Bayesian Model Averaging, Mixture Models, and Rating Crop Insurance Contracts 1

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1 Bayesian Model Averaging, Mixture Models, and Rating Crop Insurance Contracts 1 Preliminary Draft # 1: Please do not cite without permission. Alan P. Ker Yong Liu Tor N. Tolhurst University of Guelph March, 2015 Some key words: multiple density estimation, Bayesian model averaging, rating crop insurance contracts Abstract The Agricultural Act of 2014 solidified insurance as the cornerstone of U.S. agricultural policy. The Congressional Budget Office (2014) estimates this Act will increase spending on agricultural insurance programs by $5.7 billion to a total of $89.8 billion over the next decade. In 2014, total government liabilities associated with the crop insurance program exceeded $108.5 billion. In light of the sizable resources directed toward these programs, accurate pricing or rating of insurance contracts is of utmost importance to farmers, private insurance companies, and the federal government. Unlike most forms of insurance in which sufficient information exists to accurately estimate the probability and magnitude of losses (i.e. the underlying density), agricultural insurance is plagued by a paucity of spatially correlated data. A novel application of Bayesian Model Averaging and mixture models is used to estimate a set of possibly similar densities that offers greater efficiency if the set of densities are similar while seemingly not losing any if the set of densities are dissimilar. Standard simulations indicate that finite sample performance -- in particular small sample performance -- is quite promising. The BMA methodology is relatively easy to implement, does not require knowledge of the form or extent of any possible similarities, admits correlated data, and can be used with either parametric or nonparametric estimators. The proposed methodology is used to estimate U.S. crop insurance premium rates for area-type programs. A simulated out-of-sample game between private insurance companies and the federal government highlights the policy implications for a number of crop-state combinations. Two tests are developed to evaluate the efficacy of the proposed methodology. Also of policy relevance is the expansion of area-type programs, via the Supplemental Coverage Option, to crops and states with significantly less historical data. To this end, the empirical analysis is also undertaken for a variety of reduced sample sizes. As with the standard finite sample simulations, the performance of the proposed methodology for rating area-type crop insurance contracts -- in particular small sample performance -- appears quite promising. 1 Contact info: Alan P. Ker, Professor and Director (Institute for the Advanced Study of Food and Agricultural Policy), Department of Food, Agricultural and Resource Economics, University of Guelph, aker@uoguelph.ca. Yong Lui is a Ph.D. student, Department of Food, Agricultural and Resource Economics. Tor Tolhurst is a research associate, Department of Food, Agricultural and Resource Economics.

2 2 1. Introduction Federally regulated crop insurance programs have been a part of U.S. agricultural policy for upwards of 80 years. In 2014, total liabilities held by the federal government exceeded $108.5 billion. Over the past twenty years, crop insurance has spread from traditional crops like corn, soybean, and wheat to speciality crops including, for example, walnuts, mustard as well as crops for biofuels like camelina. The Agricultural Act of 2014, commonly referred to as the farm bill, solidified publicly subsidized insurance as the cornerstone of U.S. agricultural policy and the primary venue to funnel monies to the agricultural sector. The Congressional Budget Office (2014) estimates this Act will increase spending on agricultural insurance programs by $5.7 billion to a total of $89.8 billion for the next decade. The federal crop insurance program is administered by the United States Department of Agriculture s Risk Management Agency (RMA). This agency is responsible for a number of aspects of operating the program but most notably setting the premium rates. Interestingly, RMA is not the delivery agent to the farmers; that is done by private insurance companies through a reinsurance agreement with RMA. RMA operates 17 different programs with additional programs being developed as a result of the 2014 Act. While most of the crop policies sold are at a farm or sub-farm level, RMA also operates area yield and revenue insurance programs. The actuarially fair premium rate (π) is defined as the expected loss as a percentage of total liability. That is, defining the random variable crop yield as Y, the actuarially fair premium rate for insurance coverage below a yield gaurantee, denoted y G, is: (1) π = yg 0 (y G y)f Y (y I)dy/y G where the conditional yield density f Y (y I) needs to be estimated and I is the set of information at time of rating. Unlike most forms of insurance in which sufficient information exists to accurately estimate the probability and magnitude of losses (i.e. the underlying density), agricultural insurance is plagued by a paucity of spatially correlated data. The standard approach in the literature is to take a series of historical yields for the area of interest and (i) estimate the temporal process; (ii) if necessary adjust the residuals for heteroskedasticity; and (iii) estimate the conditional yield density of interest using the prediction and (adjusted) residuals. Rates are derived using the estimated density and equation 1. Deterministic and stochastic approaches have been considered in estimating the temporal process (i.e. technological change) of yields. Deterministic approaches have dominated the literature and include a simple linear trend, two-knot linear spline (Skees and Reed, 1986), and polynomial trend (Just and Weninger, 1999). Stochastic approaches include the Kalman filter (Kaylen and Koroma, 1991) and ARIMA(p, d, q) (Goodwin and Ker, 1998). More recently, Ozaki and Silva (2009) and Claassen and Just (2011) incorporated spatial information into their temporal model. Finally, Tolhurst and Ker (2015) using mixtures allowed for heterogeneous rates of technological change across subpopulations of the yield distribution.

