ABSTRACT. RAMSEY, AUSTIN FORD. Empirical Studies in Policy, Prices, and Risk. (Under the direction of Barry Goodwin and Sujit Ghosh.

Transcription

1 ABSTRACT RAMSEY, AUSTIN FORD. Empirical Studies in Policy, Prices, and Risk. (Under the direction of Barry Goodwin and Sujit Ghosh.) This dissertation is composed of essays that explore aspects of agricultural policy or farm structure. At this time, the major feature of agricultural policy in the United States is the federal crop insurance program. This program provides multiple forms of subsidized crop insurance to American farmers. The first two-thirds of the dissertation deal with statistical and econometric methods for pricing insurance policies with an eye towards obtaining more accurate premium rates. The last study considers the agricultural sector in Japan and permanent exit of farmers from the agricultural enterprise. As conclusions are reached by way of observed data, the essays in this dissertation are empirical. The first essay examines the effects of relaxed actuarial assumptions on premium rates for revenue insurance policies offered through the federal crop insurance program. An important component of revenue insurance is dependence between the crop yield observed in a year and the relative price of the crop at harvest time. Yields are measured at the county level while prices are obtained from trading of futures contracts at the Chicago Mercantile Exchange. Yield and price are combined to form a distribution of revenue that is used to generate insurance premium rates. In practice, the Risk Management Agency of the United States Department of Agriculture makes several assumptions in obtaining the distribution of revenue. Correlation between yields and prices is assumed to be fixed across all counties within a state. Furthermore, the dependence relationship is assumed to be adequately characterized by a Gaussian copula. In this application, the bivariate copula model is allowed to vary across counties and selected according to fit criteria. The effect of this increased flexibility on the price of insurance is then determined. The second essay uses copula methods to price a crop insurance supplement with losses determined by more than two variables. Private insurers, not able to compete on price for policies in the federal program, recently began offering privately sold supplemental policies. These policies typically extend one or more aspects of the policies sold in the federal program. Supplemental policies that provide additional price coverage are one such category. This paper develops a rating methodology for crop insurance policies that pay out on the maximum or average price observed over an interval. Because the loss is a function of multiple stochastic prices, multivariate copula methods are utilized to construct a joint distribution for the underlying prices. Results indicate important dependencies between prices and yields and in the serial behavior of prices. The final essay examines the factors affecting exits from farming in Japan. Off-farm employment opportunities are thought to have an effect on farm exit rates, though evidence on the sign of this effect has been mixed. This essay considers farm exits in the context of Japanese agriculture, which is

2 characterized by high rates of farm exit, land abandonment, and part-time farm work. Based on data from the Japanese Census of Agriculture in 2000 and 2005, the decision to exit farming is shown to be related to off-farm income as a share of household income, and more specifically to the nature of off-farm work. Off-farm employment with income less than farm income is most strongly associated with continued farm operations. Off-farm work where the farmer is self-employed is associated with accelerated exits. Comparison is made between two models of farm exits: a hierarchical Bayesian linear model and a hierarchical Bayesian Poisson model. Both models perform adequately in terms of predicting exit rates across the prefectures of Japan, although minor preference is given to the Poisson.

3 Copyright 2017 by Austin Ford Ramsey All Rights Reserved

4 Empirical Studies in Policy, Prices, and Risk by Austin Ford Ramsey A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Economics Raleigh, North Carolina 2017 APPROVED BY: Denis Pelletier Xiaoyong Zheng Barry Goodwin Co-chair of Advisory Committee Sujit Ghosh Co-chair of Advisory Committee

5 DEDICATION To my parents and grandparents. ii

6 BIOGRAPHY Ford Ramsey was born in Charlotte, North Carolina and grew up in Marvin, North Carolina. He holds the Bachelor of Music with a Second Major in Economics and Master of Arts in Teaching from the University of North Carolina at Chapel Hill. Ford entered the Ph.D. program in economics at North Carolina State University in His primary area of study is agricultural economics. In the fall of 2017, Ford will join the faculty in the Department of Agricultural and Applied Economics at Virginia Polytechnic Institute and State University. iii

7 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES v vi Chapter 1 How High the Hedge: Relationships between Prices and Yields in the Federal Crop Insurance Program Ratemaking in Federal Crop Insurance Measures of Association and Dependence Copulas Empirical Application Conclusion Chapter 2 Rating Exotic Price Coverage in Crop Revenue Insurance Federal Crop Insurance and Revenue Coverage Copulas for Multivariate Dependence Empirical Results Conclusion Chapter 3 Saying Sayonara to the Farm: Exits from Farming in Japan Entry/Exit, Off-Farm Work, and Government Policy Theory Data and Statistical Models Results Conclusion BIBLIOGRAPHY iv

8 LIST OF TABLES Table 1.1 Properties of Copulas Table 1.2 Frequencies of Selected Copulas Table 2.1 Density Fit Statistics by County Table 2.2 Elliptical Copula Parameter Estimates Table 2.3 Hierarchical Archimedean Copula Parameter Estimates Table 2.4 Probabilities of Loss and Insurance Rates Table 3.1 Town Level Variables and Summary Statistics Table 3.2 Prefecture Level Variables and Summary Statistics Table 3.3 Predicted Exits within Prefectures v

