CANKER RISKS: IMPLICATIONS FOR INSURANCE

Transcription

1 SPATIOTEMPORAL MODELING OF ASIAN CITRUS CANKER RISKS: IMPLICATIONS FOR INSURANCE AND INDEMNIFICATION FUND MODELS BARRY K. GOODWIN AND NICHOLAS E. PIGGOTT Asiatic citrus canker is a disease that poses a significant hazard to commercial citrus production. This article examines spatiotemporal models of the risks of citrus canker transmission. Models evaluate risks and are used to price annual contracts, which would pay producers a prespecified indemnity in the event that their grove is found to be infected with canker. Implications for risk management policy are discussed. Key words: insurance premium rates, invasive species, risk models, zero-inflated generalized Poisson. Florida had 621,000 acres of commercial citrus groves in This represented a significant decrease over the most recent high of nearly 858,000 acres in 1996 (a decline of 17%) (Florida Agricultural Statistics Service 2007b). The total value of sales on-tree was estimated at approximately $1.3 billion for Florida is the largest citrus growing state in the United States and accounts for about 79% of total U.S. citrus production. Total production in the crop-year amounted to 162 million boxes comprised of 129 million boxes of oranges (79.6%), 40.9 million boxes of grapefruit (16.8%), and 5.8 million boxes of other types of fruit (3.1%) (Florida Agricultural Statistics Service 2007a). The prominent role played by Florida in citrus production has been threatened in recent years by widespread outbreaks of Asiatic citrus canker (ACC) an infectious, invasive plant disease thought to have originated in Asia. Citrus canker disease affects plants in varieties of citrus species and citrus relatives. 1 The Goodwin is William Neal Reynolds Professor in the Departments of Agricultural and Resource Economics and Economics at North Carolina State University and Piggott is professor in the Department of Agricultural and Resource Economics at North Carolina State University. This research was supported by the North Carolina Agricultural Research Service and by a grant under the PRESIM program of the Economic Research Service of the U.S. Department of Agriculture. We are grateful to Debra Martinez and Glen Gardner of the Division of Plant Industry in the Florida Department of Agriculture and Consumer Services for assistance with the data and our analysis. We are also grateful for the helpful comments and suggestions provided by three anonymous reviewers, Paul Preckel, and by seminar participants at Wageningen, the University of Wisconsin, and at the Economic Research Service. 1 The following citrus species have been identified as being highly susceptible : grapefruit, key/mexican lime, Palastine sweet disease is caused by a bacterial pathogen, Xanthomonas axonopodis pv. citri. Plants infected by citrus canker develop lesions on leaves, stems, and fruit. These lesions ooze bacterial cells making canker highly contagious. Canker can be spread rapidly by wind-driven rain, movement of equipment, or workers that have come into contact with infected trees, or movement of infected or contaminated plants. These vectors of transmission, involving significant weather events, and idiosyncratic movements of workers or people carrying contaminated plants, makes containment a significant challenge (USDA-APHIS 2005a). A canker eradication program in Florida began in 1995 and evolved into a program which attempted to manage separate infestations and different strains over numerous Florida counties. The initial infestation of an Asiatic strain of citrus canker has been traced to an infection in a residential area near Miami International Airport. 2 Despite control efforts, citrus canker continued to spread over the next several years. It was suspected that the disease was transferred via human activities from nearby residential areas to the north. In October 1999, the USDAs Risk Management Agency (RMA) announced that the 2000 Florida Fruit Tree Pilot Crop Insurance Program, which has been in existence since 1996, lime, and trifoliate citrus, sweet orange cultivars: Hamlin, Navel, and Pineapple (Schubert et al. 2001). 2 Gottwald et al. (2001) estimate that the 1995 Miami discovery near the airport spread from an initial 14-square-mile area to over 1,005 square miles in the metropolitan area, plus an additional 260 square miles of urban and commercial citrus areas through the state. Amer. J. Agr. Econ. 91(4) (November 2009): Copyright 2009 Agricultural and Applied Economics Association DOI: /j x

2 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1039 would be revised to allow producers to insure against losses to citrus trees arising from ACC. The coverage area was expanded to include 29 counties, making the pilot available to most commercial tree growers. By 2005, crop insurance liability in the citrus pilot program had increased to $1.141 billion. In February 2000, the Florida Commissioner of Agriculture announced the implementation of a significant canker eradication program. This program included efforts to decontaminate workers and equipment moving between groves, removal of all trees within a 1,900 feet radius of an infected tree, and a voucher program that provided residents with affected trees with $100 vouchers for the purchase of noncitrus replacement trees (USDA-APHIS 2005b). An interim rule was published in October 2000 providing eligible producers of commercial citrus payments to replace trees removed because of citrus canker (USDA-APHIS 2000). The payment was in the amount of $26 per tree, up to a maximum of between $2,704 and $4,004 per acre, depending on the variety. It was estimated that this program would compensate producers approximately $18.8 million, and an estimated 723,800 trees were destroyed. Owners of commercial citrus groves were eligible for payments if trees were removed because of a public order between 1986 and 1990 or on or after September 28, 2005 (USDA-APHIS 2002). Limes had the largest payment at $6,503 per acre for lost production and a maximum payment of $4,004 per acre for tree replacement. Next were oranges, Valencia oranges, and tangerines, with a payment of $6,446 per acre for lost production and a maximum payment of $3,198 per acre for tree replacement. Payments on navel oranges were slightly less, with $6,384 per acre for lost production and a maximum of $3,068 per acre for tree replacement. Grapefuit and other mixed citrus fruits had considerably lower payment levels, with a lost production payment of $3,342 per acre and a maximum tree replacement payment of $2,704 per acre. Note that the total value of payments associated with the loss of an acre of citrus production (lost production and tree replacement costs) is approximately $10,000 for all citrus except grapefruit, which has a value of about $6,000 per acre. Finally, if a producer purchased ACC crop insurance coverage and received an indemnity payment, lost production payments would be reduced by the amount of the indemnity payment. If the producer failed to purchase insurance if it was available, the per-acre production payment was reduced by 5%. Invasive species such as ACC pose significant hazards to agricultural production and thus warrant consideration of the development of a specialized class of single-peril crop insurance products. The potential benefits from this pursuit are twofold. First, such an approach may result in more actuarially appropriate rates, which in turn have the potential to increase insured participation against the specific hazard. Second, the potential economic benefits stemming from mitigation and the reduction of the risk of further spread of the hazard may be greater under such a specific-peril plan. The first benefit stems from the notion that understanding the spatiotemporal aspects of a particular hazard s transmission and likelihood of infection warrants the study and development of a class of single-peril products that can more accurately measure the risks of infection than is the case with multi-peril products. If more actuarially fair rates can be implemented in a single-peril policy, one should expect increased participation and a larger number of producers wanting to purchase this insurance, especially when infection can be catastrophic to production. In addition, more actuarially sound rates reduce the problems associated with adverse selection. The second benefit stems from the notion that a better understanding of the spatiotemporal aspects of a hazard s transmission and recognition of the potential spatial externalities that result from mitigating spread of the disease have the potential to provide significant economic benefits to producers, insurance providers, and society in general. Fully capturing these benefits requires a careful study of the spatiotemporal aspects of a hazard s transmission. Further, an effective policy must include design and implementation of singleperil insurance products that limit adverse selection and moral hazard concerns and induce incentive-compatible actions by farmers to prevent further spread of the hazard. Crop insurance products that cover invasive species hazards might also need to consider compliance policies that introduce mitigating behavior in the case of indemnification in order to prevent further spread of the hazard. An example could be curative treatments or idling of production in the affected areas on detection in order to prevent further proliferation of the hazard. Actuarially sound contracts can be designed to include the costs of such mitigating