3 3 With the exception of Just and Weninger (1999) and Harri et al. (2011), heteroskedasticity has received little attention in the literature considering the magnitude of its effect on crop insurance rates. Harri et al. (2011) found that arbitrarily imposing proportional heteroskedasticity or homoskedasticity in premium rate setting for crop insurance limited actuarial soundness. They proposed a methodology whereby they estimate, rather than assume, the degree of heteroskedasticity. A wide variety of density estimation approaches have been proposed. In 1958, Botts and Boles first suggested the use of Normal curve theory to determine crop insurance premium rates. Day (1965) argued crop yield densities displayed non-normal attributes such as significant skewness. In response, Gallagher (1987) suggested the gamma distribution while Nelson and Preckel (1989) suggested the beta distribution. Later parametric specifications included the logistic (Atwood, Shaik, and Watts, 2003) and Weibull distributions (Sherrick et al., 2004). Inverse sine transformation methods were used by Moss and Shonkwiler (1993); Ramirez (1997); Ramirez, Misra, and Field (2003); Ramirez and McDonald (2006). Maximum entropy yield models have been forwarded by Stochs and LaFrance (2004); Wu and Zhang (2012); Tack, Harri, and Coble (2012); Tack (2013). Normal mixtures have been used by Ker (1996); Goodwin, Roberts, and Coble (2000); Woodard and Sherrick (2011); Tolhurst and Ker (2015). Goodwin and Ker (1998) proposed nonparametric kernel density methods while a semi-parametric approach was forwarded by Ker and Coble (2003). The above approaches do not make use of abundant -- albeit spatially correlated -- extraneous yield data in estimating the density for an area of interest. That is, while there exists historical corn yield data for all counties in Iowa, when estimating the density for say Adams county Iowa, only the yield data from this county is used. The yield data in the other 98 counties is ignored. 2 Given that yields are highly influenced by climate, production practices, and soil type, all of which should be relatively homogeneous across the 99 Iowa counties, the conditional yield densities while not identical, are likely quite similar in shape. The purpose of this manuscript is to propose a methodology -- Bayesian Model Averaging (BMA) -- to estimate a set of possibly similar yield densities that offers greater efficiency if the set of densities are similar while seemingly not losing any if the set of densities are dissimilar. The methodology must accommodate correlated data and not require knowledge as to the form or extent of any similarities. The intuition of the proposed methodology is as follows. Consider if the set of yield densities were known to be identical, the efficient approach would be to pool the sample data and estimate one density. Conversely, if the set of densities were known to be quite dissimilar, the efficient approach would be not to pool nor use any of the sample realizations from the other densities. Bayesian model averaging, which assigns weight to every model in a set of models based on BIC (which itself is data driven) is ideal for this situation and, as needed, does not require knowledge of the extent or form of the similarity given that is unknown. 2 It is worth noting that after the initial rates are recovered, RMA does undertake credibility weighting which smooths the rates across the various counties. The justification for this comes from Stein s Paradox. Because the RMA s loss function is not based on a single county but rather all the counties, premium rates are shrunk toward the mean. The approach here is to improve the density estimate of each county and thereby improve the accuracy of the premium rates before the credibility weighting takes place. As a result, credibility weighting does negate the use of an improved density estimate nor does an improved density estimate negate the use of credibility weighting.

4 4 Consider for example a set of Q crop yield densities with sample realizations {y 11,..., y 1n1,..., y Q1,..., y QnQ }. The estimated densities using sample data from its own density are denoted ˆf 1,..., ˆf Q. The BMA estimate for density i is given by f i = Q j=1 ωi ˆf j j. The weight, ωj i, is a function of the BIC evaluated using the yield realizations from yield density i, that is y i1,..., y ini, evaluated at estimated density ˆf j. The weights necessarily sum to 1 across j. Consider for example the two extreme scenarios, identical and very dissimilar. Note, if the set of densities are very dissimilar, the BIC evaulated using density estimate j and yield data y i1,..., y ini will be high unless j = i. The result will be that ωj i 0 for i j and ωi j 1 for i = j. That is, the BMA estimate and the individual estimate will be quite close. Conversely, if the set of densities are identical, the BIC evaluated using density estimate j and yield data y i1,..., y ini will be relatively constant j. The result will be that ωj i 1/Q i, j. That is, the BMA estimate will be the average of the individual estimates and will resemble a pooled estimate. To the extent that set of densities are similar, the weights will reflect this. 2. Bayesian Model Averaging Model averaging, including both frequentist model averaging (Hjort and Claeskens, 2003) and Bayesian model averaging (BMA), is a popular technique to take model uncertainty into consideration by combining estimates across different models. Early work related to model averaging includes Roberts (1965), who proposed a distribution model which combines the opinions of two experts. Leamer and Leamer (1978) contributed the basic paradigm for BMA by suggesting a Bayesian approach to calculate the posterior probabilities of all the competing models. Kass and Raftery (1995) reviewed BMA and discussed the costs of ignoring model uncertainty. Their work also provided several important techniques for computing Bayes factors, which is the core ingredient of BMA. Madigan and Raftery (1994) presented a treatment for model selection and model uncertainty in graphical models by the standard Bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Draper (1995) proposed a Bayesian approach to assessing the structural uncertainty in model selection. Draper s approach was based on the idea of expanding models. That is, starting with a single best model indicated by the observational data, expanding model space to accommodate all the models that are suggested by the context of research or other considerations, finally averaging over the whole expanded model space. We consider the crop yield model estimated by the county specific data to be the best model while the models estimated using extraneous county data as the set of expanded models filling out the model space. Draper s model expansion idea could be practically implemented by employing Occam s Window. BMA has been applied to many statistical models such as linear regression (Raftery, Madigan, and Hoeting, 1997), generalized linear models (Raftery, 1996), and survival analysis (Volinsky et al., 1997), showing improved predictive performance in all these cases. As a summary of previous works, Hoeting, Madigan, Raftery, and Volinsky (1999) provided a general guidance for the implementation of BMA, including