9 LIST OF FIGURES Figure 1.1 Assumed Correlations in Federal Crop Insurance Figure 1.2 Pearson Correlation: Price vs. Yield Figure 1.3 Spearman Correlation: Price vs. Yield Figure 1.4 Kendall s Tau: Price vs. Yield Figure 1.5 Hoeffding s D: Price vs. Yield Figure 1.6 Distance Correlation: Price vs. Yield Figure 1.7 Best Fitting Copula by County Figure 1.8 Difference in Premium Rates Figure 1.9 Difference in Premium Rates Figure 1.10 Histogram and Kernel Density of Rate Differences Figure 1.11 Histogram and Kernel Density of Rate Differences Figure 2.1 Local Linear Regression of Corn and Soybean Yields Figure 2.2 Matrix Plots of Detrended Yields and Relative Prices Figure 2.3 Hierarchical Structures for Atchison County, KS Corn Figure 2.4 Hierarchical Structures for Adams County, IL Soybeans Figure 2.5 Rates for MOI Exotic Price Coverage Figure 3.1 Net Exit Densities Figure 3.2 Income Parameters Figure 3.3 Structure Parameters Figure 3.4 Age Parameters Figure 3.5 Prefecture Parameters Figure 3.6 Prefecture Intercepts Figure 3.7 Scatterplot of True Exits vs. Predicted vi

10 CHAPTER 1 HOW HIGH THE HEDGE: RELATIONSHIPS BETWEEN PRICES AND YIELDS IN THE FEDERAL CROP INSURANCE PROGRAM Markets for agricultural commodities are generally thought to be competitive. Market participants, of which there are many, have nearly perfect knowledge, and the underlying products are homogeneous and divisible. A large infrastructure consisting of grading and quality assurance systems, transportation networks, and price reporting services, has evolved to support trade in agricultural commodities. Most agricultural products are also perishing. Depending on the speed at which the good perishes, this behavior results in a vertical supply curve in the sort run. If demand conditions are perfectly stable, market prices and quantities can then be used to precisely trace the demand curve for the good in question. Under these admittedly strict assumptions, one can obtain the causal effect of a change in quantity supplied on market price. The idea that prices respond to clear the market has led to the 1

11 study of inverse demand systems (Barten and Bettendorf, 1989; Huang, 1988). Similar results can be obtained for agricultural yields as the forcing variable, assuming that the quantity of land in production remains fixed. Yield is defined as the total quantity of the commodity produced in a unit of time divided by the amount of land used in production. As price falls when quantity rises, so should price fall when yields rise. The inverse relationship between prices and yields, implicit in simple models of agricultural markets, is often referred to as the natural hedge. The natural hedge protects farm revenues when prices or yields decline and is a direct consequence of a downward sloping market demand curve. Studies by Timoshenko (1928) and Finger (2012) empirically established the strength of the natural hedge. The degree of inverse dependence between yields and prices is of practical importance in the federal crop insurance program. This program, which provides subsidized insurance to agricultural producers, is now the most expensive instrument of agricultural policy in the United States (Congressional Budget Office, 2014). The vast majority of the liability in the program derives from revenue insurance policies. Rating such policies requires a distribution function for farm revenue, which is constructed in the federal program as a joint distribution function across yield and price. The Risk Management Agency (RMA), which is charged with ensuring the actuarial fairness of policies offered through the federal crop insurance program, assumes that correlation between yields and prices is fixed across counties within a state. RMA also assumes that the dependence structure between yields and prices can be adequately captured by a Gaussian copula. In effect, the same copula model is used for all counties within a state. Both assumptions constitute a priori beliefs about the natural hedge. This paper investigates the practical consequences of these assumptions on crop insurance premium rates. The natural hedge will only hold as far as the market under consideration has been adequately defined. In this case, prices are obtained from the Chicago Mercantile Exchange while yields are measured in counties across the country. Because the market is somewhat artificially constructed, there is little reason to assume that prices and yields will be strongly or inversely related. The relationship is expected to vary across areas that are of differing importance in the price formation process for the Chicago futures market. In spite of this variation, legislation has called for revenue insurance policies to be offered in as many counties as possible. With little theory on which to base actuarial assumptions, specifying the relationship between prices and yields in the federal crop insurance program demands an empirical solution without rigid presuppositions. A pragmatic approach for measuring dependence between yields and prices is to make use of copula functions; copulas allow for modeling of the dependence structure and the marginal distributions to be separated. With these two components, a joint distribution can be formed. Because the marginal behavior of prices and yields has been extensively investigated, our interest 2

12 lies primarily in the dependence relationships between these variables. For policy purposes, it is important to know how this dependence varies over space, and whether the relationship can be adequately described by Pearson correlation or a Gaussian copula. Spatial variation in dependence may manifest itself in terms of correlation or rank correlation, but could also involve differences in other dependence concepts like radial symmetry or tail dependence. We consider a number of different copulas to characterize the dependence structure of the data. Bozic et al. (2014) addressed similar questions in the context of dairy margin insurance. Our methods and insurance application are closer to Goodwin and Hungerford (2015). However, Goodwin and Hungerford (2015) only considered a subset of counties in the corn belt where correlation is expected to be strongest and one can reasonably assume negative dependence. Our application considers all corn producing counties in the United States and thus we utilize a wider variety of copulas. Prices are still taken from the Chicago Mercantile Exchange so any spatial variation comes from differences in county yields. In particular, the finding of positive dependence between prices and yields in some counties requires a number of different copula rotations. Counties tend to be clustered by the strength and direction of dependence between prices and yields. Similar clustering occurs in terms of the best fitting copula, implying spatial differences in tail symmetry and tail dependence. Premium rates for revenue insurance are ultimately determined by both strength of dependence and the form of the copula used to construct the joint distribution. Our results allow us to determine whether insurance rates differ systematically across space as a result of clustering of dependence elements and copula properties. A standard problem with pricing crop insurance in the United States is the limited availability of historical yields. Ker, Tolhurst, and Liu (2015) designed a Bayesian procedure to pool information from yield densities in different locations. As the copulas are distribution functions, our results suggest that there could be similar gains from methods that pool information from similar copulas. An easily implemented method would be to pool yield data using concentric circle smoothing. In the sense that concentric circle smoothing can be viewed as a strict form of model averaging, more delicate approaches might be designed. As shown in this study, fixing the strength and structure of dependence within states leads to large differences in premium rates for some counties. Given these differences, and a policy focus on provision of federal crop insurance in as many counties as possible, we suggest the flexible approach to ratemaking developed herein. 1.1 Ratemaking in Federal Crop Insurance The federal crop insurance program functions as a public-private partnership where private insurers service the policies offered through the program. As of the 2017 fiscal year, there were sixteen 3