3 1040 November 2009 Amer. J. Agr. Econ. measures and thereby bestow economic benefits on all producers and society in general. This most recent outbreak of citrus canker presents an ideal case study for modeling the risks associated with infection by invasive species since extensive data relating to transmission and the factors underlying risks have been collected. The State of Florida implemented an inspection program that checked commercial groves from one to several times each year. This article examines spatiotemporal models of the risks of citrus canker transmission during the period of , using data from over 338,000 inspections. To model the spatial and temporal aspects of the risks of citrus canker transmission, it is critical to understand vectors of infection, the symptoms, rates of dispersion, and other important characteristics which impact the spatial and temporal aspects of infection. An extensive literature, summarized by Alexander, Cartwright, and McKinney (1988) and Rothenberg and Thacker (1992), has investigated spatial aspects of disease transmission. The epidemiology of citrus canker involves bacteria spreading from lesions during wet weather and dispersal at short range by splash, at medium long range by windblown rain and at all ranges by human assistance. We use statistical models to quantify infection risks and estimate associated indemnity/insurance fund contribution rates. The risks and associated premium rates are estimated for annual risk-management contracts which would pay producers a pre-specified indemnity in the event that their grove is found to be infected with canker. Our results confirm the significance of fruit type and other citrus grove characteristics as factors associated with the risk of canker infection. We also demonstrate how zero-inflation and over-dispersion may impact count data models of the sort typically used in such applications. Implications for more sophisticated models of spatial/temporal risk relationships and risk management policies are also discussed. Risk Models and Insurance/Indemnity Fund Contracts Risk management, disaster relief, and subsidized crop insurance programs have played an important role in U.S. agricultural policy in recent years. Crop insurance programs face a number of challenges that may limit their actuarial performance. This is especially true of all-risk (multiple-peril) crop insurance programs, which may face difficulties associated with measuring the risks from all possible hazards. 3 In the case of inaccurately measured risks, insurance programs may face problems associated with adverse selection, as high-risk agents tend to overinsure while low-risk agents underinsure, leaving an adversely selected insurance pool. Insurance programs may also suffer from moral hazard when the provision of insurance, which is often highly subsidized, leads producers to assume more risk. Moral hazard may range from simple changes in production practices to outright fraud and may be difficult for the risk modeler (and therefore the insurer) to quantify. Accurate measurement and pricing of risks is essential for addressing the problems associated with adverse selection and moral hazard. As we have noted, several government programs have been directed toward providing compensation for those citrus producers affected by citrus canker. In the case of disaster relief, the assistance has been of an ad hoc nature, with state and federal policy makers providing disaster payments in response to larger scale infections. Crop insurance protection has been part of an all-risk insurance plan. An alternative to all-risk insurance and ad hoc indemnification plans is a specific-peril plan of protection. In this case, the task of quantifying risks is limited to a single peril. Protection is offered only for losses caused by this peril, and thus, actuarial considerations are limited to modeling only the risks associated with the particular peril being covered. Examples of specific peril policies include hail, flood, and cancer insurance. The key element to any effective insurance or indemnification plan is comprehension of the risks associated with the hazards being covered. In insurance contracts knowledge of this risk underlies the actuarially fair insurance premium (or premium rate). 4 The actuarially fair rate corresponds to the rate (expressed in terms of total premiums as a percentage of total liability) that sets total premiums equal to total expected indemnities. For example, if I 3 See Goodwin and Smith (1996) for a detailed discussion of contract design issues associated with all-risk crop insurance plans. 4 It is common to normalize premiums by dividing by liability (the maximum possible payout), which yields a premium rate representing the amount of premium charged per dollar of liability coverage. Although the terms are often used interchangeably, premium refers to the total amount charged for insurance protection, while the premium rate expresses this as a proportion (or percentage) of liability.

4 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1041 expect to pay $1,000 in a typical year on an insurance contract that covers up to $10,000 in total liability, the actuarially fair premium rate is 0.10 (or 10%). In the case of an indemnification fund that could be funded by an assessment (or tax) on producers, the actuarially fair premium rate is analogous to the rate that must be charged in order to equilibrate expected payouts with contributions into the indemnification fund. The risk models needed to measure the actuarially fair premium or checkoff rate usually are expressed in terms of the conditional probability density or cumulative distribution function underlying the outcomes being considered. For example in the case of crop yield insurance, one is generally concerned with obtaining an estimate of the density describing crop yields. Most crop insurance contracts guarantee a certain proportion of expected yield and pay indemnities that will compensate losses due to yields below the guarantee. In many insurance programs loss occurs as an all-or-nothing event. For example, life insurance policies will pay a fixed amount only in the event of death, with no other provisions that could generate partial payments. Such structure simplifies the construction of insurance premium rates, since the payout amount is predefined. Likewise, the actuarially fair premium rate is equal to the probability of a loss occurring. Such a contract is suitable for situations, such as the citrus canker case, where any exposure corresponds to a complete loss. In general terms the actuarially fair premium is set equal to expected loss, which is given by the product of the probability of a loss and the expected loss, given a loss occurs. If we define z = 1 to denote a loss event (i.e., z = 1 if a loss occurs and z = 0 otherwise) and express the probability of a discrete loss event (z = 1) conditional on observable values of covariates X and a set of parameter estimates as Prob(z = 1 X ) = F(X ), expected loss is given by (1) E(Loss) = F(X ) E(Loss z = 1). In the case of coverage that pays a fixed amount when a loss event occurs, E(Loss z = 1) is predetermined, and one must only measure the probability of a loss, given by Prob(z = 1 X ). In such a case the expected loss is given by the product of the probability of a loss and the fixed payment made in the event of a loss. Modeling the probability of loss is thus central to the determination of the actuarially fair level of premium. Several important issues underlie such riskmodeling problems. Specifically, important questions pertain to the probability function F(X ). A specific choice of the form of the probability function must be made. Goodwin and Ker (2001) discuss specification issues related to the distributional assumptions that must be made in modeling insurance contract parameters. As they note, one may choose to employ nonparametric density estimation techniques in cases where prior information about the parametric family governing the data generating process is absent. Alternatively, a wide variety of parametric distributions are commonly applied to model parameters of insurance contracts. For example, crop yields commonly exhibit negative skewness, reflecting the natural biological constraints that govern maximum crop yields. Thus, a common choice for modeling crop yields is the beta distribution, which is capable of representing the negative skewness often observed for crop yields. Recognition of the factors that loss events should be conditioned on is also an important component of risk models. For example, crop yields have exhibited significant trends over time, and such trends must be explicitly recognized when assessing the risk of crops using data collected over time. To the extent that observable, deterministic factors are pertinent to risk, more accurate premium rates can be constructed by taking these factors into consideration. In the case of contracts to insure citrus canker risks, we know that factors such as fruit type and characteristics of the grove are important determinants of the risk of infection, and thus models of risk should be conditioned on such factors in order to produce accurate assessments of risk. There are several operational considerations that must be considered when contemplating an insurance or indemnification program. One important factor involves the insurance period. A common crop insurance period is the calendar or crop year, where the terms of a contract are set prior to the beginning of the year and protection begins and ends with the beginning and ending of the year. In our analysis we assume an insurance period corresponding to a calendar year. The period of insurance is important to how one models risk, since risks can only be conditioned on information available prior to the beginning of the insurance period. For example, it is widely