5 5 several typical methods and related software. Within the context of mixture models, BMA was employed by Wei and McNicholas (2012), but their work focused on clustering and classification instead of density estimation. The general framework of BMA includes three parts. Suppose there are models M 1, M 2,..., M J to be considered and is the quantity of interest. The posterior distribution of given data D is (2) pr( D) = J pr( M i, D)pr(M i D), i=1 where pr( M i, D) is the posterior distribution of under model M i, and pr(m i D) is the posterior model probability of model M i. As it can be seen, pr( D) is an average of the posterior distributions under each of the models considered, weighted by their posterior model probability. The posterior probability of model M i is given as (3) pr(m i D) = pr(d M i )pr(m i ) J i=1 pr(d M i)pr(m i ), and (4) pr(d M i ) = pr(d θ i, M i )pr(θ i M i ) dθ i, where θ i is the vector of parameters of model M i, pr(θ i M i ) is the prior for θ i under model M i, and pr(m i ) is the prior probability of model M i. Although BMA can give better predictive performance than using any single model (Madigan and Raftery, 1994), there are two major difficulties in the implementation of BMA. One is that the number of models in the summation of Equation (2) can be enormous, therefore an exhaustive summation may be practically impossible. The other is that the posterior model probabilities pr(m i D) are difficult to compute since generally the computation involves high dimensional integrals. In this manuscript we restrict the number of models to the number of counties in the data set for that crop. This alleviates the first difficulty. The second difficulty is handled using normal mixtures as per Tolhurst and Ker (2015) to model crop yields. In the context of Normal mixture model, the integral of the Equation (4) can be approximated by using the BIC (Dasgupta and Raftery, 1998). A Laplace approximation to the integral gives: (5) ln pr(d M i ) = ln pr(d θ i, M i ) k ln n. Combine with Equation (??), pr(d M i ) can be computed through the BIC, that is { (6) pr(d M i ) = exp 1 } 2 BIC i,

6 6 where BIC i is the BIC value of model M i. Assuming each model has equal prior model probability, pr(m i D) is computed through Equation (6), i.e., (7) pr(m i D) = { } exp 1 2 BIC i { }. J i=1 exp 1 2 BIC i Equation (7) gives the weights of each mixture model considered, and the final estimate of the yield density is a weighted average of these five mixture densities. That is, (8) f BMA (y) = 5 pr(m i y)f i (y) j=1 where f i (y) is the density of individual mixture Normal, characterized by the parameters estimated using the Expectations-Maximization (EM) algorithm. Intuitively, assume there are Q unknown densities with realizations (x 11,..., x n11,..., x 1Q,..., x nq Q). The standard approach to estimate f i is to use data x 1i,..., x nii and construct some parametric or nonparametric estimate. Conversely, the BMA approach initially estimates each density with data from only that density, denote these f 1,..., f Q. These represent the set of candidate models for the BMA. The weights for posterior estimate for density j, are based on the BIC from using data from density j only but evaluated for all the candidate models f 1,..., f Q. Denote these ω 1j,..., ω Qj. The posterior estimate for density j is then the weight sum of the initial estimates, ˆf j = Q i ω ij f i. While the EM algorithm exhibits monotonic convergence and admissible estimates when the initial values are within the admissible range, the likelihood function is unbounded because it goes to infinity when one of its component variances goes to zero (Karlis and Xekalaki, 2003; Chen and Li, 2009). Therefore, to ensure the admissibility of its estimates, we initialize the algorithm over a number of different sets of starting values and use the penalized likelihood of Chen and Li (2009) to choose the best estimate over the different initializations. For a J component mixture, their penalized likelihood is defined as (9) pl n (λ, µ, σ) = l n (λ, µ, σ) + p n (σ 1 ) + + p n (σ J ) + p n (λ 1 ) + + p n (λ J 1 ) where the penalty on the component variance is (10) p n (σ) = {s 2 n/σ 2 + log(σ 2 /s 2 n)} where s 2 n is the sample variance 3 and the penalty on the J 1 mixture weights are given by (11) p n (λ) = log(1 1 2λ ). Intuitively p n (σ) is maximized at p n (σ)=s 2 n, which prevents under estimation of σ, while p n (λ) is maximized at the boundaries when either λ = 0 or λ = 1. These penalties were chosen to depend on the data and 3 i.e. s 2 n = 1/n n i=1 (X i X) 2 with X = 1/n n i=1 X i.