13 insurers approved to provide insurance coverage under the United States Department of Agriculture s (USDA s) Standard Reinsurance Agreement. The Risk Management Agency, which is a division of the USDA, sets the parameters of the underlying policies and also determines which policies will be offered through the federal program. The public-private aspect of federal crop insurance has been a component of the program since 1981 and is often cited as a major reason for the growth of insurance uptake since that time (Glauber, 2004). Many types of insurance are offered through the federal program and the availability of policies differs by location. The most popular types of policies are revenue insurance policies that pay out on lost revenue. Revenue is the product of price and yield. In federal crop insurance, revenue policies are priced by constructing a joint distribution function of prices and yields. The joint distribution function maps to a distribution function of revenue. If price and yield have an inverse relationship, then revenue variance will be less than revenue variance under independence of these components (Bohenstedt and Goldberger, 1969). The loss on the most basic revenue insurance policy is given by Loss = max(0, Y P P P λ Y H P H ) (1.1) where Y P and P P are expected (planting time) yields and prices and Y H and P H are realized (harvest time) yields and prices. 1 λ is a coverage level between 0 and 1. At the time the policy is sold, the only stochastic variables in Equation 1.1 are Y H and P H. The marginal distributions of these quantities can be reasonably specified based on previous studies. Details of actuarial aspects of the federal crop insurance program can be found in Coble et al. (2010). Prior to the development of revenue insurance, yield insurance was the main policy purchased through the federal program. Losses under yield insurance policies are determined only by yield shortfalls. Price variation alone cannot trigger an indemnity. These policies are offered at both farm and county levels, and insuring on area yields minimizes adverse selection and moral hazard problems (Skees, Black, and Barnett, 1997). Botts and Boles (1958) suggested that yields followed a normal distribution. Later studies focused on distributions that could accommodate skewness and other non-normal features. Gallagher (1987) suggested a gamma distribution while Nelson (1990) and Nelson and Preckel (1989) utilized the beta distribution. The beta distribution was used to model yields for federal crop insurance until the advent of the common crop insurance policy (COMBO) rating method, though Sherrick et al. (2014) found that the Weibull distribution performs well in terms of out-of-sample bias and efficiency. Goodwin and Ker (1998) and Ker and Goodwin (2000) applied variants of nonparametric kernel density methods 1 The revenue insurance policy developed here is closest to the Revenue Protection Harvest Price Exclusion and Area Risk Protection Insurance policies offered through the federal program. The actuarial problem that we address is equally applicable to more complicated policies such as Revenue Protection. 4

14 to achieve greater flexibility in the modeling of yields. Tolhurst and Ker (2015) used parametric mixtures to bridge the gap between parametric and nonparametric approaches. They also exposed interesting features of the evolution of yields over time, suggesting that yields may follow two underlying distributions corresponding to normal and catastrophic situations. Earlier work by Tack, Harri, and Coble (2012), using maximum entropy approaches, reinforced the point that weather and irrigation variables have significant effects on higher moments of yield distributions. The Risk Management Agency s COMBO rating methods are based on censored normal distributions (Coble et al., 2010). Most density estimation methods cannot be directly applied to observed yields because agricultural yields are subject to both trends and heteroskedasticity. In many applications, the estimation of yield densities follows a two step process. The first step corrects for trends and heteroskedasticity. Corrected yields are used as inputs for standard density estimation procedures in the second step. Gallagher (1987) used linear regression to adjust for trends. Goodwin and Ker (1998) utilized a number of autoregressive integrated moving average (ARIMA) models to detrend observed yields. Examples of some different approaches to the detrending problem can be found in Miranda and Glauber (1997) and Zhu, Goodwin, and Ghosh (2011). More recently, Hungerford and Goodwin (2014) and Goodwin and Hungerford (2015) applied nonparametric methods to account for trend and eliminated conditional heteroskedasticity by recentering yields. Due to the popularity of the model of Black and Scholes (1973), it is often assumed that financial prices are distributed lognormally. Whether this is the appropriate distribution for prices has been debated. Research by Yang and Brorsen (1992) and Hsieh (1989) document features of price distributions that violate normality. Goodwin, Roberts, and Coble (2000) considered these issues in the context of crop revenue insurance and found mixed evidence for assumptions of lognormality. Modeling of prices must be balanced against the practical needs of insurers. At present, revenue insurance actuarial procedures assume lognormally distributed prices in accordance with the Black-Scholes model. Volatility is estimated from observed options prices. An additional task in developing rates for revenue insurance is modeling the relationship between crop yields and the normalized average price of a futures contract at the Chicago Mercantile Exchange. The normalized price is constructed by dividing a realized monthly average price by the monthly average price for the crop before the growing season begins. This normalized price can be thought of as a return over the growing season. According to the theory of the natural hedge, the normalized price should be inversely related to yields. An underlying assumption is that farmers do not strategically adjust their inputs or effort in the middle of the insurance policy period. While most studies couch the natural hedge in terms of Pearson correlation, there is little to suggest that dependence between these two quantities must be linear. Some estimates of the natural hedge, 5