5 1042 November 2009 Amer. J. Agr. Econ. recognized that hurricanes, with their characteristic high winds and precipitation, are an important causal factor related to citrus canker infection. However, in that it is impossible (or at least very difficult) to predict the occurrence of a hurricane at any single location in the following year, knowledge that prior infections were correlated with hurricane strikes is of little use in constructing insurance contracts. In contrast we know that different fruit types have varying levels of infection risk. The type of fruit to be insured in year t + 1 is known at time t, and thus, the parameters of an insurance contract can be conditioned on fruit type. An insurance contract must also specify the unit of insurance. Because of the diversification that comes with increasing size, risks are often lower as more aggregate units of insurance are defined. However, in cases such as citrus canker, where any exposure corresponds to a total loss, it is important that the unit be defined at a level consistent with the extent of loss upon exposure. Our data on canker inspections are given in terms of multiblock units, which roughly correspond to individual commercial citrus groves. The eradication program mandates that each multiblock unit be inspected at least once a year, with some groves being inspected more than once in a year. Larger groves are more likely to receive multiple inspections. The annual acreage inspected for multiblock units in our data averages acres and ranges from 0.05 to 2,526 acres. 5 In measuring risk and specifying insurance contract parameters, one must also decide on the level at which risks will be measured. Alternative levels of aggregation may vary in terms of the stability of the premium rates implied as well as the accuracy of individual rates. Estimates made at a very fine level of resolution may be excessively volatile as one moves across units. Insurers typically smooth individual premium rates across neighboring contracts to address such volatility. In contrast highly aggregated units may fail to reveal heterogeneity among the units making up the aggregate and thus may provide less accurate measures of risk. Aggregation may also greatly simplify risk modeling. In our case we consider neighbor effects intended to capture the increased risk due to nearby infections. Quantifying this neighbor effect may be complicated by the dimensionality of the problem that arises in comparing each unit to all others in the sample. 6 In light of this dimensionality problem, we chose an alternative, hierarchial modeling approach based on conditional probabilities and aggregated units. This approach is also merited by the relative rarity of canker infections. We consider a two-step approach to modeling canker infection risks. In the first we construct aggregate units and estimate models describing the risks associated with infections within the aggregate units. In this case, we measure the frequency (count) of infections in each aggregate unit. In the second step we consider a model of the conditional probability that any individual multiblock within a positive aggregate unit will be infected. Consider an individual multiblock i, which is a member of a larger grouping of multiblocks into an aggregate unit k. The probability that this individual multiblock unit i is infected can be expressed as the product of the probability that the aggregate unit k has one or more infected multiblock units and the conditional probability that the individual multiblock unit i is infected, given that it is in an aggregate unit k that has one or more positive infections among individual units. Note that this approach requires estimates of the infection probabilities at two levels for the aggregate units and for multiblocks within positive aggregate units. We considered two possible levels of aggregation. We initially considered aggregate groupings based on the Township-Range- Section (TRS) definition, which defines local political boundaries in Florida. However, the TRS system in Florida does not encompass the entire citrus growing region. In addition irregularity in the size and shape of TRS units may also make their use for defining units of homogeneous risk questionable. In light of the limitations associated with the TRS units, we chose to identify and develop our own aggregate units based on an evenly spaced grid that covers the entire commercial citrus growing region of Florida. We chose a grid defined by 10 km 2 units. As is true of the TRS designations, the groupings are ad hoc, and other possible group definitions could have advantages. 5 These annual totals include cases of multiple inspections of a single unit. The physical size of multiblock units averages acres and ranges from 0.05 to 510 acres. Dormant units are not counted in this total as no citrus acreage exists on dormant multiblocks. 6 Specifically, the distance between every pair of units must be calculated. In our application, which is based on 62,183 multiblock units (observed through multiple inspections over multiple years), this would imply 1.9 billion pairwise distance measures.

6 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1043 However, this approach was compared to grids of alternative sizes and found to perform well in the analysis that follows and produce robust results. Finally, our approach requires that we adequately incorporate any measurable factors that can be used to condition the risk of infection. Recall that only those factors that can be measured prior to the beginning of the insurance period are useful in conditioning the risk of infection. An important aspect of citrus canker, as with any communicable disease, is that infection is spread through exposure. We know that infection risk is subject to important spatial and temporal correlation factors. In particular proximity in a spatial and temporal sense to existing infections may affect the likelihood that a grove will be infected. We capture this relationship by considering the infections recorded in the previous year in all aggregate units having centroids that are less than 30 km from the centroid of the unit in question. 7 Under these assumptions, we can view our risk modeling approach to involve attempts to measure the conditional probability associated with citrus canker infection at the aggregate unit level and for individual multiblocks within positive units. If we define y kt to be the integer count of the number of positive multiblock units in year t for aggregate unit k, which is surrounded by n other aggregate units, the conditional probability of one or more infections for aggregate unit k can be expressed as ( ( ) ) n (2) Pr y kt 0 y kt 1 + y jt 1, X kt j=1 where y jt 1 is the positive infection status count of neighboring aggregate unit j in year t 1, X kt represents other predetermined factors relevant to the likelihood of canker infection, and the summation is over the set of n surrounding units. A similar expression is considered for the second-stage model of multiblocks, although this specification includes a different set of explanatory factors. Empirical Models and Results In order to make the transition to an empirical analysis, we must choose specific empirical 7 The geographic centroid is the center of gravity of a geographic shape. In our units the centroids are the exact centers of the 10km 2 units. This is analogous to considering two concentric circular groups of units that surround each unit of interest. models of the likelihood of infection both at the aggregate unit level and the individual multiblock unit level. Our measure of infection is the status of a particular multiblock unit at the time of its inspection a discrete 0/1 indicator. However, our aggregate units are made up of multiple multiblocks and, thus, the extent of infections is reflected in a count of the number of positive multiblocks within the aggregate unit. One caveat associated with any empirical model based on historical patterns of risk should be acknowledged. It is always possible that changes in production practices can change risk patterns, making estimates based on historical data less accurate. This may reflect exogenous changes in the structure of the industry or, alternatively, moral hazard that results from changes in behavior that occur once insurance coverage is taken. Specifically, insured agents may exercise less care and protection against covered risks once insurance is available. Commercial insurers often load premium rates to account for such possibilities and to reflect other unobserved factors associated with risk. We do not apply such loading factors in our analysis, although the potential for changes in risk patterns should be recognized when interpreting the results that follow. Further, changes in regional patterns of production could also change the overall portfolio of risks. Model Specification and Econometric Procedures Given our assumed modeling framework of applying the models to aggregated 10 km 2 units, our measure of infection for the aggregate unit is the simple count of infections within the unit. Two alternative approaches to modeling the risk of infection at the aggregate unit level are obvious. We are primarily interested in the status of an aggregate unit (i.e., whether canker is present or not). This suggests a standard discrete dependent variable model such as a logit. However, we also have information about the degree of infection in the counts of positive inspections within each 10 km 2 unit. Applying a discrete dependent variable model to the status of infection ignores potentially valuable information available in the integer counts. This suggests that a count-data model, such as a Poisson regression, may offer advantages. We initially consider a standard Poisson model. However, a standard Poisson specification may not be appropriate in cases where a significant proportion of the counts are zero as is the case in our application.