7 7 satisfy a number of conditions presented in Chen and Li (2009) for p n (σ) and Li, Chen, and Marriott (2009) for p n (λ). 3. Finite Sample Simulations Prior to estimating crop yield densities, a first step would be to investigate the performance of the proposed approach for estimating a set of generic densities (i.e. true densities are known). To this end, we consider the finite sample performance with a best and worst case scenario. The best case scenario assumes the densities are identical while the worst case scenario uses the first nine Marron and Wand (1992) test densities. 4 These densities, which represent a large variety of more or less realistic density shapes, are commonly employed to assess finite sample performance of density estimators (see Hjort and Glad (1995), Jones et al. (1995), Jones and Signorini (1997) among others). The logic is that if the proposed estimator performs well across these various densities, then one can be reasonably assured that the estimator will perform well in an empirical setting. For each test density, 500 samples of size n = {25, 50, 100, 500} were taken. For our models we assume a mixture of normals, estimated using the EM algorithm as outlined above. Minimizing BIC is used to choose the number of mixtures; note this can vary across each sample and density. For the set of densities, the standard approach uses only data from the density of interest and we denote this the individual estimate (denoted above as f i ). The first simulation exercise was constructed to represent a worst case scenario, that is, a situation where the set of densities are very dissimilar and the proposed estimator would not readily come to mind. One could erroneously employ the proposed estimator in an empirical setting believing the densities are similar when in fact they are not. At what cost does this come relative to the alternative estimator, using only the individual data? 5 The MISE and the average weight put on the individual density are located in Table 1 for each of the nine test densities. The BMA column is the standard approach described above. The BMA rs column is where the data from each sample is rescaled first to the sample mean and variance of the from the density of interest. By doing this, the BMA is only applied to the upper moments of the distribution since all candidate distributions will have equal first two moments. This is easily done and may be warranted in some empirical applications where one believes the upper moments are similar but desires to use only the individual sample data to estimate the location or location-scale of the density. There are a number of results: (i) MISE of the BMA and BMA rs is essentially equal to the MISE of the individual estimate; (ii) the average weight on its own candidate distribution is much greater with BMA than with BMA rs as expected; (iii) MISE decreases as the sample size increases; (iv) average weight on the individual density increases as sample size increases; 4 The densities are: (i) standard normal; (ii) skewed unimodal; (iii) strongly skewed unimodal; (iv) kurtotic unimodal; (v) outlier; (vi) bimodal; (vii) separated bimodal; (viii) asymmetric bimodal; (ix) trimodal. The supplementary on-line appendix has the exact densities. 5 Given the flexibility of the proposed estimator, one can transform the samples to have equal mean and variance prior to estimation thereby using the BMA and extraneous data to influence the estimated shape (moments 3) while using the individual mean and variance. If one believes the estimated densities to be of similar shape but different location and scale, this is easily accommodated. We do such a transformation in the worst case scenario).

8 8 (v) density 5 which is very different has the most weight on itself. The results are most significant in that they suggest that the lower bound for the error of the BMA methodology is the individual methodology and thus there is very little risk in applying the methodology, even in the situation where the set of densities are dramatically different. The second simulation, summarized in Tables 2 (MISE) and 3 (individual weights), considers the ideal situation in which all densities are identical (N(0,1)). We consider various sizes of the set of densities to be estimates, Q = {2, 5, 10, 25}. The inidividual estimate as well as the pooled estimate (based on simply pooling all the data) are also included. The results are as expected in a number of important ways: (i) the MISE of the BMA is bounded above by the individual estimate; (ii) the MISE of the BMA is bounded below by the pooled estimate; (iii) the MISE of the BMA realizes efficiency gains as Q increases compared to the individual estimate; (iv) spatial correlation while decreases the efficiency gains of both the BMA and pooled estimate relative to the individual estimate, the efficiency of the BMA estimate relative to the pooled estimate increases; (v) the weight on the individual estimate decreases as the sample size increases but quite slowly; (vi) the weight on the individual estimate decreases as Q increases; and (vi) as spatial correlation increases the weight on the individual estimate slightly decreases. Overall, these simulations suggest that the BMA does no worse than the individual estimate when the underlying densities are quite different and can realize significant efficiency gains, relative to the individual estimate, when the set of densities are identical. The weight on the individual estimate versus the other candidate models is as expected: approaches 1 when the underlying densities are very dissimilar and decreases dramatically when the densities are identical. While one might at first expect that the weights should be close to 1/Q, the weights are based on BIC which is a simple transformation of the likelihood, which itself is dominated by location. The individual data will necessarily place relatively more weight on its own candidate model as that is the model that maximizes the likelihood of realizing the sample data. Spatial correlation can only effect the BMA through the weights. It is not surprising that the weights on its own candidate model decrease as correlation increases because the other estimated candidate models are closer to the individual estimate. However, the amount of additional weight is quite marginal and correlated data effects the MISE of the BMA to a much lesser extent than the pooled estimate. This is expected because while the pooled estimate is comprised entirely of the correlated sample (the data from each of the densities gets weight 1/Q) whereas the BMA puts greater weight on the individual estimate which does not suffer from correlation.