15 using a variety of prices and quantities, can be found in Li and Vukina (1998); Hanson, Myers, and Hilker (1999); Adhikari, Belasco, and Knight (2010); and Finger (2012). There are two assumptions made in the pricing of federal revenue insurance that relate to the natural hedge, or more generally to the structure of dependence. Dependence between prices and yields is assumed to be adequately described by a Gaussian copula, further implying that yields and prices are (asymptotically) tail independent. It is also assumed that the correlation matrix for the Gaussian copula is fixed within states. For instance, CME prices and corn yields in Erie County, Pennsylvania have the same dependence relationship as CME prices and corn yields in Bucks County, Pennsylvania. In a majority of states, these assumptions imply that the variables are independent of one another. Because the underlying relationship between prices and yields almost certainly does not respect state boundaries, but rather follows a smooth spatial process, such assumptions are likely to result in flawed pricing. The Risk Management Agency faces several constraints when designing and implementing policies through the crop insurance program. They are mandated to design policies within reason that are actuarially fair: the premium is equal to the expected loss. Because there are ongoing efforts to expand crop insurance coverage to additional crops and locations, the agency also faces the practical concern of designing policies that can be implemented in the face of varying administrative costs, producer demand, and differences in geographic units. On the first point, strict assumptions in pricing, which are not borne out in the data, have the potential to seriously undermine the actuarial fairness of the program. Assessing the suitability of these assumptions is thus a basic issue of policy research. At the county level, historical information on crop yields may only be available for a small number of years. The situation for individual farm yield and revenue histories is even more problematic. An obvious solution to the problem is to pool data from surrounding counties or units. Nearby counties are more likely to be subject to similar weather and pests. In other words, they face similar perils. Zhu, Goodwin, and Ghosh (2014) used spatial autoregressive models to investigate systemic spatial correlation in yield distributions. Several papers have considered data pooling or model averaging in the context of area yield insurance. Ker, Tolhurst, and Liu (2015) used Bayesian mixture models to pool information from estimated densities, whereas Park, Brorsen, and Harri (2016) pooled information prior to the estimation of the density. These findings raise the question of whether dependence relationships between prices and yields are clustered geographically. As dependence is almost surely subject to some spatial correlation, efficiency in pricing might be obtained through similar pooling schemes. This paper makes two essential contributions towards the pricing of revenue insurance in the federal program. The first is to determine the effect, on premium rates, of allowing the dependence 6

16 structure between prices and yields to be driven by the data. This requires measuring those aspects of dependence that are important from an insurance perspective and providing flexible approaches for modeling. By considering data from all corn producing counties, the second contribution is to expose the geographic nature of dependence. These spatial linkages would not be evident if our dependence models were not suitably flexible. Taken together, these contributions suggest changes in actuarial methods that could result in improved premium rates, improved loss ratios, and a greater variety of insurance offerings. 1.2 Measures of Association and Dependence Many distribution-free measures have been developed to measure association and dependence between two or more variables. These measures vary in their ability to capture different types of dependence and in their computational simplicity. Increasingly large datasets have made it essential to be able to detect associations of all types in short time and at low computational cost. Statistics of association are useful tools for exploring the dependence structure without making assumptions on distributional form. They can also be used to parameterize copula models. In our later empirical application, the statistics described in this section are applied to fully analyze dependence between prices and yields. Dependence between prices and yields is often measured in terms of the Pearson correlation coefficient. With random variables X and Y, and a random sample of size n, the population and sample Pearson correlations are ρ P = C o v (X, Y ) i ρ P = (x i x )(y i ȳ ) V a r (X ), V a r (Y ) i (x i x ) 2 i (y (1.2) i ȳ ) 2 where x and ȳ are the sample means of X and Y. Colloquially, the terms correlation and dependence have become nearly synonymous. And when correlation is discussed, Pearson correlation is usually implied. This in spite of the fact that, from elementary statistics, it is widely known that Pearson correlation is only appropriate for measuring linear relationships. The magnitude of Pearson correlation is only invariant to linear transformations of X and Y. It is fallacious to say that a joint distribution can be defined from its marginal distributions and Pearson correlation (Embrechts, McNeil, and Straumann, 2002). The notion is false, except in some special cases such as when the joint distribution of the variables is Gaussian. Assumptions of normality are often based on appeals to the Central Limit Theorem. But in many cases, normality appears to be accepted for the convenience of the analyst, even if there is no economic or statistical justification. Pearson correlation provides an adequate description of dependence only in certain 7

17 situations. A full description of dependence between variables requires more general measures of association. Other nonparametric measures of association can capture nonlinear and nonmonotonic dependence relationships. Monotone dependence occurs when, if one variable increases, the other variable tends to increase as well. Criteria for measures of monotone association, or bivariate concordance, were developed by Scarsini (1984). Spearman s rank correlation and Kendall s tau satisfy these criteria, while Pearson correlation does not. Suppose now that X and Y have a given joint distribution function and let (X 1, Y 1 ) and (X 2, Y 2 ) be independent and identically distributed random vectors with the given joint distribution function. Then the population and sample Spearman rank correlations, due to Spearman (1904), have the form ρ S =3(P [(X 1 X 2 )(Y 1 Y 2 ) > 0] P [(X 1 X 2 )(Y 1 Y 2 ) < 0]) i ρ S = (R i R )(S i S) i (R i R ) 2 i (S i S) 2 (1.3) where R i and S i are the ranks of x i and y i respectively. R and S are the means of the ranks. From Equations 1.2 and 1.3 it should be clear that Spearman s rank correlation has the same formula as Pearson correlation, but with the calculation based on ranks instead of the levels of the variables. Because Spearman rank correlation is only a function of ranks of the data, it is invariant to monotone increasing transformations of X and Y and does not rely on an assumption of linearity. The statistic, which can take any value in [ 1,1], obtains the values -1 and 1 when the variables are monotone decreasing or increasing functions of one another. Kendall s tau is another rank correlation statistic with population and sample versions formulated as τ =P [(X 1 X 2 )(Y 1 Y 2 ) > 0] P [(X 1 X 2 )(Y 1 Y 2 ) < 0] i < j τ = (sgn(x i x j )sgn(y i y j )) (T0 T 1 )(T 0 T 2 ) (1.4) with T 0 = n(n 1)/2, T 1 = k t k (t k 1)/2, and T 2 = l u l (u l 1)/2. In this case, t k is the count of tied x values in the k th group of tied x values, and u l is the number of tied y values in the l th group 8