7 1044 November 2009 Amer. J. Agr. Econ Positive Finds 2000 Frequency Count (a) Count Frequencies Positive Finds 200 Frequency Count Actual (b) Count Frequencies and Poisson Distribution Figure 1. The preponderance of zeros problem Recent research has recognized the fact that many of the parametric specifications used to model count data may be inappropriate in cases where the data do not fit the underlying distribution. A particular case and one that characterizes our data arises when the count data reflect a preponderance of zero counts. Forcing a specific distribution to conform to actual counts that involve such a preponderance of zeros may result in important misspecification. Panel (a) of figure 1 presents a histogram of our aggregate unit count data. The extreme preponderance of observations with zero infections is apparent. Panel (b) of figure 1 illustrates a Poisson distribution, fit to our data (absent any conditioning on covariates) alongside the positive infection counts. The degree of potential misspecification is apparent in the figure, where the distributional aspects of the observed count data result in a significant overstatement of the probability of small infection counts. The Poisson specification is unable to accurately fit the distribution of the counts. Poisson Various mixture models have been developed to address this misspecification issue. Lambert (1992) introduced the zero-inflated Poisson (ZIP) model, which involves a mixture of a discrete model, representing the discrete 0/1 threshold represented above by our probit models and a model appropriate for the count data above zero. 8 A second important specification issue inherent in models of integer count data pertains to the distributional specification characterizing the integer counts. A standard Poisson model assumes that the mean and variance of the dependent variable are identical. This may be a strong assumption that could potentially result in significant biases and misleading inferences. A variety of models that 8 A comprehensive survey of zero-inflated models and their application is presented by Ridout, Demetrio, and Hinde (1998). Alternative versions of count data models have been developed to address various specification issues, including the preponderance of zeros issue. See, for example, the rapid decay models and alternative specifications presented by Sarker and Surry (2004).

8 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1045 allow the variance to exceed the mean the case of over-dispersion have been developed and applied to count data. One important example is the negative binomial specification, which allows for a fixed degree of overdispersion. However, this specification may also be unnecessarily restrictive in that it assumes a constant level of over-dispersion. A more general alternative that allows the degree of over-dispersion to vary and potentially depend on covariates was introduced in the form of the generalized Poisson model by Consul and Jain (1973). More recently, Czado et al. (2007) introduced a zero-inflated, generalized Poisson (ZIGP) model. This model encompasses both of the specification concerns by explicitly modeling over-dispersion and zeroinflation. We adopt the ZIP model of Czado et al. (2007), which is defined by a mixture of a logit model that captures the discrete infection/noinfection distinction and a Poisson model that represents the positive infection counts. The dependent variable of interest y it is the number of positive infection counts in unit i in year t. The Poisson model is generalized by endogenizing the degree of over-dispersion and allowing it to vary with exogenous covariates. This results in a three-parameter distribution y it ZIGP( it, it, it ), where each of the three parameters are allowed to vary through functional relationships with observable covariates. The ZIGP probability density function of Czado et al. (2007) is given by (3) Pr(y it = Y it, it, it ) [ ] = it it + (1 it )e it it + (1 it ) [(1 it ) it( it + ( it 1)Y ) Y 1 Y! Y it e 1 it ( it +( it 1)Y ) where it equals one if y it = 0 and is zero if y it > 0. The mean and variance of y it, conditional on a set of observable exogenous covariates x it, are given by (4) E(y it X = x it ) = (1 it ) it ] The three parameters defining the zeroinflated, generalized Poisson distribution are related to covariates through the following expressions: (6) (7) (8) E(y it ) = it ( ) := e x it > 0 it ( ) := 1 + e z it > 1 ev it it ( ) := (0, 1). 1 + e v it The model is then specified using the following logarithmic link functions: (9) (10) (11) it ( ) = log( it ( )) it ( ) = log( it ( ) 1) ( ) it ( ) it ( ) = log. 1 it ( ) The values x it, z it, and v it represent covariates (not necessarily mutually exclusive) that describe the mean, dispersion, and zero-inflation processes, respectively. Note that the ZIGP model reduces to a standard Poisson model when = 1 and = 0. Likewise, when only = 0, the model corresponds to the generalized Poisson model of Consul and Jain (1973). Finally, if only = 1, the ZIGP model corresponds to the standard zero-inflated Poisson model of Lambert (1992). Czado et al. (2007) note that the link function for the zero-inflation level does not allow for no zero-inflation, and thus, the models are not nested. As a result, standard nested specification testing through a comparison of loglikelihood function values cannot be pursued. Thus, we adopt the nonnested specification testing approach of Vuong (1989) to evaluate the alternative specifications. We use standard maximum likelihood (ML) estimation methods. 9 Additional specification considerations apply to the empirical modeling approach. Specifically, the exogenous conditioning factors to be included in each portion of the empirical models must be defined. For the purposes of establishing the terms of an insurance or indemnity contract to be offered at the beginning of the calendar year, conditioning (5) 2 it = Var(y it X = x it ) = E(y it X = x it ) ( 2 it + it it ). 9 Estimation was conducted using the R statistical language. The R packages zigp (Erhardt 2008) and pscl (Jackman et al. 2008) were used in estimation.

9 1046 November 2009 Amer. J. Agr. Econ. factors must satisfy several conditions. First, the conditioning factors must be observable prior to the start of the insurance period (i.e., prior to the beginning of the insurance year). Second, the covariates should be exogenous to the risks of infection. Recall that the canker eradication program mandated that each multiblock be inspected at least once per year. In our inspections data about 83% of the multiblocks were inspected only once per calendar year, while the other 17% (mainly comprised of larger units) were inspected more than once per year. We should also note, however, that the scope of the area inspected in the state expanded during the early years of the program, such that some multiblock units and thus some of our aggregate 10 km 2 units were not included in the inspection program until 2000, after which about 50,000 70,000 inspections were performed each year. We use the total area of grove space (including dormant land) in each of our aggregate 10 km 2 units as well as the total acreage of citrus and the share of citrus acreage planted in each type of fruit (oranges, grapefruit, and tangerines) as explanatory factors. The infectious nature of citrus canker suggests that a more dense concentration of citrus trees will correspond to a higher risk of infection. However, it should also be noted that the area of groves does not precisely correspond to the area inspected each year, since some units are inspected more than once per year. The area actually inspected each year may not be a valid explanatory factor because it may be endogenous to infection risk if inspections are targeted toward areas with more risk. Further, the area actually inspected cannot be known prior to the beginning of the calendar year and thus cannot be used as a conditioning factor in setting insurance or indemnity fund premium rates. However, the actual acreage of groves and shares of grove space devoted to each type of fruit are exogenously determined and are fixed over time. It should be noted that inclusion of the total area in groves accounts not only for the higher risks associated with denser citrus production but also for differences in inspection intensity a factor known as the degree of exposure, in count data models of this type Alternative models that included the area and composition of grove space actually inspected were also considered and produced very similar empirical results. Further, we considered instrumental variables versions that accounted for the endogenous level of inspections and reached very similar conclusions. However, such The second step of our analysis involves estimation of the risks of infection at the individual multiblock unit in order to measure the conditional probability that any individual multiblock within a positive aggregate unit will be infected. To this end we estimate a standard logit model of the multiblock infection status using the subset of multiblocks lying within a positive aggregate unit. Discussion of Data Our empirical analysis is based on inspections data collected under the Florida Citrus Canker Eradication Program. The inspections data span Data describing characteristics of the multiblock units and inspections reports were obtained from the Florida Department of Agriculture and Consumer Services Division of Plant Industry. The survey data report on the results of periodic inspections, which are made an average of 1.3 times per year on each multiblock. The data consist of reports on 333,892 multiblock inspections. Our unit of observation for the first stage of our empirical analysis is the 10 km 2 unit of aggregation. The dependent variable in our analysis is the count of positive canker finds revealed by inspections over a calendar year. 11 The existing scientific evidence suggests that several observable factors may be relevant to the likelihood of infection. In particular certain fruit varieties are believed to be more susceptible to canker infection than others. Limes, lemons, and grapefruits tend to be more susceptible than oranges and tangerines. We consider four variables representing the proportions of the citrus grove acreage in each aggregate unit devoted to particular fruit types oranges, tangerines, grapefruit, and all other fruits (which consist of limes, lemons, carambolas, and other minor citrus fruit varieties). It is also the case that groves frequently have dormant acreage grove space for which no citrus is currently being produced but which models are of limited usefulness in constructing insurance rates because inspection rates cannot be known in advance of the insurance year and thus cannot be used to condition the terms of insurance contracts. As a reviewer has noted, one could also consider fixed-effects models. The application of a fixed-effects specification within the context of zero-inflated models with over-dispersion remains an important topic for future research. 11 We also considered an alternative specification having positive canker status as the dependent variable, where positive status was defined as any inspection coming within the two-year period following a canker find on an individual multiblock. The results were very similar for this specification and thus are not presented here. These results are available from the authors on request.