9 9 Table 1. Simulation Results: Worst Case Scenario Sample Size MISEx1000 Average % Weight Individual BMA BMA rs BMA BMA rs n=25 Density Density Density Density Density Density Density Density Density Average n=50 Density Density Density Density Density Density Density Density Density Average n=100 Density Density Density Density Density Density Density Density Density Average n=500 Density Density Density Density Density Density Density Density Density Average Note: BMA1 is Baysian model averaging without rescaling data. BMA2 is Bayesian model averaging with data rescaled.

10 10 Table 2. MISE x 1000: Best Case Scenario n=25 n=50 n=100 n=500 No Correlation Individual Proposed Q= Proposed Q= Proposed Q= Proposed Q= Pooled Q= Pooled Q= Pooled Q= Pooled Q= Correlation = 0.25 Individual Proposed Q= Proposed Q= Proposed Q= Proposed Q= Pooled Q= Pooled Q= Pooled Q= Pooled Q= Correlation = 0.75 Individual Proposed Q= Proposed Q= Proposed Q= Proposed Q= Pooled Q= Pooled Q= Pooled Q= Pooled Q=

11 11 Table 3. Weight of Individual Density (%): Best Case Scenario No Spatial Correlation n=25 n=50 n=100 n=500 Weight Q= Weight Q= Weight Q= Weight Q= Spatial Correlation = 0.25 Weight Q= Weight Q= Weight Q= Weight Q= Spatial Correlation = 0.75 Weight Q= Weight Q= Weight Q= Weight Q=

12 12 4. Application: Rating Area-yield Crop Contracts The motivation for the proposed estimator came from the desire to improve the accuracy of the estimated premium rates for area-type programs in the U.S. crop insurance program. Area yield and revenue insurance has attracted significant attention in the agricultural economics literature as a means to avoid the problems of moral hazard, adverse selection, and the transaction costs associated with individual coverage crop insurance policies. A number of studies discuss the strengths and weaknesses of area-type programs in detail, for example Halcrow (1949); Miranda (1991); Bourgeon and Chambers (2003); Glauber (2013). Fundamentally, an area design triggers based on the observed yield for an area that encompasses several producers. By doing so, moral hazard is mitigated because a producer directly controls inputs for a relatively small percentage of total production. Adverse selection in area yield insurance is argued to be reduced relative to individualyield insurance because of a reduction in private information about the data generating process underlying the yield that triggers indemnities (Ozaki, Ghosh, Goodwin, and Shirota, 2008). Ultimately the functioning of an area yield design is conditional upon the actuarial process used to rate the product. Several studies have either explicitly or implicitly proposed systems to rate area yield designs (for example see Skees, Black and Barnett, and Ker and Goodwin). While these area programs have not been very popular in the past, this will change with the new Farm Bill. The 2014 Agricultural Act introduced an option to supplement individual coverage with an area-type policy allowing farmers to protect against a portion of their individual policy deductible. This option, termed Supplemental Coverage Option, is expected to account for nearly 30% of the estimated increase in government crop insurance spending (Congressional Budget Office (2014)). Actuarial methods for area-type insurance products will be used to rate the supplemental coverage option and as such the proposed estimator will be applied to rating area-yield contracts. A most interesting feature of the U.S. crop insurance program is that government uses private insurance companies, termed intermediaries, to deliver the program to farmers/producers. While the government sets the premium rates for each insurance policy, insurance companies sell the contracts, conduct claim adjustments, and participate in the underwriting gains and losses of those contracts. An underwriting gain is realized if indemnity payments (claims) are less than premiums paid. Conversely, an underwriting loss is realized if indemnity payments are greater than premiums paid. To elicit the participation of the insurance companies, two mechanisms are provided by the government. First, there is a mechanism in which the insurance companies can cede the majority of the liability of an undesirable policy. In a private market, the insurance company would simply not offer such a policy. Second, because the premium rates are set equal to the estimated expected losses, that is they are estimated to be actuarially fair, there needs to be a mechanism in which the insurance companies receive a return to their capital. To that end, the insurance companies share, asymmetrically, the underwriting gains and losses of the contracts sold with the government. While the actual agreement between the private insurance companies and government is somewhat complex (and outlined in the Standard Reinsurance Agreement), it is structured such that insurance companies