18 of tied y values (Kendall, 1938). The sgn( ) function is defined as 1 if x > 0 sgn(x ) = 0 if x = 0 1 if x < 0 (1.5) Kendall s tau measures the number of concordant and discordant pairs of observations in the data. If all pairs are concordant then the statistic in Equation 1.4 will equal 1. This indicates perfect concordance and positive dependence. If the pairs are perfectly discordant, then the value of Kendall s tau is -1, indicating negative dependence. Both Kendall s tau and Spearman rank correlation are empirical estimators of the concordance function of Nelsen (1993). They differ only by a normalizing constant. Both Spearman s rho and Kendall s tau can be used to test for independence. However, such tests have little power against alternative hypotheses of nonmonotonic dependence. Hoeffding s D dependence coefficient facilitates tests of independence when the alternative is nonmonotic. It was introduced in Hoeffding (1948) and the population and sample versions are D = (F X Y F X F Y ) 2 d F X Y D = 30 (n 2)(n 3)D 1 + D 2 2(n 2)D 3 n(n 1)(n 2)(n 3)(n 4) where D 1 = i (Q i 1)(Q i 2), D 2 = i (R i 1)(R i 2)(S i 1)(S i 2), and D 3 = i (R i 2)(S i 2)(Q i 1). The term Q i is the bivariate rank of point i, which is 1 plus the number of points with both x and y less than the value of the i th point. If the data do not include any ties among the observations, then Hoeffding s D is bounded in [.5, 1] and takes a value of 1 in cases of complete dependence. Hoeffding s D was developed from the definition of independence, which is that two variables are independent when F (x, y ) = F (x )F (y ). Hoeffding (1948) considered the distance between the joint distribution and the product of the marginals, which should be zero when the variables are independent. Hoeffding s D is an unbiased estimator of this distance and thus can be used to test for independence against a wide variety of alternatives. The final measure we consider is the distance correlation proposed by Székely, Rizzo, and Bakirov (2007) and applied by Székely and Rizzo (2009). Generalize X and Y by allowing them to take values in a multidimensional Euclidean space. Let X be distributed according to µ and Y be distributed according to v. With a µ (x ) = E [ X x ] and a v (y ) = E [ Y y ] then (1.6) d µ (x, x ) = x x a µ (x ) a µ (x ) + E [a µ (X )] (1.7) 9

19 The term d v (y, y ) is analogously constructed. The square of the population distance covariance is then Population distance correlation is d C o v 2 (X, Y ) = E [d µ (X, X )d v (Y, Y )]. (1.8) d C o r (X, Y ) = d C o v (X, Y ) d V a r (X )d V a r (Y ) (1.9) The sample distance correlation can be obtained by using sample estimators of the terms in Equation 1.9. Beginning with the Euclidean distance matrices a k l = ( x k x l ) and b k l = ( y k y l ), let A k l = a k l ā k. ā.l + ā.. (1.10) where ā k. = 1 n n a k l, i =1 ā.l = 1 n n k=1 a k l, ā.. = 1 n 2 n a k l (1.11) k,l =1 The term B k l is similarly obtained. The sample distance covariance is then defined as V 2 = 1 n 2 n A k l B k l (1.12) k,l =1 with the sample distance correlation given by V 2 (x,y ) V 2 (X )V 2 (Y ) > 0 ρ D = V 2(x )V 2 (Y ) 0 V 2 (X )V 2 (Y ) = 0 (1.13) Distance correlation lies in the interval [0,1] with the value zero corresponding to independence. Distance correlation can be used to test for independence against all alternatives with finite second moments. The payment for being able to detect nonmonotonic dependence, in the case of both Hoeffding s D and distance correlation, is that the statistic loses its ability to inform on the direction of dependence. Another appealing property is that although not particularly useful in this bivariate case distance correlation is well defined for random variables in any dimension. Distance correlation is a generalization of Pearson correlation, and for a bivariate normal distribution, distance correlation is a deterministic function of Pearson correlation. Much like Spearman rank correlation, distance correlation can be computed on ranks of the data leading to a rank distance correlation statistic. Because distance correlation is capable of finding complicated dependence 10

20 structures, there is little gain in transforming the data to ranks. As demonstrated in Székely and Rizzo (2009), distance correlation performs well in detecting nonlinear dependence even when the sample size is relatively small. Simon and Tibshirani (2014) found distance correlation to have more power compared to Pearson correlation and the maximal information coefficient. Of the measures of dependence considered here, distance correlation is the most general and widely applicable. 1.3 Copulas Even if measures of association provide some clue as to the magnitude and direction of dependence, it is often still the case that a joint distribution must be formed and evaluated. One approach to forming the distribution and incorporating nonlinear dependence is through the use of copula functions. The theorem of Sklar (1959) is the fundamental existence theorem for copulas, although work on standardized distributions predates Sklar s theorem. Let F be a joint distribution function with univariate marginal distribution functions F 1,..., F d. Then there exists a copula function C : [0, 1] d [0, 1] such that F (x 1,..., x d ) = C (F 1 (x 1 ),..., F d (x d )) (1.14) where x 1,..., x d are random variables. Provided that the marginal distribution functions are continuous, the copula function is unique. Additionally, if the variables forming the joint distribution are continuous, the copula is a function of univariate marginals that are distributed Uniform(0,1). Copulas provide a way of constructing joint distributions and simultaneously describing scale-free or rank dependence. By inversion of the joint distribution in Equation 1.14, the copula function can be written as C (u 1,..., u d ) = F (F 1 1 (u 1 ),..., F 1 d (u d )) (1.15) where F1 1,..., F 1 d are one dimensional quantile functions and u 1,..., u d [0, 1]. The copula is parameterized by a vector θ consisting of dependence parameters. Given that the copula is itself a joint distribution function, it satisfies all of the criteria for a joint distribution function; its corresponding density function can be similarly derived. The copula density c (u 1,..., u d ), provided that it exists, can be obtained by taking partial derivatives such that c (u 1,..., u d ) = d C (u 1,..., u d ) u 1 u d (1.16) There are three important and foundational copulas that warrant exposition and to which we will refer later. The first is the independence copula. In the bivariate case, two variables with continuous 11