10 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1047 still remains classified as commercial grove space. 12 About 9% of the multiblocks, representing about 12% of grove space, was classified as being dormant and thus had no actual citrus production. Such dormant acreage could serve as a buffer against infection, at least to the extent that it insulates the fruit-bearing trees from the boundaries of the multiblock units. We include the proportion of total grove area that is dormant. Finally, we utilize a count of the total number of positive multiblocks found in the aggregate unit and in neighboring units in the previous calendar year. Recall that neighboring units are defined as any unit whose centroid is less than 30 km from the unit of interest. Table 1 presents summary statistics for measures of infection and other relevant explanatory factors. We present variable definitions and summary statistics both for the individual multiblock (grove) units and for the aggregate 10 km 2 block units in table 1. The data consist of 333,892 inspections of 62,183 multiblock units observed between 1997 and This resulted in 259,343 multiblock-year combinations. Aggregation to the 10 km 2 units resulted in 2,379 annual aggregate block unit observations. Note that about 2.5% of the aggregate observations have positive finds. About 75% of citrus production is in the form of oranges; 9% is represented by grapefruits; 5.5% is in tangerines; and other minor citrus crops account for small proportions of production. Aggregate Unit Results The overarching goal of our models is to provide measures of the risk of canker infection which could be applied in the construction of insurance or indemnification plans. To this end we used ML procedures to estimate standard Poisson and logit models, the ZIP model, and finally the zero-inflated generalized Poisson model, which allows for a variable degree of over-dispersion across the sample. In interpreting the logit model results, it is important to recognize that the probabilities correspond to the case of no infection (i.e., the positive event being modeled by the logit specification 12 Dormant space includes groves that may have been previously cleared (because of canker, tree age, or for other reasons) and grove space otherwise not currently engaged in production. Dormant space does not include land that has been placed into development, such as in housing or roadway construction. Land that once had commercial citrus production but has been cleared often has shoots which can harbor canker, and thus, inspection of such dormant land was an important component of the program. is that y it = 0 where y it is the count of infections). Each model was estimated for positive canker status and positive canker inspections at the aggregate unit level. We applied Vuong (1989) nonnested specification tests to evaluate each model relative to its alternatives. Table 2 presents ML parameter estimates for the models of positive canker finds. The results reveal the significance of several conditioning factors in terms of explaining the risks of canker infection. Coefficients for the logit selection mechanism, reflecting the probability that no infections will be found in an aggregate unit, are quite similar across the standard logit and the ZIP model. An exception occurs for grapefruit share and total acreage, which were smaller and not statistically significant in the ZIP model. When a variable degree of overdispersion is allowed, the parameter estimates are quite different. Each specification is tested against the two alternatives using the test of Vuong (1989), and test results and associated p-values are presented in the table of results. Vuong s tests strongly favor the ZIP model over the standard logit/poisson model combination and likewise favor the zero-inflated generalized Poisson model over the ZIP model that does not allow for over-dispersion. Thus, the specification testing results strongly favor the ZIGP specification. The results suggest that the probabilities that one or more positive inspections will occur are lowest for units with a large share of tangerines. The probability of positive finds is similar for units comprised of oranges and grapefruits. The default category, consisting of lemons, limes, and other minor citrus commodities, clearly has a higher likelihood of infection than is the case for oranges. This is in accordance with expectations in that such fruit varieties have been shown to be more susceptible to canker infection than is the case for oranges, tangerines, and grapefruits. These differences in infection probabilities across different citrus types suggest that it might be appropriate to vary risk mitigation efforts according to the makeup of citrus types within particular units. That is, for units made up of a large amount of lemons, limes, and other minor citrus types, it might be appropriate to intensify mitigation efforts. In contrast it might be appropriate to have a less intensive mitigation effort in units with large amounts of oranges due to lower probabilities of canker infections. Table 2 also presents elasticity estimates for the preferred specification the zero-inflated

11 1048 November 2009 Amer. J. Agr. Econ. Table 1. Variable Definitions and Summary Statistics a Variable Definition N Mean Std. Dev. Individual Multiblocks Multiblock size Multiblock size (in 10,000 m 2 ) 62, Orange acres (all multiblocks) Average orange acreage on all 62, multiblocks Grapefruit acres (all multiblocks) Average grapefruit acreage on all 62, multiblocks Tangerine acres (all multiblocks) Average tangerine acreage on all 62, multiblocks Other citrus acres (all multiblocks) Average other citrus acreage on all 62, multiblocks Oranges 0/1 Indicator of oranges on citrus 62, multiblocks Tangerines 0/1 Indicator of tangerines on citrus 62, multiblocks Grapefruit 0/1 Indicator of grapefruit on citrus 62, multiblocks Other citrus 0/1 Indicator of other citrus on citrus 62, multiblocks Dormant multiblock status 0/1 Indicator of dormant status on 62, multiblocks Positive find in positive units 0/1 Indicator of positive find in 11, aggregate units with finds Orange acres (multiblocks with Orange acreage on subset of 45, oranges) multiblocks with oranges Grapefruit acres (multiblocks with Grapefruit acreage on subset of 9, grapefruit) multiblocks with grapefruits Tangerine acres (multiblocks with Tangerine acreage on subset of 2, tangerines) multiblocks with tangerines Other citrus acres (multiblocks with Other citrus acreage on subset of 4, other citrus) multiblocks with other citrus Positive find 0/1 Indicator of positive canker find 259, in calendar year 10 km 2 Unit Aggregates Positive find count Count of number of positive finds in 2, aggregate unit Acreage Total citrus acreage (10,000 acres) 2, Orange share of acreage Orange share of total unit citrus 2, acreage Grapefruit share of acreage Grapefruit share of total unit citrus 2, acreage Tangerine share of acreage Tangerine share of total unit citrus 2, acreage Other citrus share of acreage Other citrus share of total unit citrus 2, acreage Dormant share of total area Dormant status share of total area 2, Positive status 0/1 Indicator of positive canker 2, status in unit Positive find 0/1 Indicator of positive canker find in unit 2, a The data consist of 333,892 inspections of 62,183 multiblock units, which yielded 259,343 multiblock-inspection-year observations and 2,379 aggregate-unityear observations. generalized Poisson model. The results indicate that increasing the share of oranges in an aggregate unit by 1% raises the probability that no canker will be found by 1.50%, relative to the default category of other citrus fruits. In the case of grapefruit and tangerines, an additional percentage increase in the share of acreage raises the probability that no canker will be found by 0.20%. An interesting discovery is that a positive finding of canker in