13 13 can either cede or retain the majority of the underwriting gains/losses of a policy it sells. Obviously, insurance companies attempt to predict which policies will return an underwriting gain and which will return an underwriting loss. That is, which policies are overpriced and which are underpriced. To do so, insurance companies could, and likely do, re-estimate the premium rates using an alternative methodology and compare their rates to those of the government. For those policies with rates larger than the government rates they cede back to the government as they believe them to be underpriced. Conversely, for those policies with rates lower than the government rates they retain them as they believe them to be overpriced. If there methodology is superior in terms of estimating rates, the policies retained should have a lower loss ratio (ratio of claims to premiums) than the policies ceded. This mechanism allows us to act as an insurance company to evaluate the proposed methodology with respect to the current methodology employed by RMA. This metric is particularly relevant because there exists an avenue in which insurance companies can engage in such adverse selection activities to garner significant profits at the expense of the public program Data and Yield Model We choose corn, soybean, cotton, and winter wheat to evaluate the efficacy of our proposed methodology. For corn and soybean, we use county level yield data from Illinois, Indiana, Iowa, Minnesota, Missouri, Ohio, and Wisconsin. For cotton, we choose Arkansas, Georgia, Louisiana, Mississippi, and Tennessee. Finally, for winter wheat we choose Illinois, Indiana, Kansas, Maryland, Michigan, Missouri, Ohio, Oklahoma, and Tennessee. Historical NASS yields are available from 1955 to present for each county. The purpose of the manuscript is to propose an alternative rating methodology that improves on the current RMA rating methodology. To that end, we employ BMA to make use of abundant extraneous yield data in estimating the density for an area of interest. While BMA can be applied to both nonparametric and parametric models, we follow Tolhurst and Ker (2015) by modeling crop yields using normal mixtures with embedded trend functions to account for potentially different rates of technological change in different components of the yield distribution. For a vector of yields y indexed over time t: (12) y t J λ j N(h j (t), σj 2 ) j=1 where the unknown parameter vectors λ j, σj 2 and functions h j(t) are estimated with a maximum likelihood approach using the EM algorithm for the J components of the mixture. Tolhurst and Ker (2015) indicate the mixture model offers many advantages: it can approximate many of the distributional structures associated with conditional yield densities; embedding possibly unique trend functions within each mixture does not restrict the effect of technological developments to the first two moments of the yield distribution; and a mixture model exhibiting different rates of technological change in different components will lead to a nonconstant variance with respect to time (heteroskedasticity is common in yield data). BIC is used to choose the number of mixtures. The normal mixture model with embedded trend functions is used as the individual

14 14 model. We compare the BMA to the individual model as well as the current rating methodology of RMA. RMA models the temporal process of yields with a two-knot robust linear spline with spatial and temporal priors on the knots. The estimated residuals are then adjusted for heteroscedasticity. The RMA rate is the empirical rate Simulated Game We conduct a repeated game of out-of-sample rating accuracy. To this end, we estimate the 1994 premium rates using both the proposed methodology and the RMA methodology for each of the 102 counties using data from only. The decision rule is to cede those counties where the RMA estimated premium rate is lower than the estimated premium rate using the proposed methodology. We repeat this for 1994,...,2013 using data only from ,..., Using actual realized yields we calculate the loss ratio (defined as total indemnities divided by total premiums) for the county-year combinations we retained as well as the county-year combinations we ceded. The loss ratio for a set of policies, denoted A, is defined as: (13) LossRatio A = where ˆπ RMA is the RMA estimated premium rate. j A max(0, λye j y j) j A ˆπ RMA We conduct this case study at the 90% coverage level (λ) because roughly 95% of the recent policies sold are at this coverage level. In addition, we calculate the percent of total county-year combinations retained using our methodology. To ascertain the statistically significance of our results, we use a randomization test. That is, we randomly select the same percent of contracts as those retained under the decision rule and calculate the loss ratio for the corresponding set. We repeat this 5000 times. The 5000 loss ratios represent the distribution of the loss ratio from the policies we retained using our decision rule under the null that the decision rule is ineffective in identifying more profitable policies Empirical Results The simulated game results are presented in tables 4 and 5. Table 4 presents the out-of-sample rating game between the proposed methodology -- two-trend normal mixture BMA model -- and the current RMA methodology. For the 28 state-crop combinations, economically significant rents can be generated by insurance companies in 27 of the 28 cases of which 23 are statistically significant. Table 5 presents the out-of-sample rating game between the proposed methodology -- two-trend normal mixture BMA model -- and the individual two-trend normal mixture model. For the 28 state-crop combinations, economically significant rents can be generated by insurance companies in 27 of the 28 cases of which 22 are statistically significant. In table 6 we present the average weight on the individual estimate by state and crop. The individual weights seem to be fairly constant across crops and states with corn being marginally less and cotton being marginally higher suggesting more homogeneity of yield densities for corn than for cotton across