21 margins will be independent if and only if the copula has the form C (u 1, u 2 ) = u 1 u 2 (1.17) This follows naturally from the definition of independence; two variables are independent if and only if their joint distribution is equal to the product of the marginal distributions. The copula in Equation 1.17 is often referred to as the product copula. If two variables are functionally dependent with x 1 = h(x 2 ), and h( ) a generic increasing function, then their copula takes the form C (u 1, u 2 ) = min(u 1, u 2 ) (1.18) Any draws from the copula will have rank correlation of 1. Two variables are comonotonic if and only if their corresponding copula is given by Equation If the function characterizing the relationship between the variables is decreasing, then in two dimensions the copula takes the form C (u 1, u 2 ) = max(u 1 + u 2 1, 0) (1.19) resulting in rank correlation of -1. Two variables are countermonotonic if and only if their copula is given by Equation The copulas in Equations 1.18 and 1.19 are known as the Frechet-Hoeffding upper and lower bounds respectively. Any bivariate copula will satisfy the inequality max(u 1 + u 2 1, 0) C (u 1, u 2 ) min(u 1, u 2 ) (1.20) Because all copulas must lie within the Frechet-Hoeffding bounds, these bounds can be viewed as characterizing the most extreme forms of dependence. Some copulas do not attain the bounds for any value of their dependence parameters, and thus are limited in their ability to realize strong positive or negative dependence. Copulas allow for a joint distribution function to be described in terms of the marginal behavior of the underlying variables and their dependence structure. Several features of the dependence structure may be of interest. Popular measures of dependence like the Pearson correlation coefficient are often incapable of discriminating between differences in these features. From an empirical perspective, the choice of a parametric copula function involves a priori assumptions about the nature of dependence. Some of the most important properties relating to copulas are symmetry, radial symmetry, joint symmetry, associativity and Archimedeanity, max-stability, and tail dependence (Li and Genton, 2013). 12

22 A bivariate copula is symmetric if C (u 1, u 2 ) C (u 2, u 1 ) = 0 (1.21) for all (u 1, u 2 ) [0, 1] 2. In the two dimensional case, this amounts to equal copula densities at points reflected on either side of the diagonal bisecting the unit square. Many copulas satisfy the condition of Equation 1.21, so it is usually more informative to determine whether the copula satisfies stricter symmetry concepts. The copula is radially symmetric if C (u 1, u 2 ) C (1 u 1, 1 u 2 ) + 1 u 1 u 2 = 0 (1.22) for all (u 1, u 2 ) [0, 1] 2. Moreover, the copula is jointly symmetric if C (u 1, u 2 ) + C (u 1, 1 u 2 ) u 1 = 0 (1.23) C (u 1, u 2 ) + C (1 u 1, u 2 ) u 2 = 0 (1.24) for all (u 1, u 2 ) [0,1] 2. A detailed discussion of these, and other symmetry concepts, can be found in Nelsen (1993). Radial symmetry is often called tail symmetry. Two points that are equidistant from the middle of the unit square, and lie on rays pointing in opposite directions from the middle of the unit square, will have the same copula density. Joint symmetry is a form of symmetry that most copulas do not satisfy; the product copula is one of the rare copulas that is jointly symmetric. One rule linking these symmetry concepts is that jointly symmetric copulas are radially symmetric. Joint symmetry does not imply symmetry and vice versa. Similarly, radial symmetry does not imply symmetry. Intuitively, the most interesting type of symmetry is radial symmetry because radially asymmetric copulas have different dependence relationships in the lower and upper tails of the distribution. These differences in dependence can often be motivated by appeals to economic theory. From a practical standpoint, insurers and financial analysts are usually interested in behavior in the lower tail as this is where worst case portfolio losses occur. An Archimedean copula is defined as a function C : [0, 1] d [0, 1] where C (u 1,..., u d ) = Ψ(Ψ 1 (u 1 ) + + Ψ 1 (u d )) (1.25) and the function Ψ : [0, ) [0, 1] possesses several properties. Namely, Ψ(0) = 1, Ψ( ) = 0, and ( 1) i Ψ i 0, for i = 1,...,. These conditions imply that Ψ is a strictly decreasing differentiable function. This function is referred to as the generator of the copula. There are many functions which 13

23 satisfy the requirements on the generator, and so there are many different copulas within this class. It should be clear that all Archimedean copulas are symmetric. They are also associative with C (C (u 1, u 2 ), u 3 ) = C (u 1, C (u 2, u 3 )) (1.26) for all (u 1, u 2, u 3 ) [0, 1] 3. Max-stable copulas arise as the limiting copulas for componentwise maxima of independent and identically distributed random variables. Copulas are extreme-value copulas if C (u 1,..., u d ) C h (u 1/h 1,..., u 1/h d ) = 0 (1.27) for any h > 0. It is difficult to find mention of the term copula in the extreme value literature. In fact, the copula concept was independently discovered in literature on extremes, but was termed the dependence function (Deheuvels, 1978). This application does not concern extreme values, but there are many uses for extreme value copulas in insurance and finance more broadly (Coles, 2001). Copulas can also be distinguished by their tail dependence. The upper tail dependence coefficient is defined as λ U = lim u 1 P r (U 1 u U 2 u) = lim u 1 1 2u + C (u, u) 1 u while the lower tail dependence coefficient is defined as (1.28) λ L = lim u 0 P r (U 1 u U 2 u) = lim u 0 C (u, u) u (1.29) A copula has tail dependence if and only if the respective coefficients in Equations 1.28 and 1.29 are nonzero. Tail dependence is an asymptotic concept, and when the tail dependence coefficient is close to one, large values of the variables occur together. For a given copula model, the tail dependence coefficients are functions of the underlying dependence parameters. The choice of the copula family is intrinsically a choice of theoretical tail dependence. Li and Genton (2013) developed a nonparametric test to determine the structure of copulas from given data. The basic idea is that a parametric copula family can be selected using the structure suggested by the test. Their test is based on the asymptotic distribution of the empirical copula function; it is only appropriate for reasonably large sample sizes. The test does not address tail dependence. The list of copula properties in this section is not exhaustive and various authors have been able to exploit other interesting features of copulas such as separability and homogeneity (Hennessy and Lapan, 2002). The five most popular parametric copulas are the Gaussian, t, Gumbel, Clayton, and Frank. 14

