Empirical Bayes Nonparametric Density Estimation Of Crop Yield Densities: Rating Crop Insurance Contracts

EMPIRICAL BAYES NONPARAMETRIC DENSITY ESTIMATION OF CROP YIELD DENSITIES: RATING CROP INSURANCE CONTRACTS A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by ANAS RAMADAN In partial fulfilment of requirements for the degree of Masters of Science August, 2011 c Anas Ramadan, 2011 ABSTRACT EMPIRICAL BAYES NONPARAMETRIC DENSITY ESTIMATION OF CROP YIELD DENSITIES: RATING CROP INSURANCE CONTRACTS Anas Ramadan University of Guelph, 2011 Advisors: Professor Alan Ker This thesis examines a newly proposed density estimator in order to evaluate its usefulness for government crop insurance programs confronted by the problem of adverse selection While the Federal Crop Insurance Corporation (FCIC) offers multiple insurance programs including Group Risk Plan (GRP), what is needed is a more accurate method of estimating actuarially fair premium rates in order to eliminate adverse selection The Empirical Bayes Nonparametric Kernel Density Estimator (EBNKDE) showed a substantial efficiency gain in estimating crop yield densities The objective of this research was to apply EBNKDE empirically by means of a simulated game wherein I assumed the role of a private insurance company in order to test for profit gains from the greater efficiency and accuracy promised by using EBNKDE Employing EBNKDE as well as parametric and nonparametric methods, premium insurance rates for 97 Illinois counties for the years 1991 to 2010 were estimated using corn yield data from 1955 to 2010 taken from the National Agricultural Statistics Service (NASS) The results of this research revealed substantial efficiency gain from using EBNKDE as opposed to other estimators such as Normal, Weibull, and Kernel Density Estimator (KDE) Still, further research using other crops yield data from other states will provide greater insight into EBNKDE and its performance in other situations iv Acknowledgments I would like to thank Dr Alan Ker, my advisor, for his support and guidance His professional expertise in the topic and his patience has been a tremendous help and encouragement during my time at FARE I would also like to thank my committee members, Dr Radhey Singh and Dr Getu Hailu, for their input and insights Special thanks to the entire faculty at FARE who have contributed greatly in my academic endeavour Also, many thanks to the staff who were very supportive throughout the past two years I would also thank all my peer students who were my family away from my family Finally, I thank my parents for their unconditional love and support, and my wife for her extreme patience and positive attitude v Table of Contents List of Tables vii List of Figures viii Introduction Insurance challenges 2.1 Adverse selection 2.2 Moral hazard 2.3 Modeling yield densities and 2.4 Concluding remarks premium rates Methods 3.1 Parametric methods 3.1.1 Normal distribution 3.1.2 Weibull distribution 3.2 Nonparametric 3.2.1 Bandwidth selection 3.3 Empirical Bayes Nonparametric Kernel Density Estimator 3.3.1 Variance estimation of a kernel 3.4 Summary 7 11 13 15 17 18 18 19 21 24 27 29 30 Data 31 Empirical work 5.1 Temporal process of corn yields 5.2 Heteroskedasticity 5.2.1 Heteroskedasticity tests 5.2.2 Correcting for Heteroskedasticity 5.3 Density estimation 5.4 Summary 35 35 37 37 41 44 52 Methods evaluation 6.1 Remarks 54 61 Conclusion 63 vi Bibliography A Appendix A.1 Normal MLE A.2 Bayesian posterior mean A.3 The functions file A.4 Body code A.5 Data 66 71 71 74 76 83 103 vii List of Tables 5.1 5.2 5.3 5.4 5.5 Heteroskedasticity results: Total detections on ǫˆ Heteroskedasticity results: Total detections on ǫ˜ Heteroskedasticity results: In percentage on ǫˆ Heteroskedasticity results: In percentage on ǫ˜ Summary of premium rates 42 44 50 51 52 6.1 6.2 Results of simulated game EBNKDE1 Results of simulated game EBNKDE1 56 57 A.1 A.2 A.3 A.4 A.5 A.6 Data Data Data Data Data Data 1/6 2/6 3/6 4/6 5/6 6/6 106 107 108 109 110 111 viii List of Figures 1.1 1.2 1.3 FCIC aggregated programs summary GRIP program summary GRP program summary 4 3.1 3.2 KDE of random data points Oversmoothing and undersmoothing 23 25 4.1 4.2 4.3 Annual yield for two randomly sampled counties Annual mean yield for corn of Illinois counties High yield variation at higher yield levels for two random counties 32 33 34 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 LS applied on a random county GQ data sectioning Inflation process applied on a random county Errors before and after heteroskedasticity correction Correct scaling preserves the shape Multiple estimated densities on a county Different bandwidth KDE on a county EBNKDE1 and EBNKDE2 KDE1, EBNKDE1, and µ ˆ for a random county KDE1, EBNKDE1, Normal 36 40 43 43 43 45 46 47 48 49 6.1 Zoom-in of Figure 5.10 58 Chapter Introduction Farmers can face reduced or possibly total loss of income due to poor yields caused by significant weather variations (Ozaki, Goodwin, and Shirota 2008) They seek affordable protection against the financial burden resulting from poor yields Insurance providers, whether public or private, require estimates of the expected loss, or, equivalently, the actuarially fair premium rate, in order to price the insurance contract The actuarially fair premium rate is the rate at which an insurance contract has no expected profit (i.e., premiums equal expected indemnities) The estimation of actuarially fair premium rates for crop insurance requires the estimation of the underlying crop yield densities In the United States, the federal crop insurance program is the primary source of protection for farmers as there is no private crop insurance (with the exception of private insurance for hail) The literature suggests that private crop insurance does not exist for two main reasons: (i) adverse selection which occurs when highrisk farmers buy more insurance than low-risk farmers at a given price (Skees and Reed 1986; Quiggin, Karagiannis, and Stanton 1993); and (ii) moral hazard which occurs when farmers undertake riskier farming practices (Chambers 1989; Horowitz and Lichtenberg 1993; Smith and Goodwin 1996) Multi-peril crop insurance (MPCI) (an individual farm yield-based insurance) has traditionally provided primary coverage to farmers The government continues to face pressure from farm lobbies to introduce many insurance bundles The Federal Crop Insurance Corporation (FCIC) has introduced a variety of risk management tools and programs as instruments to address problems associated with crop insurance including adverse selection and moral hazard (Harwood et al 1999) FCIC created diverse insurance bundles to give farmers greater flexibility in choosing appropriate coverage Despite this, FCIC continues to deal with moral hazard and adverse selection problems (Knight 2009) Farmers currently have a variety of insurance bundles that provide revenue and yield coverage at both the farm and county level The types of insurance that target farm revenue-based losses are Crop Revenue Coverage (CRC), Revenue Assurance (RA), and Income Protection (IP) Group Risk Income Protection (GRIP) is an example of coverage that is area revenue-based rather than farm revenue-based The thesis focuses on Group Risk Plan (GRP) Under GRP, farmers receive indemnity payments based on the county’s yield shortfall instead of individual farm yield This solves the problem of moral hazard but not that of adverse selection Knowing the actuarially fair premium rates would enable FCIC to successfully eliminate the problem of adverse selection (Skees and Reed 1986; Goodwin 1994; Just, Calvin, and Quiggin 1999; Knight and Coble 1999; Babcock, Hart, and Hayes 2004) Eliminating adverse selection would result in reduced subsidies, increased participation rates, reduced costs to the program, and, subsequently, reduced cost to taxpayers (Coble and Knight 2002) 98 ] / ( c l ∗Exp y [ z , i ] ) r a t e group [ z , i ,27]