12 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1049 Table 2. Poisson, Logit, and Zero-Inflated Models: Positive Finds a Model 1: Model 2: Model 3: ML Poisson and Zero-Inflated Poisson Zero-Inflated Generalized Logit Models Model Poisson Model Parameter Estimate Std. Error Estimate Std. Error Estimate Std. Error Elasticity Logit Selection Mechanism for Positive Finds Intercept Orange share Grapefruit share Tangerine share Positive neighbors (t 1) Total acreage Dormant share Poisson Model of Positive Counts for Positive Finds Intercept Orange share Grapefruit share Tangerine share Positive neighbors (t 1) Total acreage Dormant share Dispersion Equation for ZIGP Model for Positive Finds Intercept Positive neighbors (t 1) Multiblock area Summary Statistics Number of observations 2,379 Log-likelihood Poisson Log-likelihood Logit Log-likelihood ZIP Log-likelihood GZIP Vuong test (1) vs. (2) 6.11 [< 0.01] Vuong test (1) vs. (3) 6.08 [< 0.01] Vuong test (2) vs. (3) 3.57 [< 0.01] a Asterisks indicate statistical significance at the = 0.10 or smaller level, and p-values are given in brackets. Elasticities are evaluated at the mean values of the explanatory factors. Logit elasticities for the kth variable are given by f ( X ) k, where f ( ) = e X /(1 + e X ) 2. The analogous Poisson elasticities are given by X k k. previous years is associated with a lower probability that an infection will be found in the current year. Specifically, a 1% increase in the number of positive finds of canker raises the probability that no canker will be found by 7%. This likely reflects the significant control and mitigation actions undertaken after a positive canker find. Mandatory spraying of fungicides and treatment of workers and equipment prior to entering and leaving groves is typically undertaken after positive canker finds. The ZIGP model also suggests, however, that among those aggregate units with one or more positive canker finds, the expected number of positive finds increases in response to positive finds in the previous year in neighboring units. Each additional percentage increase in canker finds in neighboring units in the previous year raises the expected level of finds by 0.19% in the Poisson portion of the ZIGP model. Aggregate units with larger shares of dormant grove space appear to be significantly more likely to have positive canker finds. In particular each percentage point increase in the area that is in a dormant state lowers the probability that no canker will be found by 0.56%. Conversely, in areas were one or more canker finds occur, the opposite effect of dormant area is true. Each additional percentage point of dormant area lowers the expected number of finds by 0.31%. These results may reflect the fact that dormant groves may have suffered prior infections that resulted in the clearing of citrus trees, thus resulting in a dormant status in the inspection year. An interesting result is that the total citrus acreage within an aggregate unit does not appear to be statistically significant in determining the probability of one or more infections but is significantly associated with a higher positive infection count in units with one or more infections. A 1% increase in the total number of acres in an aggregate unit raises the expected number of infections by 1.27% in units with one or more positive finds. The specification tests of Vuong (1989) strongly support the ZIGP model over more restrictive alternatives. The generalized model

13 1050 November 2009 Amer. J. Agr. Econ. allows the degree of dispersion to vary in response to changes in certain covariates. Preliminary testing suggested a specification that allowed the degree of dispersion (the variance of the count of positive finds) to depend on the total grove area in each aggregate 10 km 2 unit and the number of positive canker finds in the previous year in neighboring units. 13 The results suggest that positive finds in the preceding year and a larger concentration of grove space correspond to a lower variance in the count of positive canker finds. A final consideration in our econometric modeling involved the potential for residual autocorrelation in the spatial dimension. In particular it is common to correct for spatial autocorrelation in models applied to spatial data of the sort considered here. Spatial autocorrelation typically reflects the effects of omitted variables that are themselves correlated in the spatial dimension. We have made a direct attempt to capture such effects in a structural sense by including infections in neighboring units in the preceding year. However, other nonstructural approaches to correcting for spatial correlation are also conceivable, although such approaches may be difficult to implement in the models adopted here. In addition one could argue that a correction for such correlation, which is analogous to conditioning probability forecasts on contemporaneous errors in neighboring units, is not appropriate in our insurance application in that it inherently involves conditioning forecasts on contemporaneous, unobservable factors. Specifically, it is not appropriate to condition our predicted infection probabilities on anything that cannot be observed at the time the predictions are made. In order to consider the potential effects of such spatial correlation, we considered a block-bootstrapping estimation approach. Specifically, we applied the moving block bootstrap procedures of Künsch (1989) and Liu and Singh (1992). Under this approach individual observations are selected at random, and all observations within the same time period that fall within a predefined proximity to the selected unit are also included in the bootstrap sample. We defined blocks as all contiguous units (those with centroids within 10 km) 13 Covariates for the dispersion equation were chosen through a consideration of statistical significance, log-likelihood function values, convergence of the estimates, and the validity of the predicted range of dispersion. Some specifications that were considered implied unreasonably large ranges for the over-dispersion parameter and thus were eschewed in favor of alternative specifications. and reestimated the models. The results (not presented here to conserve space) for the infection probability models were nearly identical with one notable exception. The variable representing infections in neighboring units in the preceding year was no longer statistically significant. Intuitively, this would seem to suggest that the inclusion of previous-year infections in neighboring units does capture the effects of spatial correlation. When an explicit correction for such correlation is considered, this variable is no longer statistically significant, although the results for all other variables are nearly identical. 14 Multiblock Unit Logit Results The first step of our empirical modeling involved estimating the probability that an aggregate 10 km 2 unit will have one or more finds of canker. A next step in deriving an estimate of the actuarially fair insurance premium is to estimate the probability that individual multiblocks within positive aggregate units will be infected. Two possible approaches to measuring this probability are obvious. First, the simplest and most straightforward approach to estimating the probability of infection, conditional on a multiblock being within a positive aggregate unit, is to consider the proportion of multiblocks that are positive within the positive aggregate units. The proportion of multiblocks with positive finds within a year within aggregate units that are positive is Note that this approach does not allow for differences in infection probabilities across individual multiblocks, which may consist of different sizes and fruit types. A more refined approach to estimating the probabilities of canker infections involves estimation of a second logit model that relates the probability of infection for individual multiblocks within positive aggregate units to a set of covariates representing the composition and makeup of the multiblocks. Note that the former approach of using fixed probabilities results in more stable premiums across multiblock units within an aggregate unit. In particular all multiblocks within a 10 km 2 unit are assumed to have the same risk of infection and thus the same premium. In contrast the second step logit model allows each multiblock 14 As a reviewer correctly notes, spatial correlation may have important implications for the systemic risk carried by an insurer in a portfolio of insurance policies. To the extent that risks are highly correlated across space, the degree of diversification offered in a portfolio of canker insurance contracts may be diminished.