15 15 Table 4. Out-of-Sample Rating Game Results Number of Retained by Loss Ratio Loss Ratio Crop-State Counties Payouts (%) Private (%) Government Private p-value Two Trend BMA Method versus RMA Method Corn Illinois Indiana Iowa Minnesota Missouri Ohio Wisconsin Soybean Illinois Indiana Iowa Minnesota Missouri Ohio Wisconsin Cotton Arkansas Georgia Louisiana Mississippi Tennessee Winter Wheat Illinois Indiana Kansas Maryland Michigan Missouri Ohio Oklahoma Tennessee Note: Counties with incomplete yield histories are excluded. Statistical significance of lower private (government) loss ratios indicated by -10% ( -10% ), -5% ( -5%), and -1% ( -1%), respectively. To account for heterogeneity in county acreage, calculations are weighed by share of total crop-state acreage in space. Overall, these results are quite strong and suggest that insurance companies could make significant economic rents using the BMA methodology Reduced Sample Size While our simulated game used the all available historical data ( ), this may be of limited use practically to RMA for two reasons. First, in many places where area-yield rates are now required under the 2014 Agricultural Act, there is significantly less historical data. Second, it can be argued that given technological change, specifically seed technology, that using yield data of more than 20 years may be harmful

16 16 Table 5. Out-of-Sample Rating Game Results Number of Retained by Loss Ratio Loss Ratio Crop-State Counties Payouts (%) Private (%) Government Private p-value Two Trend BMA Method versus Two Trend Individual Method Corn Illinois Indiana Iowa Minnesota Missouri Ohio Wisconsin Soybean Illinois Indiana Iowa Minnesota Missouri Ohio Wisconsin Cotton Arkansas Georgia Louisiana Mississippi Tennessee Winter Wheat Illinois Indiana Kansas Maryland Michigan Missouri Ohio Oklahoma Tennessee Note: Counties with incomplete yield histories are excluded. Statistical significance of lower private (government) loss ratios indicated by -10% ( -10% ), -5% ( -5%), and -1% ( -1%), respectively. To account for heterogeneity in county acreage, calculations are weighed by share of total crop-state acreage in as the underlying density has changed dramatically beyond corrections for the mean and variance. We choose Illinois corn, Illinois soybean, Georgia cotton, and Kansas winter wheat to evaluate the BMA for lower sample sizes. That is, we repeat the empirical game in tables 4 and 5 by assuming we only have 25, 20, and 15 years of historical data. That is, to recover the estimate for 2002 and using only 15 years of historical data, the rates are based on data from only. Table 7 presents the out-of-sample rating game between the BMA methodology and the current RMA methodology for reduced sample sizes. For the 12 state-crop-sample size combinations, economically and

17 17 Table 6. Own-County BMA Weight During Out-of-Sample Simulation Corn Soybean Cotton Winter Wheat Missouri 53.9% Wisconsin 56.5% Georgia 75.3% Missouri 68.3% Wisconsin 43.9% Missouri 55.8% Louisiana 69.7% Tennessee 61.9% Minnesota 43.4% Minnesota 54.9% Mississippi 63.8% Indiana 56.4% Illinois 38.8% Ohio 52.1% Tennessee 56.6% Maryland 54.3% Ohio 37.7% Illinois 49.6% Arkansas 47.7% Illinois 49.6% Indiana 35.4% Iowa 40.0% Ohio 49.2% Iowa 28.8% Indiana 38.9% Michigan 49.0% Oklahoma 47.4% Kansas 42.7% Sensitivity to Number of Years Illinois Corn Illinois Soybean Georgia Cotton Kansas Wheat Sample (39+) 38.8% Sample (39+) 49.6% Sample (39+) 75.3% Sample (39+) 42.7% 25 Years 11.0% 25 Years 17.6% 25 Years 40.9% 25 Years 21.7% 20 Years 8.2% 20 Years 14.2% 20 Years 35.3% 20 Years 16.7% 15 Years 5.7% 15 Years 10.6% 15 Years 29.9% 15 Years 12.5% statistically significant rents can be generated by insurance companies in all 12 cases. Table 7 also presents the out-of-sample rating game between the BMA methodology and the individual two-trend methodology for reduced sample sizes. For the 12 state-crop-sample size combinations, economically and statistically significant rents can be generated by insurance companies in 11 of the 12 cases. In the reduced sample sizes, the results are quite strong and suggest that insurance companies could make significant economic rents using the BMA methodology. Note, that as the sample size is reduced less average weight is placed on the individual candidate model Efficacy of the Proposed Methodology Unfortunately, while the above results suggest that the BMA approach may provide better premium estimates than the individual or current RMA methodology, this is not necessarily the case. 6 While this seems counter-intuitive, there is advantage to the insurance company in that the RMA reveals its rate first. As a result, two effects are necessarily aggregated in the results of the out-of-sample game. The first is that insurance company which reacts to the government rate, will make positive economic rents because of this additional information (denote this the revelation effect). Second, the more efficient estimator will also lead to positive economic rents (denote this the methodology effect). The above tables show the aggregate of the revelation and methodology effects and thus, despite BMA s strong performance, does not indicate its relative efficacy compared to the individual methodology or RMA methodology. 6 Consider the situation in which the RMA estimated rate is normally distributed with mean equal to the true premium rate and variance 0.05 whereas the insurance company estimated rate is normally distributed with mean equal to the true premium rate and variance The two rates are uncorrelated. If the RMA estimated rate is greater than the true rate there is a greater than 50% probability that the insurance company rate is less than the RMA rate and it retains this contract that is expected to make a profit. Conversely, if the RMA estimated rate is less than the true rate there is a greater than 50% probability that the insurance company rate is greater than the RMA rate and it cedess this contract that is expected to make a loss. Therefore, the insurance company is able to retain or cede the contract after the RMA rate is revealed, they may make a profit.

18 18 Table 7. Out-of-Sample Rating Game Results: Sensitivity to Number of Years Number of Retained by Loss Ratio Loss Ratio Crop-State Counties Payouts (%) Private (%) Government Private p-value Two Trend BMA Method versus RMA Method Illinois Corn Sample (39+ Years) Years Years Years Illinois Soybean Sample (39+ Years) Years Years Years Georgia Cotton Sample (39+ Years) Years Years Years Kansas Winter Wheat Sample (39+ Years) Years Years Years Two Trend BMA Method versus Two Trend Individual Method Illinois Corn Sample (39+ Years) Years Years Years Illinois Soybean Sample (39+ Years) Years Years Years Georgia Cotton Sample (39+ Years) Years Years Years Kansas Winter Wheat Sample (39+ Years) Years Years Years Note: Counties with incomplete yield histories are excluded. Statistical significance of lower private (government) loss ratios indicated by -10% ( -10% ), -5% ( -5%), and -1% ( -1%), respectively. To account for heterogeneity in county acreage, calculations are weighed by share of total crop-state acreage in 2011.

19 19 We construct two tests so as to isolate the methodology effect. The first is based on the fact that the revelation effect necessarily leads to positive rents for the insurance company. Because we know the direction of this effect, if the insurance company losses while using methodology A versus the government using methodology B we can conclude that methodology B is more efficient for that state-crop combination. Tables 5, 6, and 8 indicate that the insurance company does not statistically significantly lose economic rents to the government using the BMA methodology. Marginally positive results for the BMA methodology as this is a situation where the null could not be rejected. We repeat the analysis of tables 5 and 8 assuming the insurance company uses the current RMA methodology and the RMA uses the BMA methodology (see Table 9 hypothesis test 1). In this case only for Illinois winter wheat does the insurance company lose statistically significant economic rents using the RMA methodology while the RMA uses the BMA methodology. Again, marginally positive results for the BMA. We repeat the analysis of tables 6 and 8 assuming the insurance company uses the individual two-trend methodology and the RMA uses the BMA methodology (see Table 9 hypothesis test 2). In this case, 15 of the 28 crop-state combinations lose statistically significant economic rents using the two-trend methodology while the RMA uses the BMA methodology. This results are much stronger in favor of the efficacy of the BMA model compared to the individual model. Table 11 shows the results for the restricted sample sizes and, not surprisingly, provides more evidence in favor of the BMA methodology. In two of the 12 cases where the insurance company uses the RMA methodology while the RMA uses the BMA methodology, the insurance company realizes statistically significant losses. In 8 of the twelve cases where the insurance company uses the two-trend methodology while the RMA uses the BMA methodology, the insurance company realizes statistically significant losses. The results from hypothesis test 1 show marginal but positive support for the efficacy of the BMA methodology relative to the current RMA methodology and fairly strong support for the efficacy of the BMA methodology relative to the individual two-trend methodology, particularly in reduced sample sizes. Hypothesis test 1 is relatively weak in that the methodology effect must be so pronounced as to overcome the revelation effect as well as the inherent randomness in the results. If we assume that the revelation effect is either fixed or random but independent of the rating methodology, a stronger test can be constructed. If the methodology effect is zero, that is the efficacy of the two methodologies are equivalent, then the ratio of the revelation effect when the insurance company uses methodology A and RMA uses methodology B to the revelation effect when the insurance company uses methodology B and RMA uses methodology A. That is, the ratio of the two revelation effects should have median 1. We compute this every year for the 20 years of out-of-sample simulations. Assuming yields are independent across years (not space), then we have 20 independent draws with probability 0.5 of being below or above 1 under the null. The total number of draws above 1 is distributed under the null as a binomial with n = 20 and p = 0.5 and thus we can then calculate the probability of observing our total number of draws above 1 under the null. Formally, to test the null that the methodology effect are equivalent between methodologies A and B we do the following. First, assume the insurance company uses methodology A and the RMA uses methodology