24 With various rotations, these copulas can capture a wide range of dependence behavior. None of the copulas are jointly symmetric, and only one is an extreme value copula. In the application that follows, none of the data are extreme values and we have no reason to believe that joint symmetry should be imposed. Therefore, we are confident in the ability of the selected copulas to adequately describe dependence as observed in the data. Both the Gaussian and t copulas are extensions of multivariate elliptical distributions. The bivariate Gaussian copula has the form C N (u,σ) = Φ(φ 1 (u 1 ),φ 1 (u 2 )) (1.30) where Φ is the bivariate standard normal distribution function parameterized by the correlation matrix Σ. The φ are distribution functions for standard normal random variables. The normal copula will generate the standard normal distribution if and only if the marginal distributions are standard normal. As the correlation parameter approaches either -1 or 1, the Gaussian copula attains the lower and upper Frechet-Hoeffding bounds respectively. When the correlation parameter is 0, the Gaussian copula is equivalent to the independence copula. As a generalization of an elliptical distribution, the Gaussian copula is both symmetric and radially symmetric. Its tail dependence coefficients are always zero except in the pathological case when Σ = 1. The Student s t copula is defined as C t (u,σ, v ) = t(t 1 (u 1 ), t 1 (u 2 )) (1.31) In this formula, t is the bivariate Student s t distribution parameterized by the correlation matrix Σ and the degrees of freedom parameter v. As v converges to infinity the t copula converges to the Gaussian. Like the Gaussian, when the correlation parameter approaches either -1 or 1, the t approaches the Frechet-Hoeffding bounds. When the correlation parameter is 0, the independence copula is not obtained. Symmetry and radial symmetry are both properties of the t copula, which has tail dependence for all positive values of v. The remaining three copulas are all Archimedean copulas. The Clayton copula is C C (u,θ ) = 2 i =1 u θ i /θ (1.32) where θ > 0. As the dependence parameter approaches zero, the Clayton copula approaches the independence copula. And as the parameter approaches infinity, the Clayton copula approaches the upper Frechet-Hoeffding bound. The Clayton is symmetric but not radially symmetric. This 15

25 Table 1.1: Properties of Copulas Properties of Five Popular Copulas and the Frechet-Hoeffding Bound Copulas Copula Tail Dependence Radial Symmetry Archimedean Associative Normal No Yes No No t Yes Yes No No Gumbel Yes No Yes Yes Clayton Yes No Yes Yes Frank No Yes Yes Yes Fr.-H. Upper Yes Yes No No Fr.-H. Lower No Yes Yes Yes radial asymmetry is manifested in terms of tail dependence with dependence in the lower tail but independence in the upper tail. The Gumbel copula has the form C G (u,θ ) = e 2 i =1 ( log u i ) θ 1/θ (1.33) with θ > 1. As the dependence parameter approaches one, the copula approaches the independence copula. As the parameter approaches infinity, the copula approaches the upper Frechet-Hoeffding bound. As with the Clayton, the Gumbel is only able to capture negative dependence and thus cannot obtain the Frechet-Hoeffding lower bound. The Gumbel is symmetric but not radially symmetric, and of the copulas considered here, it is the only extreme value copula. The upper tail dependence coefficient is positive, while the lower tail dependence coefficient is always zero. The Frank copula is given by C F (u,θ ) = θ log i =1 e ( θ u i ) 1 + e ( θ ) 1 (1.34) with θ /0. As the parameter approaches zero, the copula devolves to the independence copula. As it approaches negative infinity and infinity, the copula approaches the Frechet-Hoeffding lower and upper bounds. The Frank can be distinguished from its fellow Clayton and Gumbel Archimedean copulas by its ability to capture negative dependence. The Frank is both symmetric and radially symmetric with tail independence in both the upper and lower tails. Table 1.1 lists major properties of the Gaussian, t, Clayton, Gumbel, and Frank copulas as well as the Frechet-Hoeffding upper and lower bounds. Note the many types of dependence structures that can be obtained from these copulas. 16

26 Copula parameters can be estimated from observed data by calibration or maximum likelihood approaches. With parametric marginal density functions f (x i θ i ), the density function of the joint distribution F can then be given as d f (x 1,..., x d ) = c (F 1 (x 1 ),..., F d (x d )) f i (x i ) (1.35) For an independent and identically distributed random sample of size n the log-likelihood function is then n L = log f (x 1,..., x d ) (1.36) j =1 Substituting into Equation 1.36 from Equation 1.35 results in n d n L = log c (F 1 (x 1 ),..., F d (x d )) + log f i (x i ) (1.37) i =1 j =1 i =1 j =1 The first term on the right of Equation 1.37 is the contribution of the dependence structure to the log-likelihood and the second term is the contribution of the univariate marginal distributions. One could estimate the joint distribution by maximizing the likelihood in Equation 1.37 over the parameters of the copula and the univariate marginal distributions. Joe and Xu (1996) suggested an alternative procedure, now known as Inference for Margins (IFM), where estimates of the parameters of the marginal distributions are first obtained by maximum likelihood estimation. Then the copula contribution to the likelihood is maximized holding the univariate parameters fixed at their estimates from the first stage. Both approaches result in estimators that are consistent and asymptotically normal. A related approach to IFM is to estimate the joint distribution using semiparametric maximum pseudo-likelihood. This method is discussed in Genest, Ghoudi, and Rivest (1995) and is applicable for continuous variables with no covariates. The copula contribution to the likelihood is maximized using nonparametric estimates for the marginal distributions. As with IFM and maximum likelihood approaches, the semiparametric estimator of the copula parameters is consistent and asymptotically normal. If the estimates of the copula parameters obtained under IFM and maximum pseudolikelihood techniques diverge, this may suggest an inadequacy in terms of the models for the marginal distributions or the copula. Calibration approaches to copula estimation rely on the one-to-one relationship between the parameters of a copula and nonparametric measures of association such as Kendall s tau. Given the single copula parameter θ, the theoretical values for Kendall s tau are: θ /(θ + 2) for the Clayton 17

27 copula, 1 1/θ for the Gumbel copula, and 1 4θ 1 (1 D (θ )) where D = θ 1 θ 0 t /(e t 1)d t when θ > 0 and D = θ 1 θ 0 t /(e t 1)d t + 0.5θ when θ < 0 for the Frank copula. It is fairly simple to invert the first two formulas, and while the inversion for the Frank copula has no closed form, it can be inverted numerically. In the case of elliptical copulas, τ = 2 π arcsin(θ ) where θ is the correlation parameter. Note that the degrees of freedom parameter of the t is independent of its theoretical value for Kendall s tau. Because there are many functions which meet the criteria of a copula, there are many choices for modeling dependence. A general modeling approach is to compare several parametric copulas or to utilize a fully nonparametric copula. Consideration of strictly parametric copulas is often limited to a small range of copulas that may or may not be capable of capturing important aspects of the dependence structure. While several goodness of fit tests for parametric copulas exist, these tests rarely provide any guidance as to the appropriate choice of copula if the proposed copula is rejected (Genest, Remillard, and Beaudoin, 2009; Huang and Prokhorov, 2014; Kojadinovic, Yan, and Holmes, 2011). Such tests may also require large samples when based on comparisons of the estimated copula with empirical copulas. As noted in Joe (2015), the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) can be used to compare parametric copula models. Because the copula has tail dependence and radial symmetry features, in addition to its general strength of dependence, AIC and BIC provide measures of fit that take all dependence aspects into consideration. However, they only provide a ranking of proposed copulas. It could be the case that all of the models under consideration are inadequate for a given problem. A related likelihood-based test from Vuong (1989) has been applied to copulas. The Vuong test can be used to determine if one of two copula models is a better fit, but like AIC and BIC, it does not assess the adequacy of the models under consideration. 1.4 Empirical Application Payouts and premium rates for revenue insurance policies depend on the joint distribution of yield and price. This distribution, in turn, can be classified in several ways depending on its structure. It may have differing types of dependence in the tails of the distribution, asymmetries across tails, and more broadly may vary in exchangeability. There is little economic intuition to guide choice of the joint distribution of prices and yields, and perhaps even less to guide the imposition of differing dependence structures. The only seemingly agreed upon result is that the natural hedge should lead to negative dependence between prices and yields when markets are properly defined. Our data come from the National Agricultural Statistics Service and include annual yields in all corn producing counties between 1980 and In some counties, yields are either not reported or 18

28 zero, often to protect the identity of producers who might be indentifiable. Counties with zero or missing observations were dropped from the analysis. Raw yields in each county were detrended using robust regression and then recentered about the yield in the last year for which the county had data (Huber, 1973). Counties with less than ten years of available historical yields were also removed from the analysis. Prices were obtained from the Chicago Board of Trade and were normalized as the logarithm of the October average price (P O ) for a December corn futures contract divided by the February average price (P F ) for the same December futures contract. The joint distribution is estimated over this normalized price (ln(p O /P F )), or return, and the detrended county yield in each year. The Risk Management Agency makes two assumptions when rating revenue insurance policies. First, the correlation between prices and yields within a state is assumed to be fixed across counties. Even though counties vary in yields, each county in a state is assumed to have the same yield-price correlation as all other counties in a state. Second, the dependence structure in all counties is specified using a Gaussian copula. We investigate the suitability of these assumptions with respect to the actuarial fairness of the federal crop insurance program and provide estimates of insurance rates under a model that allows for county variation in dependence structures. Figure 1.1 shows the assumed correlations for four different crops with revenue insurance available through the federal program. The Pearson correlations used in constructing premium rates for corn range from 0 in most states to -0.4 in Illinois and Iowa. The pattern of correlation follows, very roughly, the total quantities of corn that are produced in different states in the U.S. Figure 1.2 shows estimated Pearson correlations between yields and prices for all corn producing counties. The estimated correlations are similar to those assumed by RMA in states like Iowa and Illinois, but there are large differences in the South and along the Mississippi River. RMA assumes zero correlation in all counties in Mississippi, while counties near the Gulf of Mexico appear to have strong negative dependence. A strange finding is positive Pearson correlation in many counties in eastern North Carolina, South Carolina, Minnesota, and Wisconsin. Similar positive relationships hold when dependence is measured using Spearman s rho or Kendall s tau as in Figures 1.3 and 1.4. There are several reasons why a given county could have yields that are positively dependent with respect to the Chicago Mercantile Exchange price. Instabilities in demand or supply could generate such results, and more broadly, the price formation process in Chicago is determined by many factors. The importance of any single county yield in determining the futures price is practically zero. The farther away the market implied by the revenue policy is from the ideal market underlying the theory of the natural hedge, the less likely the natural hedge is to hold. Copulas were fitted to the data from each county using semiparametric maximum pseudo- 19

29 Figure 1.1: Assumed Correlations in Federal Crop Insurance 20