14 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1051 Table 3. Logit Model Estimates of Conditional Probabilities at Multiblock Level a Parameter Estimate Std. Error Elasticity Intercept Orange share Grapefruit share Tangerine share Dormant share Total acreage Number of observations 11,334 Likelihood ratio test McFadden s LRI a Asterisks indicate statistical significance at the = 0.10 or smaller level. to have its own risks, which depend on the composition of the multiblock unit as well as the makeup of the aggregate unit to which the multiblock belongs. Of course insurers must weigh the utility of individual rate refinements against the programmatic shortcomings associated with premium rates that vary substantially across neighboring individuals. We estimated a standard logit model of positive canker infection finds for the subsample of multiblocks within aggregate units having a positive infection find. The resulting parameter estimates are presented in table 3. These estimates demonstrate that further rate refinements are possible by accounting for factors specific to an individual multiblock unit. Specifically, the logit estimates suggest that, within aggregate units having a positive canker infections, groves with tangerines are the least likely to realize infection, followed by oranges and then by grapefruits. The highest rates of infection are again implied for the miscellaneous fruits, including lemons and limes. At the multiblock level infection probabilities are shown to vary directly with the size of the groves. An indicator variable corresponding to a dormant status is not statistically significant. Insurance/Checkoff Premiums The ultimate goal of our analysis is to use the estimated risk models to construct measures of actuarially fair premiums for an insurance or indemnity fund. In the context of our analysis, the actuarially fair premium will be set equal to expected loss, which is given by (12) E{Loss ijt }=Payment Pr(Unit Jt = Positive X Jt ) Pr(multiblock ijt = Positive Unit Jt = Positive, Z ijt ) where Unit Jt represents 10 km 2 unit J, observed in year t, having conditioning factors X Jt, and multiblock ijt represents multiblock i in unit J, observed in year t, having conditioning factors Z ijt. Payment represents the payment to be made per acre in the event of a positive canker infection. It is common to present premiums in a unitless rate form by dividing expected loss by the total amount of the payment, such that the rate corresponds to dollars paid in premiums for each dollar of insurance liability. Thus, the actuarially fair premium rate is given by (13) Rate ijt = F J (X Jt ) G ijt (Z ijt ), for i J where F J (X Jt ) is the probability associated with one or more infections in aggregate unit J and G ijt (Z ijt ) is the probability that multiblock i within positive aggregate unit J will be positive for canker in year t. In light of the calculations associated with payments made to compensate growers for canker losses under the tree replacement payment program discussed earlier, a reasonable assumption for defining premiums and indemnity payments is that a unit of citrus stock is worth approximately $10,000 per acre. However, it is likely that individual units of citrus are of a heterogenous quality and age and thus are valued differently, even in the case of a common fruit type or variety. Hence, we limit our evaluation to a consideration of actuarially fair premium rates. As discussed earlier, we consider two different measures of the multiblock-level infection probabilities G ijt ( ) the proportion of positive multiblocks in our sample and the probabilities implied by a logit model. Table 4 contains summary statistics for the estimated premiums for individual

15 1052 November 2009 Amer. J. Agr. Econ. Table 4. Summary Statistics on Premiums Rates (Premium as a Percentage of Total Liability) Standard Model Mean Deviation Min. Max. Standard Logit Model Multiblock Logit Fixed Probability Zero-Inflated Poisson Model Multiblock Logit < Fixed Probability Zero-Inflated Generalized Poisson Model Multiblock Logit < Fixed Probability < Note: Fixed probability refers to the proportion of positive multiblocks within the aggregated units (0.0173). Premium rates are expressed in percentage terms. multiblocks. The mean premium levels are quite similar across the standard logit and ZIP models. In the case of the ML estimates of the standard logit model, the estimated premium rates average 0.05% for the model based on a fixed probability of infection at the multiblock level (0.0173). In the case where the multiblock-level logit model is used to allow rates to vary at the multiblock level on the basis of grove characteristics, rates average 0.038%. As expected, allowing rates to vary at the individual grove level (rather than holding them fixed across all multiblocks within an aggregate unit) results in a much greater degree of variability. In particular for the fixed probability case, rates range from % to 0.76% while rates based on the multiblock logit model vary from to 8.20%. 15 Premium rates based on the ML estimates of the ZIP model are very similar to those presented for the simple logit model. The rates average for the fixed probability case and when based on the multiblock logit estimates. Again, the rates which are allowed to vary with covariates in the multiblock logit model are much more variable. The two preceding cases one based on a simple logit model for one or more canker finds and the second, which incorporates a Poisson model to explain positive levels of infection may be misleading if a degree of 15 As a reviewer correctly notes, using the model with fixed probabilities could result in significant adverse selection since high risk agents may be underpriced if everyone in an aggregate unit is assumed to have the same probability of infection. This wide range of actuarially fair rates which reflect the underlying probabilities of infections could lead to the high risk regions which have much higher rates to reduce production, which could lead to a reduction in the status of disease. Such a result could lower overall risk and present a positive externality for lower risk areas, which would be expected to increase production. over-dispersion is present in the count of positive canker inspections. Alternative models allowing for over-dispersion may offer improvements if the degree of over-dispersion varies across the sample. To address this possibility, we also estimated the zero-inflated generalized Poisson models. Premium rate estimates based on the ZIGP model are also presented in table 4. It is notable that the average premium rates are much larger than was the case where no over-dispersion was allowed. Specifically, the rates average 0.27% for the multiblock logit model and 0.24% for the fixed probability model. The ZIGP model rates are as high as 32%, suggesting that ignorance of the over-dispersion in a standard ZIP or logit model would result in significant underpricing of insurance coverage and thus would result in poor indemnity fund performance. The ZIGP rates are approximately five times as large as those implied by the other model specifications. An even higher degree of rate volatility is implied by the GZIP model estimates. Such volatility is not an unusual result when rating insurance contracts using empirical models. Rare events may imply implausibly low rates. Likewise, risks may be very high for certain units and thus may imply very high premium rates. 16 In all the empirical models provide precise measures of the risk associated with citrus canker infection and suggest actuarially sound premiums for specific-peril insurance against canker risks. In addition to providing improved understanding of the factors associated with canker infection risks, these estimates should be valuable in alleviating adverse selection 16 Consider, for example, base reference premium rates for corn crop insurance in Adair County Iowa (2.1%) and Lake of the Woods County Minnesota (38.5%).

16 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker 1053 concerns associated with all-peril insurance programs. Of course agents could change their behavior under any particular insurance program, and thus, the rates presented here would require monitoring and possible loading to account for the potential of moral hazard actions by insuring agents. Particular moral hazard concerns may pertain to intentional infection of groves that are less productive (due to age or other environmental factors) or that are subject to development pressures to obtain indemnities. Such issues, which may even involve fraud in some cases, remain a challenge for any insurance or indemnification program and may require particular loss-adjustment, monitoring, or other underwriting considerations to address. These factors are not explicitly addressed in our empirical models but should be recognized in the implementation of any insurance program and may require additional adjustments to the rates or terms of coverage. Conclusions This analysis presents empirical models of the infection risks associated with ACC in Florida citrus. A brief overview of the history of citrus canker outbreaks in Florida is provided. We also highlight biological aspects of citrus canker that are important for measuring infection risk. We discuss methodological issues associated with the design of specific-peril crop insurance and/or indemnification plan programs that would provide risk protection for Florida citrus producers. We present a specificperil program that offers to indemnify only those damages associated with citrus canker infections. As we have noted, specific-peril insurance plans may permit a more precise measure of risk and thus may be less vulnerable to adverse selection issues than is the case with traditional multi-peril plans. The overarching goal of our analysis is to construct empirical risk models that allow us to quantify the risks of canker infection and identify actuarially fair premiums or checkoff charges that should be paid for this protection. Our empirical models of risk are based on the results of a series of citrus grove inspections undertaken as a part of the Citrus Canker Eradication Program in Florida. We consider models of the frequency of canker infections. Canker infection is a relatively rare event, at least over the period of our study. Thus, standard count data models, such as a Poisson regression model, may not be appropriate due to the preponderance of zero counts within our data. Zero-inflated models, such as the zero-inflated Poisson (ZIP) model of Lambert (1992) may offer advantages in addressing this issue but may also suffer from specification issues. In particular standard ZIP models assume a fixed relationship between the mean and variance of infection counts. As an alternative, we adopt the zero-inflated generalized Poisson model of Czado et al. (2007), which permits the degree of over-dispersion to vary according to certain explanatory variables. We present nonnested specification testing results that strongly support the ZIGP specification over a standard logit/poisson and ZIP specification. Our models reveal that the risks of infection varies across different types of fruit. The risk is lowest for oranges, followed then by tangerines and grapefruits. Minor citrus commodities, including limes and lemons, are found to face the highest risk of infection with canker. Our empirical models also reveal important spatiotemporal aspects of infection. The model estimates are used to rate insurance/indemnity fund plans. Actuarially fair premium rates calculated from the standard models range from about 0.04% to 0.066%. However, when the ZIP specification is generalized to permit variable over-dispersion, average rates rise considerably and range from 0.24% to 0.27%. This suggests that indemnification plans could be seriously underfunded if the contribution rates were to be based on the standard models. Furthermore, the actuarial performance of the programs could be subject to adverse selection due to inaccurate rates if rates were to be based on standard models. The models suggest that the risks and thus premiums for protection are highest in the southern regions of Florida. This area is notable in that it has realized the highest incidence of canker infection. Our empirical models and associated measures of actuarially fair premiums provide explicit measures of the expected costs associated with citrus canker at the individual grove level. These estimates may be of significant value to U.S. policy makers who face the task of formulating cost-effective risk management programs for U.S. agricultural producers. Specifically, the models reveal differences in risks across different regions, different types of fruit, and different production units. If insurance or indemnity plans are not priced according to this underlying risk, inaccurate pricing will lead to distortions in participation

17 1054 November 2009 Amer. J. Agr. Econ. and the distribution of program benefits across producers. In particular inaccurate rates based on some aggregate risk measure typically overcharge low risks and undercharge high risks, resulting in skewing the benefits in favor of higher risks (the case of adverse selection). In addition to the obvious implications for pricing insurance or for establishing contribution rates for an indemnification pool, our estimates provide policy makers with a mechanism for quantifying the effects of certain factors on the risk of canker infection. Knowledge of these risk factors may be beneficial in targeting risk mitigation measures and tailoring other program parameters in a manner that makes them more efficient. For example, knowledge of the differences in the expected loss due to canker for individual areas or fruit types may assist policy makers in prioritizing eradication efforts by focusing on eliminating infected stock in areas posing the greatest risks of transmission. The methodology applied here could also be employed to measure other agricultural risks, such as soybean rust and karnal bunt, in an effort to gauge the cost of a single-peril program to address these risks. References [Received July 2006; accepted March 2009.] Alexander, F.E., R.A. Cartwright, and P. McKinney A Comparison of Recent Statistical Techniques of Testing for Spatial Clustering: Preliminary Results. In P. Elliott, ed. Methodology of Enquiries into Disease Clustering. London: London School of Hygiene and Tropical Medicine, pp Consul, P., and G.C. Jain A Generalization of the Poisson Distribution. Technometrics 15: Czado, C., V. Erhardt, A. Min, and S. Wagner Zero-Inflated Generalized Poisson Models with Regression Effects on the Mean, Dispersion and Zero-Inflation Level Applied to Patent Outsourcing Rates. Statistical Modelling 7: Erhardt, V The ZIGP Package. The Comprehensive R Archive Network. Available at: index.html (accessed September 9, 2008). Florida Agricultural Statistics Service. 2007a. Citrus Summary Florida Department of Agriculture and Consumer Services, Tallahassee FL, September b. Commercial Citrus Inventory Florida Department of Agriculture and Consumer Services, Tallahassee FL, February. Goodwin, B., and A. Ker Modeling Price and Yield Risk. In R.E. Just and R.D. Pope, eds. A Comprehensive Assessment of the Role of Risk in U.S. Agriculture. Norwell MA: Kluwer Academic Publishers, pp Goodwin, B.K., and V.H. Smith The Economics of Crop Insurance and Disaster Aid. Washington DC: AEI Press. Gottwald, T.R., G. Hughes, J. Graham, X. Sun, and T. Riley The Citrus Canker Epidemic in Florida: The Scientific Basis of Regulatory Eradication Policy for an Invasive Species. Phytopathology 91: Jackman, S., A. Tahk, A. Zeileis, C. Maimone, and J. Fearon The pscl Package. The Comprehensive R Archive Network. Available at: pscl/index.html (accessed September 9, 2008). Künsch, H.R The Jacknife and the Bootstrap for General Stationary Observations. Annals of Statistics 17: Lambert, D Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics 34:1 14. Liu, R.Y., and K. Singh Moving Blocks Jacknife and Bootstrap Capture Weak Dependence. In R. Lesage and L. Billard, eds. Exploring the Limits of the Bootstrap. New York: Wiley, pp Ridout, M., C.G. Demetrio, and J. Hinde Models for Count Data with Many Zeros. Paper presented at International Biometric Conference, December, Cape Town, Available at webfiles/zip/ibc fin.pdf (accessed March 16, 2009). Rothenberg, R.B., and S.B. Thacker Guidelines for the Investigation of Clusters of Adverse Health Events. In P. Elliott, P. Cuzick, D. English, and R. Stern, eds. Geographical and Environmental Epidemiology: Methods for Small Area Studies. London: Oxford University Press. Sarker, R., and Y. Surry The Fast Decay Process in Outdoor Recreational Activities and the Use of Alternative Count Data Models. American Journal of Agricultural Economics 86: Schubert, T., T. Gottwald, S. Rizvi, J. Graham, X. Sun, and W. Dixon Meeting the Challenge of Eradicating Citrus Canker in Florida- Again. Plant Disease 85: USDA-APHIS. 2005a. Citrus Canker: Background. Plant Protection and Quarantine Fact Sheet, Washington DC. Available at:

18 Goodwin and Piggott Spatiotemporal Modeling of Citrus Canker background.html (accessed May 18, 2005) b. Citrus Canker: Chronology. Plant Protection and Quarantine Fact Sheet, Washington DC. Available at: usda.gov/ppq/ep/citruscanker/chronology.html (accessed May 18, 2005) Q s and A s About Citrus Canker Lost Production Payments. Plant Protection and Quarantine Fact Sheet, Washington DC. Available at: ep/citruscanker/background.html (accessed May 18, 2005) Q s and A s About Citrus Canker Tree Replacement. Plant Protection and Quarantine Fact Sheet, Washington DC. Available at: planthealth/content/printableversion/faq/ phccanktree.pdf (accessed August 7, 2008). Vuong, Q.H Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses. Econometrica 57: