TOWARDS UNDERSTANDING CROP YIELD SYSTEMIC RISK AND ITS IMPLICATION FOR CROP INSURANCE CHOICES By Xuche Gong A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Agriculture, Food and Resource Economics Œ Master of Science 2019 ABSTRACT TOWARDS UNDERSTANDING CROP YIELD SYSTEMIC RISK AND ITS IMPLICATION FOR CROP INSURANCE CHOICES By Xuche Gong Area based insurance contracts have long been offered to crop producers as an option for risk management. However, the take -up rate f or such programs remains low. In this paper, utilizing RMA unit -level corn yield data and NASS county -level corn yield data, we investigate roles of systemic risk and premiums subsidies in producers ™ choices between area and individual insurance contracts. We find that , on average, systemic risk explains slightly more than one third of total unit yield variability. Systemic risk is high in the S outhern and Western Corn Belt s and its geographic distribution matches well the geographic distribution of county yield variance. Systemic risk increases with both beneficial and stressful heat accumulations, frequency of drought, and land quality. We also study t he lower bound on subsidy rate f or area insurance when normalized by that for individual insurance such that the expected net returns to area yield insurance equals the expected net return of individual yield insurance . We find that th is lower bound is negatively correlated with systemic risk. Producers in high systemic risk counties will require fewer subsidies to possibly choose area insurance over individual insurance. Moreover, we find that were transfer maximization a producer ™s only concern then the current area subsidy rate might be a ma jor deterrent for producers to choose low coverage level area insurance. Raising the area insurance subsidy rate might be a feasible option to induce more area insurance demand because the transfer -equalizing area insurance subsidy rate exceeds 100% for on ly a small fraction of producers. iii I dedicate this work to my parents and grandparents iv ACKNOWLEDGEMENTS I would like to thank my major professor s, Dr. Hongli Feng, and my co -major professor, Dr. David A. Hennessy, for their unremitting guidance and encouragement. This work cannot be done so accomplished without their work. I would also like to thank my other committee members, Dr. Robert Myers and Dr. Kyoo il Kim. They have provided me many useful comments and suggestions when I was writing the paper. I also want to thank my AFRE classmates and my Chinese friends. I cannot go through these two years without their support. Finally, I want to send my deepest gratitude to my parents for their support in my study abroad. v TABLE OF CONTEN TS LIST OF TABLES ........................................................................................................................ vi LIST OF FIGURES .................................................................................................................... vii CHAPTER 1. INTR ODUCTION ................................................................................................ 1 CHAPTER 2. CONCEPTUA L FRAMEWORK ....................................................................... 8 2..1 Modeling Systemic Risk ........................................................................................................ 8 2.2. Systemic Risk and County Growing Conditions ................................................................. 10 2.3. A Brief Introduction of Individual Insurance Contract and Area Insurance Contract .......11 2.4. Calibrating Threshold Relative Area Subsidy Rate ............................................................ 12 2.5. TRASR and Systemic Risk ................................................................................................... 15 2.6. TRASR and Yield Expectations ........................................................................................... 18 2.7. TRASR and Coverage Levels and Protection Factor ......................................................... 21 CHAPTER 3. DATA AND VARIABLES .................................................................................. 23 CHAPTER 4. EMPIRICAL RESULTS .................................................................................... 28 4.1. Measuring Systemic Risk .................................................................................................... 28 4.2. Systemic Risk and County Growing Conditions ................................................................. 32 4.3. Systemic Risk Estimate and Inadequate Investigating Time Horizon ................................ 35 4.4. Calibrating TRASR ............................................................................................................. 36 4.5. TRASR, Systemic Risk, and Area Insurance Demand ........................................................ 43 CHAPTER 5. CONCLUSIO N AND DISCUSSION ................................................................ 46 APPENDICES ............................................................................................................................. 49 APPENDIX A Major Tables and Figures ........................................................................... 50 APPENDIX B Supplementary Tables ................................................................................ 64 REFERENCES ............................................................................................................................ 67 vi LIST OF TABLES Table 1. Descriptive statistics for y ield variables and county growing condition variables ......... 51 Table 2. Descriptive statistics for county systemic risk and its components ................................ 51 Table 3. Regression results for equation (30) ............................................................................... 52 Table 4. Shapley values for each growing condition variables on systemic risk, % .................... 53 Table 5. Sign and significance of among 579 Corn Belt counties ......................................... 53 Table 6. Number of counties with different number of APH group s ............................................ 53 Table 7. Descriptive statistics for unit -level TRASR, conditional on MPCI coverage level and AYP coverage level ....................................................................................................................... 54 Table 8. Premium subsidy rates for individual and area insurance contracts, conditional on coverage level ............................................................................................................................... 55 Table 9. Coverage -level conditional percent of units whose TASR is higher than the current AYP subsidy rate and percent of units whose TASR is higher than 100% ........................................... 55 Table 10. Coveragel -level conditional Pearson ™s correlation test coefficients between county systemic risk and county -level TRASR variables ........................................................................ 55 Table 11. Observation loss after each data screening and merging step ....................................... 65 Table 12. Coverage -level conditional percent of units whose TASR is higher than the current AYP subsidy rate and percent of units whose TASR is higher than 100% ................................... 65 Table 13. Coveragel -level conditional Pearson ™s correlation test coefficients between county systemic risk and county -level TRAS R variables ........................................................................ 66 vii LIST OF FIGURES Figure 1. Share of acres insured by area insurance contracts for all crops, 1993 -2018 ................ 56 Figure 2. Geographic distributions of county growing conditions ............................................... 57 Figure 3. Geographic distributions of county systemic risk and its components ......................... 58 Figure 4. Geographic distributions of ™s value, and sign/significance category. ..................... 59 Figure 5. Geographic distributions of 2008 APH range and number of APH groups .................. 59 Figure 6. Geographic distributions of county -level percent of units whose TASR is higher than the current AYP subsidy rate, condi tional on MPCI coverage level = 75% ................................. 60 Figure 7. Geographic distributions of county -level percent of units whose TASR exceeds the current AYP subsidy rate, conditional on MPCI coverage level = 80% ....................................... 60 Figure 8. Geographic distributions of county -level percent of units whose TASR exceeds the current AYP subsidy rate, conditional on MPCI coverage level = 85% ....................................... 61 Figure 9. Geographic distributi ons of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 75% ................................................................................. 61 Figure 10. Geographic distributions of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 80% ................................................................................. 62 Figure 11. Geographic distributions of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 85% ................................................................................. 62 Figure 12. Geographic distribution of share of corn acres insured by area insurance contracts .. 63 1 CHAPTER 1. INTRODUCTION Since its inception in the 1930s, the Federal Crop Insurance Program (FCIP) has continuously supplied agricultural producers in the United States with insurance products to manage production risks. The program has undergone especially rapid expansion since the enactment of the Federal Crop Insurance Reform Act of 1994, which substantially increased premium subsidy levels. In 2018, crop insurance covered about 85.6% of pla nted acres for the ten major crops and the ratios were even higher for corn, soybeans and wheat 1, providing agricultural producers with a solid safety net. However, the high participation rate come s at a cost. According to the Risk Management Agency (RM A), FCIP has accounted for about $7.3 billion per annum direct costs over the period of 2009 through 2018, making it the most expensive agricultural commodity program in the U.S. 2 In addition, studies have found that the se significant subsid y levels have led to some moral hazard problems, such as changes in farmers ™ input use, crop choice, and acreage decision (Quiggin et al. 1993; Smith and Goodwin 1996; Babcock and Hennessy 1996; Goodwin et al. 2004; Goodwin and Smith 2013; Yu et al 2017), and ha ve caused su bstantial deadweight loss imposed on taxpayers (Lusk 2016). As a result, the federal government has long sought to reduce insurance program costs . A widely proposed option is for government to subsidiz es only the systemic risk part , leav ing the rest to the market (Miranda 1991; Miranda and Glauber 1997; Coble and Barnett 2008; Dismukes et al. 2010; Goodwin and Smith 2013; Congressional Budget Office 2017). 1 The ten major crops are: corn, cotton extra -long staple, cotton upland, oats, rice, sorghum, soybeans, sugar beets, sugarcane, and wheat. Planted acres data are from USDA ™s 2008 Acreage Survey and failed acres are also included. Insurance data are from RM A™s 2018 Summary of Business. 2 Direct cost equals total premiums paid by farmers minus underwriting gains paid to Approved Insurance Providers and total indemnities paid to farmers. Data are from RMA fiDirect Costs of Federal Crop Insurance Program fl at https://www.rma.usda.gov/ -/media/RMAweb/AboutRMA/Program -Budget/18cygovcost.ashx?la=en . 2 Systemic risk in crop insurance market s refers to strong correlation between individua l losses and stems from the fact that yield losses are generally driven by natural disasters that affect a wide range of farms within a given region (Miranda and Glauber 1997). The existence of systemic risk justifies a government ™s subsidies in crop insur ance market to the extent that this risk form undermines an insurers ™ ability to diversify risk across individuals, forces insurers to set premi ums at prohibitively high levels, and eventually leads the insurance market to breakdown (Duncan and Myers 2000) . However, some studies have also found that if government sufficiently refunds loss caused by systemic risk via some form of area based reinsurance contracts or commodity programs, then private insurers might be capable of deal ing with the remaining idiosyncratic risks even in the pr esence of asymmetric information problems, as occurs in other property insurance markets (Miranda and Glauber 1997; Duncan and Myers 2000; Coble and Barnett 2008). The Congressional Budget Office (2017) has also proposed to subsidize only area -based insura nce products as one option to reduce the budgetary costs of crop insurance programs. The potential for efficien tly separating systemic risk from idiosyncratic risk depends on our ability to characterize and measure systemic risk. Systemic risk is also closely related to demand f or area -based insurance. Unlike individual -based insurance, area -based insurance makes indemnity payments based on easily observed area yield or revenue loss , which are generally not influenced by the insured. Thus, costs caused by information asymmetry and high admin istration costs can be substantially reduced ( Halcrow 1949; Miranda 1991; Smith et al. 1994; Skees et al. 1997; Mahul 1999; Vercammen 2000; Barnett et al 2005; Shen and Odening 2013 ). As a result, the USDA introduced the first ar ea yield insurance program and area revenue program in 1990s, and the 2014 Farm Bill further introduced several county -level revenue -based insurance plans (Goodwin and Hungerford 3 2014). However, d emand f or area -based insurance remains low. As shown by Figure 1, the share of area insurance insured acres in total insured acres w as less than 5% in most years and was less than 20% even at its peak year, 2006. Experiences from area -based insurance progr ams and weather index programs in other countries suggest that the existence of basis risk might be an important deterren t to participating in area -based insurance programs (Elabed et al. 2013; Shen and Odening 2013; Clarke 2016; Hill et al . 2016; J ense n et al. 201 8). Basis risk exists when individual yield is poor but area average yield is good. Basis risk decreases when the correlation between individual loss and area loss increases, i.e., when systemic risk increases. Thus, precise estimates of systemic risk shou ld a lso help to identify whether basis risk is the major deterrence to the demand for area -based insurance in the United States. However, given the importance of estimating systemic risk in controlling crop insurance costs, comparatively few studies have investigat ed it. Applying a large size farm -level yield data, Barnett et al. (2005) estimated correlations between farm and county corn yields to range between 0.36 (in Michigan ) and 0.82 (in Illinois ). Using simulated data, Dismukes et al . (2010) reported average nationwide farm -state revenue correlations to be 0.55 for corn, 0.54 for soybeans, and 0.39 for cotton. Claassen and Just (2011) found that systemic variation explains about 48% of farm yield variation for corn in Illinois and 40% for wheat in Dakotas . However, these studies only investigated the correlation between individual yield and area yield. From the perspective of risk management, a more relevant issue is the correlation between individual loss and area loss. As the only ex ception, Zulauf et al. (2013) examined farm -level yield and revenue loss that is systemic with yield and revenue loss at the county, state, and nation level s. They found farm loss systemic with area loss to be large , and t o decline as the geographical aggr egate level increased. However, Zulauf et al. (2013) just simply calculated the ratio of the yield and 4 revenue loss at high geographical aggregate levels over the yield and revenue loss at farm level, and did not model the relationship between farm loss an d area loss . In this article, we apply Miranda ™s (1991) single -factor capital market model to develop a novel theoretically grounded approach to measuring and decomposing systemic risk . We also decompose this systemic component into three components. In hi s innovative study, Miranda (1991) decomposed farm -level yield variability into a systemic component that is correlated with area yield and an idiosyncratic component that is uncorrelated with area yield . Miranda (1991) measured systemic risk by the beta, which measures the sensitivity of unit yield deviation s from expected value to area yield deviation s from expect ed value . This has become a workhorse procedure for crop insurance analysis, and also farm -level policy studies more generally because farm -level data are generally unavailable and the single factor stochastic structure is very useful when simulating farm -level data from county -level data (Mahul 1999; Barnett et al. 2005; Coble and Dismukes 2008; Carriquiry et al. 2 008; Cooper et al. 2012). However, although the beta captures well the co -movement between farm yield and county yield, it does not convey the relative importance of systemic risk with respective to idiosyncratic risk, which is particularly important when determin ing risk structure in the crop insurance market. In this article, we characterize systemic risk as the proportion of unit yield variation that can be explained by county yield related variation. This measurement , the R 2 statistic, is always bounded between 0 and 1 . As we will show , the statistic is determined not only by i) the beta, but also ii) county yield variance and iii) a farm ™s idiosync ratic yield variance. Using a large -scale unit -level data set , we estimate county -level systemic risk for 589 counties across the Midwest. We find that , on average, systemic risk explains about one -third of unit yield variation and is larger in U.S. Southern and Western Corn Belt counties. Moreover, 5 we find that the geographic distribution of systemic risk wel l matches the geographic distribution of county yield variance. This finding is consistent with a strand of literature that find s yield correlations to be higher in extreme weather years (Okhrin et al. 2013; Goodwin and Hungerfor d 2014; Tack and Holt 2016; Du et al. 2017), as counti es with higher yield variance generally experience more adverse weather shocks. In addition to measurement and decomposition , we extend our analysis framework to inv estigate how systemic risk is determined by county growing conditions. Du et al. (2017) proposed and empirically tested a model linking county -level yield -yield dependence with growing conditions. They found a substitution effect between soil and benign wa ter availability levels, but a complementarity effect between soil and beneficial heat variables. Our work extends theirs by investigating how farm -county yield -yield correlation varies with county weather conditions and soil quality. We find that systemic risk is significantly increasing in county heat accumulations (both beneficial and stressful) and drought appearance. Land quality also has a significantly positive effect on systemic risk. Our study also investigates how growing condition variables affec t systemic risk through each of the three systemic risk components and how these mediation effects reinforce or counteract each other. A policy issue that cannot be separated from the character of systemic risk is how to set cost -effective subsidies for area-based programs when subsidized individual yield insurance contracts are also available . As shown in Figure 1, a substantial decline in the share of area insured acres occurred in 2009 . This decline coincided with a reduction in area insurance premium su bsidy rates and increase in enterprise unit insurance premium subsidy rates in compliance with the 2008 Farm Bill. Th e above observation provides a necdotal evidence on the importance of premium subsidies rates on area -based insurance demand. However, though many studies have found that crop insurance demand is sensitive to subsidy levels, most of the se have focus ed on 6 individual -based insurance (Coble et al. , 1996; Goodwin et al. 2004; Shaik et al., 2008; O™Donoghue, 2014; Du et al., 2016). Among the few exceptions, Deng et al. (2007) found that after considering the large premium wedge between individual yield insurance and pr emium subsidies, area yield ins urance was preferred to individual yield insurance by cotton producers but not by soybeans producers . In a theoretical mean -variance preference model, Bulut et al. (2012) found that whenever premium rates for area insurance and individual insurance are bot h actuarially fair then producers should demand full individual insurance and no area insurance. If area insurance is fully subsidized, then area insurance might replace a portion of individual insurance. Although these studies have documented the importan ce of subsidies in determining area insurance demand, question s remains open as to whether current area insurance subsidy rate s provide producers with sufficiently high compensation for their risk exposure under area -based insurance and whether raising are a insurance subsidy rates is a viable option for increasing area insurance demand. Moreover, no study has explored the relationship between systemic risk and the effective area insurance subsidy rate that will induce producers to possibly choose area insu rance over individual insurance. Areas with a high systemic risk and low effective area insurance subsidy rate are ideal places to grow area insurance demand because area insurance provides growers in these regions with comparatively good risk protection a nd they require lower subsidy levels to participate in the program. Recognizing that individual insurance provides better risk coverage in comparison with area insurance, i n this paper the task we set ourselves is to calibrate the threshold relative area subsidy rate ( TRASR ) at which individual insurance and area insurance provide the same expected level of transfer the grower . Thus, below TRASR even risk -neutral growers will not 7 choose area insurance over individual insurance. TRASR thus provides the low er bound o n the ratio of area insurance subsidy rate over individual insurance subsidy rate needed to possibly induce producers to choose area insurance over individual insurance. We then compare TRASR to the ratio of the current area insurance subsidy rate over the current individual insurance subsidy rate to ascertain whether the current subsidy rate structures deter producers from choos ing area insurance. We also check whether an area insurance subsidy rate no less than 100%, i.e., providing free area insurance or better , is a necessary condition for most producers to choose area insurance over individual insurance. Finally, we relate TRASR to systemic risk in order to establish whether there exists a good match between t he risk protection function of area insurance and the effective area insurance subsidy rate . We find that while the current insurance subsidy rate structure does deter most producers from choosing low coverage level area yield insurance contracts, this is not true for high coverage level area yield insurance contracts. For most producers, the minimum required area insurance subsidy rate to possibly choose area insurance over individual insurance is less than 100%, suggesting that when transfer maximization is the grower ™s only concern then raising area insurance subsidy rate s to level s lower than 100% might be a feasible option to induce more area insurance demand . We also find a negative correlation between systemic risk and TRASR, which supports our belie f that high systemic risk counties are indeed ideal areas to implement area insurance. 8 CHAPTER 2. CONCEPTUAL FRAMEWORK In this section, we present the conceptual framework about how we model systemic risk and how we calibrate the TRASR . Some important propositions are also derived. 2..1 Modeling Systemic Risk Our focus is on yield risk and so we assume throughout that price is non -random. In order to avoid unnecessary notation, we set output price equal to 1 and ignore it henceforth. Following Miranda ™s (1991) one factor capital market model, we apply the following model to characterize the relationship between unit yield and county yield, (1) =+()+. Here, and are individual yield and county yield variable s, respectively, while =E(), =E(), E()=0, Var ()==1, and Cov (,)=0. By way of equation (1), we decompose the unit yield deviation from expectation into a systemic component , (), which is correlated with county yield , and an idiosyncratic part, , which is uncorrelated with county yield. The coefficient measures the sensitivity of unit yield deviations from expectation to county yield deviations from expectation. Since 0 is uncommon for crop production , we assume that >0 throughout the article. Also, since E()=0 and =1, we can set 0 without loss of generality. Now =Var () measures the idiosyncratic part of unit yield variance while =Var ()=Var ()+Var ()=+, where =Var () and =Var (). Thus, by assuming no correlation between county yield and a unit ™s idiosyncra tic yield, unit yield variance can also be decomposed into two uncorrelated parts: the product of the square of unit yield ™s sensitivity to county yield and county yield variance; and unit ™s idiosyncratic yield variance. If 9 =0, then there is no idiosyn cratic risk and unit yield is completely determined by county yield in all moments. Systemic r isk, labelled as , is then modeled as the fraction of unit yield variation that can be explained by county yield related variation, (2) =Var [()]Var ()=Var ()Var ()=+ =11+/. Unlike , which is widely used as measure of farm -level systemic risk in crop insurance and farm -level policy studies (Miranda 1991; Mahul 1999; Coble et al. 200 0; Barnett et al. 2005; Coble and Dismukes 2008; Carriquiry et al. 2008; Cooper et al. 2012 ), is bounded between 0 and 1, and provides a straightforward measure of how importan t systemic risk relative to idiosyncratic risk. Values >0.5 suggest that systemic risk is the major risk source faced by the producer so that area insurance might have the potential to reduce more than half of total risks. Equation (2) also shows that systemic risk can be decomposed into three more fundamental components: i) idiosyncratic yield var iance, ; ii) county yield variance, ; and iii) the square of unit yield ™s sensitivity to county yield, . For a given insurance unit, the magnitude of systemic risk is jointly determined by these three comp onents, where Proposition 1 provides simple inferences that can be extracted from the equation . Proposition 1. Ceteris paribus , systemic risk is i) increasing in county yield variance , , and also in the square of unit yield ™s sensitivity to county yield , , ii) decreasing in a unit ™s idiosyncratic yield variance, .2F3 3 Equation (2) also conveys that systemic risk equals 0 when the uni t yield is uncorrelated with county yield or when county yield has no variability . It equals 1 when ever no idiosyncratic risk occurs . As these these extreme cases are unlikely to happen in real life , they are omitted in our discussion. 10 Proposition 1 provides some indicators for which counties and insurance units are likely to have high systemic risk. The most discernible clue is that , ceteris paribus , systemic risk will be higher in counties with large r county yield variance. Since county yield data can be easily accessed from National Agricultural Statistical Service (NASS), we can readily identify these counties. Meanwhile , although idiosyncratic yield variance data are generally not available, we might expect that units in regions where heterogeneous approaches to production are take n are more likely to display large idiosyncratic yield variance and so low systemic risk. In particular, there is reason to believe that producers who adopt in novative production practices will have a lower systemic risk so that subsidized a rea -based insurance will discourage technology adoption. 2.2. Systemic Risk and County Growing Conditions A strand of literature has found that yield correlations are higher in extreme weather years (Okhrin et al. 2013; Goodwin and Hungerfor d 2014; Tack and Holt 2016; Du et al. 2017 ). Du et al (2017) has also found that yield on better -quality land is more resilient to bad weather. I n this subsection we check whether systemic risk also vary with county growi ng conditions. Rather than direct ly model systemic risk as a function of county growing condition variables, labelled as , we allow each of the three components of systemic risk to be a function of . The aggregate effect of growing conditions on s ystemic risk can be obtained as follow s. Let ting =/, then =1/(1+) and /(1)=1/. Taking the natural log o f both sides yields (3) ln1=ln1=ln()=ln()()() =2ln [()]+2ln [()]2ln [()], 11 which shows that the logistic transformation of can be taken to be the sum of three function s of . Since the logistic transformation is monotonic, then the effect of on shares its sign with the effect of on ln[/(1)]. Equation (3) also shows that effects of county growing conditions on systemic risk pass through , , and . Thus, for a given county growing condition variable, if it has the same effect on and , then these two effects are concordant with each other . However , were a growing condition to have the same effects on and either one or other of and then these two effects w ould offset. 2.3. A Brief Introduction of Individual Insurance Contract and Area Insurance Contract Before introducing our definition of TRASR, we first outline how individual -based insurance and area -based insurance work. For simplification, we only study TRASR for yield -based insurance programs. Multiple Peril Crop I nsurance (MPCI) is picked to represent the individual yield based insurance while Area Yield Protection (AYP) is picked to represent the area yield based insurance because they are the major individual yield insurance program and area yield insurance progr ams currently implemented in the United States. A similar analysis can be conducted for revenue -based insurance programs , but empirics would be more demanding because a stochastic price variable correlated with both farm and area average yields would need to be accounted for . All premium, subsidy and indemnity variables are measured in bushels per acre (b u./ac). Producer in county with random yield who chooses to purchase MPCI will receive indemnity payments in the form (4) =max (,0), where is the realized MPCI indemnity payments, is an insurer ™s reference ‚expected ™ unit yield, or the guaranteed unit yield, as established by the RMA, and is the MPCI 12 coverage level chosen by the producer with {0.5,–,0.85} where evaluations are in 5% increments. Thus, MPCI would pay producers indemnities when realized individual yield is lower than the policy protection amount , . The lower the realized individual yield, the higher the indemnit y amount. Similarly, AYP pays indemnities when county average yield is lower than policy protected county yield level, but its indemnity function takes a more complicated form, (5) =max min ,,0, where is the realized AYP indemnity payment, is the guaranteed county average yield, is the realized county average yield, is the AYP coverage level with {0.7,–,0.9} where again evaluatio ns are in 5% increments. A protection factor, , with [0.8,1.2], is introduced to allow the producer to adjust the amount of AYP indemnities. This is important because county yield los ses do not perfectly match individual yield loss es, see in equation (1), and the protection factor allows a grower ™s choice of AYP indemnities to better match expected individua l losses. A loss limit factor, l, with fixed value 0.18, is also introduced to scale up AYP payments where the min (,) function ensures that no additional indemnity is paid whenever the realized county yield is below 18% of the expected county yield. Overall, empirical data show that AYP generally pays more indemnities than MPCI under the same guaranteed yields, coverage choices, and realized y ields . This is so because the indemnity amount is scaled up by /() , but it also charges higher initial premiums as a result (Sherrick and Schnitkey 2016). 2.4. Calibrating Threshold Relative Area Subsidy Rate In this subsection we illustrate the importance of premium subsidy in inducing producers to choose AYP over MPCI . We will also derive our model for TRASR . 13 Since MPCI provides better risk protection than AYP , under the actuarially fair premium assumption, risk averse growers will always choose MPCI over AYP whenever no premium subsidy is provided. To see this, let and denote , respectively, MPCI and AYP premium rate s, while let ting and denote respective MPCI and AYP subsidy rate s. Under the actuarial fairness assumption, i.e., =E() and =E(), the expected net returns from purchasing MPCI and AYP are (6) E( )=E[+(1)]=E()+E(), and (7) E()=E[+(1)]=E()+E(), respectively. Thus, when no premium subsidy is offered, i.e., when ==0, then the expected net returns from purchasing MPCI and AYP are the same and equal E(). Risk averse growers then will never choose AYP over MPCI as MPCI provides better risk protection. Only when premium subsid ies are introduced and the condition E()>E() is satisfied, might there be a positive probability that risk averse growers will choose AYP over MPCI. The relative subsidy rate (8) =/=E()/E() provides the lower bound of the relative subsidy rate that is required to make a risk -neutral producer indifferent between choosing AYP and MPCI. We call the threshold relative area subsidy rate (TRASR), below which even risk -neutral grower s will always choose MPCI over AYP. A risk -neutral producer with TRASR exceeding 1 would require the AYP subsidy rate to surpass the MPCI subsidy rate in order to be indifferent between these two insurance contracts. By substituting in MPCI and AYP indemnity functions, we can develop an explicit form f or TRASR. First note that by assuming the yield expectatio ns established by RMA perfectly match es the actual unit yield expectation and county yield expectation, i.e., = and =, equa tion s (1) and (4) then jointly imply =max [()(1),0], 14 which presents MPCI indemnities as a function of county yield and idiosyncratic yield. Now the MPCI indemnity function can be rewritten as (9) =()(1), whenever 1 always hold. With some simple transformations, the expected indemnities f or MPCI and AYP can be expressed as (11) E()=()()(), and (12) E()=(), respectively, where () is the cumulative density function of and () is the cumulative density function of . Thus, TRASR can be rewritten as 15 (13) =E()E()=()()()(), which presents TRASR as a function of two random variables, and , and a set of parameters. We will study this relationship by both formal analysis and numerical simulation in the sections that follow . In particular, we seek to understand how TRASR is affected by systemic risk. A natural log version of equation (13) is adopted to simplify the analysis, i.e. , (14) ln()=ln()+ln()()()ln()ln(). Thus, TRASR can be roughly decomposed into four parts: MPCI ™s indemnity scaling factor, unscaled MPCI ™s expected indemnity payment, AYP ™s indemnity scaling factor, and unscaled AYP ™s expected payments. 2.5. TRASR and Systemic Risk A comparison of (14) and (3) is in order because terms can be matched in an info rmative way. Although TRASR is not a direct function of systemic risk, it is a function of unit yield ™s sensitivity to county yield, , unit idiosyncratic yield standard deviation, , and metrics related to county yield variance .4 Differentiating ln() with respect to yields (15) ln()=M()()()()()>0. In order to demonstrate the inequality note that (M()) is a decreasing function of so that (M) puts larger weights on negative values of and smaller weights on positive 4 Both ()() () and () can be seen as integrations of cumulative distribution functions and so can be seen as a generalized measure of a variable ™s variance where an increase represents a mean -preserving spread. In that light, the firs t expression can be taken to represent the variability of unit yield and the second can be taken to represent the variability of county yield. 16 values of . By ()=0 and the covariance inequality we then have M()()<0 and ln()/>0. Inequality (15) asserts that TRASR is increasing in a unit ™s idiosyncratic yield standard deviation. This is because in deriving inequality (15) we have held () fixed. Thus, by equation (1), ceteris paribus , increase in only increases the absolute value of , or the absolute values of losses (when <0) and gains (when >0) incurred by the producer, but has no effect on the probability of incurring losses and gains. Since larger losses lead to more indemnity payments while la rger gains have no effect on indemnity payments , then the expected MPCI indemnity payment will increase . Moreover, since a change in a unit ™s idiosyncratic yield standard deviation ha s no effect on the expected AYP indemnity payment , TRASR will then increase. Inequality (15) also reveals the fact that area -based insurance might be costly to implement in areas where unit yield does not match well with county yield. Producers in these areas are still exposing to risks under the area -based insurance program and thus would require a high subsidy rate to compensate the risk exposure. Similarly, differentiating ln() with respect to yields (16) ln()=1+(1)M()()()()()+M()()()()(). The sign of equation (16) is undetermined because the first two terms on the right -hand side are positive but the third is negative as M()()<0. An increase in increases MPCI ™s indemnity scaling factor but has ambiguous effect on unscaled MPCI indemnity payment. By equation (9), o n the one hand, changes in might change the indemnity amount received by the producer where the sign of this change depends on the sign of . On the other hand, changes in might also change the trigger value, M(), thus chang ing the 17 probability of receiving indemnities. Overall, the sign of equation (16) depends on both the distribution of county yield and the distribution of a unit ™s idiosyncratic yield. But when =0, then ln()/>0, i.e., when there is no idiosyncratic risk, then TRASR is increasing in unit yield sensitivity to county yield. Equation (14) conveys limited information on how TRASR depend s on county yield standard deviation as does not enter the function , directly at least . This is because in presenting the function of , we have fixed the county yield distribution, (), where the distribution captures the variable ™s moments. Thus, it is difficult to study the relationship between TRASR and county yield standard deviation in this framework. But a simple example can help illustrate how TRASR changes with county yield standard deviation that are caused by changes in the scale of lo sses but not by changes in the probability of incurring losses or gains. Ceteris paribus , by equation (9) and (10), one -unit reduction in will increase MPCI indemnity payments by and increase AYP indemnity payments by . As loss probabilities do not change, then when >, E() will increase by more than E() and TRASR will decrease. Thus, when holding loss probabilities fixed, the effect of changes in county yield standard deviation on TRASR depends on the relative size of AYP ™s in demnity scaling factor and MPCI ™s indemnity scaling factor . Given that previous studies estimated a clustering of around 1 and AYP participants generally pick =1.2, for most producers > is more reasonable than . Thus, for most producers, TRASR is likely to decrease with an increase in county yield variance when holding county yield loss probability fixed. Prop osition 2. Ceteris paribus , TRASR is increasing in unit ™s idiosyncratic yield standard deviation. While the effect o f a unit yield ™s sensitivity to TRASR depends on both the distribution of county yiel d and the distribution of a unit ™s idiosyncratic yield, when no 18 idiosyncratic risk occurs then TRA SR is strictly increasing in unit yield ™ sensitivity to county yield. The effect of county yield standard deviation on TRA SR depends on the relative size of AYP ™s indemnity scaling factor and MPCI ™s indemnity scaling factor . When AYP ™s indemnity scaling factor is larger than MPCI ™s indemnity scaling factor the n TRSAR is decreasing in county yield standard deviation. Proposition 2 establishes that the relationship between TRASR and systemic risk depends on a large set of parameters, including the distributions of county yield and unit ™s idiosyncratic yield. But it asserts that TRASR is higher for producers with a large idiosyncratic risk and for producers in counties with lower county yield variance. Since systemic risk is decreasing in a unit ™s idiosyncratic yield variance and increasing in county yield varianc e, there might exist a negative correlation between TRASR and systemic risk. This relationship has an important policy implication. That is, AYP has the potential to simultaneously provide good risk protection and save subsidy costs in counties where syste mic risk is high . By c alibrating with appropriate data, we can easily target these counties. 2.6. TRASR and Yield Expectations We now turn to analyze how yield expectations affect TRASR . Differentiating with respect to provides (17) ln()=(1)M()()()()()<0, i.e., TRASR is decreasing in individual yield expectation. Please note the revealing presence of (1) in the expression. R elationship (17) may appear at first to be counterintuitive because for a given distribution of u nit yield, an increase in mean unit yield should increase both the probability of receiving MPCI indemnities and the amount of MPCI indemnities received, while 19 have no eff ect on AYP indemnities. Thus, expected MPCI indemnity payments should increase and TRASR should increase as well, not decrease. The conflict arises because expected yield in our model is not exogenously given but is determined by yield distributions . To se e this, by equation (1), when holding county yield, county expect ed yield , and a unit ™s idiosyncratic yield constant, an one unit increase in unit expected yield will also increase realized unit yield by one unit. Now, the MPCI indemnities will decrease by 1 (or 0), and the probability of receiving MPCI indemnities will also decrease because the unit yield distribution has shift ed to the right by one unit while the indemnity trigger value only shift ed to the right by <1 units. Thus, an increase in the unit yield expectation will decrease the expected MPCI indemnity payment while ha ving no effect on AYP indemnity payments. As a result, TRASR will decrease with an increase in unit expected yield. The above analysis also reveals a bias induced by propo rtional coverage in the current crop insurance system. Since crop insurance pays indemnities based on shortfalls from the guaranteed yield which is proportional to expected yield, producers with higher expected yield might need to experience a large yield loss, which generally has a smaller probability of occurrence on better land, in order to receive indemnities, especially when they choose a low coverage level. Thus, productive producers who have high yield expectations would prefer to either not insure o r insure at high coverage levels, i.e., to plunge (Tobin 1958). This analysis might help explain the observation that producers in Central Corn Belt where yields are high generally choose high coverage levels. Insurance incentives would be very different w here absolute yield shortfalls insured. 20 Remark . Plunging behavior is more likely for choices on better quality land than on worse quality land . Similarly, taking the first derivative of ln() with respect to , we get (18) ln()=M()() ()()()+(M)()(M)(). The sign of equation (18) is unde termined because the first two right -hand terms are positive while the third is negative. This is because, similar to the effec t of changes in unit yield expectation, changes in county expected yield also change the distribution of county yield and thus have multiple effect s on the expected AYP indemnity payments. By equation (1), holding unit yield, unit expected yield, and unit idiosyncratic yield fixed, a one unit increase in increases by one unit. By equation (10), then the maximum indemnity level increases and the range, , within which the producer receives the maximum AYP indemnity level, is also enlarged. The increase in also widens the range << within which AYP pays positive indemnities, but the indemnity level now decreases by (1). Thus, an increase in county expected yield expectation has multiple e ffects on expected AYP indemnity payments that might offset each other in signs and its effect on TRASR is undetermined. Proposition 3. Ceteris paribus , TRASR is decreasing in individual unit expected yield because an increase in individual unit expected yield decrease s both the p robability of receiving MPCI indemnit y payments and the amount of MPCI indemnities received given that a payment is made . The effect of an increase in county yield expectation on TRASR is ambiguous . Proposition 3 asserts that TRASR is lower for producers with higher expected yield, i.e., for producers with higher historical yields as yield expectation is largely determined by yield history. Thus, productive prod ucers may choose AYP over MPCI at a reasonable subsidy rate. 21 2.7. TRASR and Coverage Levels and Protection Factor Under the actuarially fair premium assumption, it is obvious that an increase in MPCI coverage level will increase TRASR because it increases expected MPCI indemnity payment and has no effect on expected AYP indemnity payments. But the relationship between AYP cover age level and TRASR is not so clear. Taking the first -order derivative of ln() with respect to , we get (19) ln()=(M)()+(). The first right -hand term in equation (19) is negative while the second is positive. Thus, the sign of equation (19) is unde ter mined. This is because , by equation (14), an increase in has a positive effect on the AYP ™s indemnity scaling factor related part but has a negative eff ect on the unscaled AYP indemnity payments part. The overall effect is then determined by the relative size of these two effects . Finally, an increase in protection factor will decrease TRASR because it increases AYP indemnity scaling factor and then incr eases expected AYP indemnity payments. Proposition 4. Ceteris Paribus , TRASR is increasing in MPCI coverage level a nd is decreasing in AYP protection factor. The relationship between TRASR and AYP coverage level is unclear. Proposition 4 asserts that TRASR is higher for producers who are willing to choose higher MPCI coverage level s because now MPCI provides more indem nity payments. Given the fact that average coverage levels have increased over the past decade (Schnitkey and Sherrick 2014) , AYP might need to set a high subsidy rate in order to attract producers to choose AYP over MPCI. The negative relationship between TRASR and the AYP protection factor suggests that allowing producers to choose a protection factor larger than the current maximum leve l might 22 induce more producers to choose AYP over MPCI . However, as Miranda (1991) has discussed, setting the protection level too high would be politically infeasible and would raise the level and variability of total indemnity outlays. 23 CHAPTER 3. DATA AND VARIABLES Unit corn yield data have been obtained from the 2008 unit -level RMA records . These data are four -to-ten -year yield historical yield data used to establish Actual Production History (APH) that is used to set up unit -level yield expectations. The historical yields are continuous unless the crop being insured is not planted in a cer tain year (Edwards 2011). If at least four successive yield records are not available, a transition yield proportional to the ten -year average county yield is substituted in for each missing year. In our study, we only keep units with ten actual yield reco rds within the period 1998 through 2007 because i) including units with transition yield records will introduce artificial correlation between county yield and unit yield and thus bias our estimate of systemic risk, ii) including years before 1998 will result in a year sample that is only comprised of units that did not plant in some years between 1998 and 2007 and will also result in few unit observations in some years that might not well capture the temporal systemic risk in some counties. 5 We also dro p units that adopt the irrigation practice and units in counties where the irrigation rate 6 exceeds 20% 7 because systematic difference might exist between irrigated land and non -irrigated land, and also between counties where irrigation rate is high and wh ere irrigation rate is low. Table 1 1 in Appendix B summarizes observation losses after each data screening step and after merging with county yield data, weather data, and land quality data. County average corn yield data are from National Agricultural St atistical Servi ce (NASS). Only counties in t welve traditional major corn production states in the Midwest and Great 5 Only keep yield records in the most recent ten previous years are also the common practice in studies adopting RMA unit -level data, see Deng et al. (2007) and Claassen and Just (2011). 6 Using RMA unit -level data, county irrigation rate is defined as the ratio of the number of units that adopt irrigation practice over the total number of insurance units in that cou nty. 7 Dropping counties whose irrigation rate exceeds 30% yields similar results. 24 Plains are kept in this study: Illinois, Indiana, Iowa, Kansas, Michigan, Minnesota, Missouri, Nebraska, North Dakota, Ohio, South Dakota, a nd Wisconsin. These states accounted for more than 88% of total corn production in the United States in 2018. 8 For the purpose of getting enough observations to estimate county yield trend, we drop counties without 50 successive yearly yield records over 1 958-2007. To make our study county representative, we further drop counties with less than 30 insurance units. We construct two sets of weather variables to reflect county -level weather conditions. The first set contains growing degree days (GDD) and stressful degree days (SDD), which are widely used in the literature to measure heat conditions (Schlenker et al. 20 06; Schlenker and Roberts 2009; Du et al. 2015). Heat data are from National Oceanic and Atmospheric Administration (NOAA) and are recorded at station level. To construct G DD and SDD, we first define i) the daily maximum temperature (in degrees Celsius) , ,, , for county c as the mean of the highest temperature s recorded by all weather stations within that county in day in year , and ii) the daily minimum temperature , ,, , for county as the mean of the lowest temperature s recorded by all weather stations. For each county, GDD, labeled as , is defined as the ten -year average of total beneficial degrees in the range [10 , 30 ] over the growing season (Neild and Newman 1990) while SDD, labeled as , is de fined as the ten -year average of the sum of excess degrees that are greater than 32.22 over the growing season (Schlenker and Roberts, 2009). Specifically, (20) =1100.5min max ,, ,,+min max ,, ,,, 8 Production data are from Crop Production 2018 Summary, USDA, 2019. https://ww w.nass.usda.gov/Publications/Todays_Reports/reports/cropan19.pdf 25 and (21) =1100.5max (,, ,)+max ,, ,, where =10,=30,=32.2, denotes county, denotes the set of days in the growing season ( ={, , , }) in year , and denotes the set of our sample weather data years ( ={1998,1979,–,2007}). The second weather variable set measures the relative moisture in a county. We use the Palmer ™s Z (PZ) index from NOAA to measure drought and excess moisture (Xu et al. 2013). PZ index measures the departure of monthly weather from the average moisture condition in a climate division level. PZ index within the range [ -2, 2.5] is viewed as normal while below indicates severe drought and above 2.5 indicates severe wetness (Karl 1986; NOAA 2014). Since different parts of a county might be covered by different climate divisions, to transfer the climate division data into county level, we first calculate the rati o of county acres covered by each intersect climate division to get a weight metrics. We then time the monthly climate division PZ index with the associated weight and sum the products across all intersect climate divisions to get the monthly county -level PZ index , ,,. We define a drought variable, (22) =,,<2, and a wetness variable, (23) =,,>2.5, where denotes month and =1 whenever the condition in the parentheses is satisfied and =0 otherwise . Thus, measures the frequency of severe short -term drought occurred in county c over 1998 -2007, and measures the frequency of severe short -term wetness occurred in county c in the same time period. 26 Land quality data are from National Resource Conservation Service (NRCS). County land quality, labeled as , is defined as the fraction of all land that is in land capability classes (LCC) I or II in that county. There are eight LCCs in total, where classes I and II are most favorable for cultivation while classes higher than II at least have plural severe limitations for cultivation. 9 Since there is little variation in land capability across years, we use the 2010 land capability as our measure of land quality for all sample years. Table 1 presents the descriptive statistics f or our variables. The mean of yearly unit yield is 149 bushels per acre and the standard devia tion is 39. 4 bushels per acre. Mean yearly county average yield is lower than mean yearly unit yield, suggesting that our unit sample might have over -weight ed the number of high productiv ity farms. County yield also has a smaller standard deviation , as exp ected. The mean of GDD and SDD are 1 ,290 and 14.8, respectively, suggesting that on average, the accumulation of beneficial degrees is plentiful for corn growth and the appearance of excessive heat is rare. The mean of mon thly severe drought occurrence is 4.72 and the mean of monthly severe wetness occurrence is 5.53. Thus, severe drought and severe wetness do not occur often in our sample counties over the sample period. The large mean for severe wetness also suggests that there was an oversu pply of moisture through 1998 to 2007. Mean fraction of county land in LCC I or LCC II is 47.3% , reflecting the fact that a large fraction of our sample counties ™ land is favorable for cultivation . In addition to the descriptive statistics, in Figure 2 we also map the geographic distribution of w eather variables and the land quality variable. Panel s A and B show that GDD and SDD generally increase as one moves south. This trend is consistent with the fact that temperature increases as one moves south. SDD also increases as one moves west. Panel C shows that 9 The classification scheme follows USDA ™s description at https://www.nrcs.usda.gov/wps/portal/nrcs/detail/national/technical/nra/?cid=nrcs143_014040 27 counties in Western and Eastern Corn Belt generally experienced more severe drought than other counties while panel D shows that counties in the north part of Corn Belt generally experienced more severe wetness than counties in the south part of Corn Belt . Counties with a large fraction of good land , as shown by Panel E, are mainly located in the Northwestern and Southeastern Corn Belt states, especially in Iowa, Illinois, Indiana, and Western Ohio. 28 CHAP TER 4. EMPIRICAL RESULTS 4.1. Measuring Systemic Risk Following equation (1), we establish the following empirical model to estimate systemic risk, (24) ,,=+(,,)+,, where a constant term is introduced to ensure that the error term has zero expectation. Unit -level systemic risk is estimated by the R -squared of OLS estimation, i.e., =Var (,,)Var (,,). However, since each insurance unit has only ten -year yie ld observations, OLS estimation with such a short time period would be highly imprecise. We instead estimate the county -level systemic risk by pooling together all units within a given county. The empirical model is now given by (25) ,,=+(,,)+,, where and denote the county -level constant term and county -level unit yield ™s sensitivity to county yield, respectively. County -level systemic risk, labelled as , is then measured by the R -squared of the OLS estimation for equation (2 5), i.e. =Var (,,)Var (,,). To estimate equation (2 5), however, we need to first estimate the yearly unit yield expectation, ,, and the yearly county yield expectation, ,. It is commonly assumed that crop yield follows some time trend that establishes yield expectation. Following Deng et al. (2007), we employ the following log -linear model to estimate county yield trend for each county, (26) log(,)=,+,(2008)+,, where , captures the inverse trend in percen t yield change starting from 2008, or the difference in log yield expectation between year t and year 2008. NASS county yield data from 1958 to 2007 are used to estimate equation (2 6) and county yield expectation for county c in year t is then calculated as ,=,=exp [,+,(2008)]. 29 For each insur ed unit, RMA sets up its yearly reference yield expectation by use of the unit ™s APH yield s, which equals the average of t he last ten years ™ actual yields. However, in our data, only 2008 APH yields are available. We proceed by assuming that unit yield expectation shares the same time trend with county yield expectation. Thus, the APHs for 1998 to 2007 are then calculated as (27) ,=, exp ,(2008), where {1998,1999 ,–,2007}. A concern with using APH yield as unit yield expectation is that the APH yield lags the true expected yield due to improved crop genetics and cultural practices (Edwards, 2012). To correct for this downward bias, RMA introduced the trend -adjusted APH yield in 2012. Basically, a trend adjustment factor is estimated for each crop and each county, which is equal to the estimated annual increase in NASS county yield. The trend adjusted factor is then used to scale up past actual yield records. Following this pr actice, we also calculate the trend -adjusted APH. Specially, the yearly county specific trend -adjustment factor, ,, is estimated as (28) ,=(,, )/(1958), where , is the yield prediction for county in year estimated by equation (2 6), and , is the 1958 actual yield for county . After obtaining ,, the yield adjustment for the year that is ahead of year then equals ( )×,. To illustrate, suppose we want to adjust the 2007 APH for unit in county and the trend adjustment factor for county in year 2007 is , =2. Then the 2006, 2005, –, and 1997 actual yield of unit will be adjusted up by 2, 4, –, and 20 bus hels per acre, respectively. The 2007 APH would be adjusted up by (2+4++20)/10=11 bushels per acre. The formula of APH with trend -adjusted can be summarized as 30 (29) _,= ,+110×, , and the trend -adjusted APH is then used as the expected unit yield in eqn. (4) before , i.e., ,=_,. We estimated model (25) and decomposed the resulting systemic risk measure according to (3). Table 2 presents the descriptive statistics f or county systemic ri sk and its three components. Mean county systemic risk is 0.37, suggesting that in general, systemic risk explains slightly more than one -third of total unit -level yield variability. But the magnitude of systemic risk varies considerably across counties as it ranges from 0.02 to 0.78. The mean of county -level unit yield ™s sensitivity to county yield is 1.05 and it ranges from 0.19 to 2.05. This fact violates Miranda ™s (1991) assertion that acre -weighted average within a given county should equal to 1. At least t wo reasons may contribute to this violation. First, our county yield data are from NASS whereas unit yield data are from RMA. NASS county yield generally do es not equal the mean of acre -weighted RMA unit yield (Zulauf et al. 2017), perhaps because of differences in survey methodologies used or because not all land is insured . Second, we have dropped many units in our data screening process thus losing the connectio n between county yield and unit yield. The means of unit idiosyncratic yield standard deviation and county yield variance standard deviation are 23.4 and 19.4, respectively, suggesting that while i diosyncratic risk and systemic risk both contribute a signi ficant amount of variability to unit yield , idiosyncratic risk contribut ions are the more important . Figure 3 map s the geographic distribution s of county systemic risk and its three components. Panel A shows that the systemic risk is generally high at the Corn Belt ™s southern and western fringe s. It also shows that systemic risk is low in Indiana but is high in some counties in Southern Minnesota. This distribution generally lines up well with the distribution of 31 county yield standard deviation s (Panel B), suggesting that county yield variance may be the most important component determin ing systemic risk. This finding has important implications for policy implementation because county yield variance data are easily accessible. Program designers can easily se lect out counties where A YP has the potential to effectively reduce yield risk by finding out counties with large yield variance. Panel C of Fi gure 3 shows that counties at the Corn Belt periphery have relatively large idiosyncratic yield standard deviation s, especially counties in Southern Wisconsin and the Eastern Dakotas. On the contrary, counties in Iowa and Illinois generally have low unit ™s idiosyncratic yield standard deviation. 10 Panel D presents evidence that Corn Belt fringe counties have r elatively large unit yield ™s sensitivity to county yield, but the pattern is not as evident because some counties in the central part also have large unit yield ™s sensitivity to county yield while coun ties in the southern fringe have low unit yield ™s sensi tivity to county yield. The Moran ™s I statistic 11 of panel D is only 0.017, though statistically significant, while the Moran ™s I statistics for panels A through C generally exceed 0.03. Thus, the distribution of unit yield ™s sensitivity to county yield is more likely to be spatially independent when compared with the distribution of systemic risk itself and the distributions of the other two systemic risk components . 10 With high mean yields and low yield variability in the Central Corn Belt, Remark 1 suggests that farmers there contemplating either individual insurance or area insurance should either not insure or take out high coverage. 11 The Moran ™s I is a frequently used correlation coefficient that measures overall spatial autocorrelation in a given dataset. Its value lies within the range [ -1, 1]. A zero -value statistic indicates the related variable is perfectly randomly distributed in space. When a positive (negat ive) value is observed, then there is a positive (negative) spatial autocorrelation across the regions. We use the spatgsa command in Stata, developed by Pisati (2001), to calculate the Moran ™s I statistics for all investigating variables. 32 4.2. Systemic Risk and County Growing Conditions To study how systemic risk is determined by county growing conditions, we now regress the three components of systemic risk on county weather and land quality variables. The following log -linear model is estimated by OLS method, (30) ln()=++++++, where {,,}, is the constant term and is the error term. Regression results f or equation ( 30) appear in Table 3. Column (1) shows that GDD has a significantly negative effect on unit yield ™s sensitivit y to county yield while SDD has a significantly positive effect. Thus, more accumulation of excessive heat increases unit yield ™s sensitivity to county yield while more accumulation of benef icial heat reduces the sensitivity. This result also explains why counties in the Southern Corn Belt , where GDD and SDD are both high , are less likely to have high unit yield sensitivity to county yield than do counties in the Western Corn Belt where SDD o nly is high. Severe wetness also has a significant negative effect on unit yield ™s sensitivity to county yield. Column (2) shows that unit ™s idiosyncratic yield variance is significantly decreasing in GDD and land quality. Thus, greater accumulation of be neficial degrees and better land may provide a higher benchmark yield shared by all units in the county and making idiosyncratic production practice less important in determining unit yield. The negative effect of land quality on unit ™s idiosyncratic yield variance also explains the low idiosyncratic yield variance in Iowa and Illinois where land quality is high. Column (3) shows that GDD and SDD both have a significantly positive effect on county yield variance. This is consistent with the observation th at counties in the southern and western parts of the Corn Belt where GDD and SDD are high generally have higher county yield variance. Severe drought also has a significantly positive effect on county yield variance while 33 severe wetness tends to reduce cou nty yield variance. Land quality has a marginally significantly negative effect on county yield variance. To find the overall effect of county growing conditions on systemic risk, and consistent with (3), we first regress ln[/(1) ] on 2ln(), 2ln () and 2ln (). As shown by column (4) of Table 3, consistent with Proposition 1, systemic risk is increasing in unit yield ™s sensitivity to county yield and county yield variance and is decreasing in unit ™s idiosyncratic yield variance 12. The overall effect of GDD on systemic risk, for example, is calculated as (0.068×1.048)+(0.050×0.625)+(0.126×1.004)=0.086. Standard errors are calculated by the delta method. Column (5) of Table 3 reports the aggregate effects of county growing conditions on systemic risk. GDD and land quality both have a significantly positive effect on systemic risk by ensuring a high benchmark yield for all units. The highly significantly positive effects of SDD and severe drought suggest that, consistent with previous findings, yie lds are more closely correlated in extreme weather years and systemic risk is higher in counties where extreme weather occurred more often. The negative effect of wetness constitutes a counterexample to the previous statement, which may possibly be because severe wetness brings excessive water supply for some farms but plentiful water supply in the soil for other farms . In addition to the sign and significance of county g rowing conditions on systemic risk, it might also be important to check which growing condition variable plays the most important role in determining systemic risk. Following Huettner and Sunder (2012), we use the Shapley 12 Equation (3) predicts that the coefficient of 2ln (), 2ln() and 2ln() on ln[/(1) ] should be exactly -1, 1, and 1. The difference between the prediction and our estimation result comes from the fact that Var (,,)Var (,,)Var (,)Var ,=. This is because in real data , and , are not constant but vary by year and are correlated with , and ,. If we regress ln[/(1) ] on 2ln (), 2ln () and 2lnVar (,,) we will obtain coefficients equal -1, 1, and 1. 34 value to measure the power of grow ing condition variables in explaining the explainable part of systemic risk. Their id ea is to remove each explanatory variable from all possible combinations of other explanatory variables and so observe the variable ™s average contribution to R 2. Column (1) of Table 4 shows that, c onsistent with the finding that the geographic distribution of systemic risk generally lines up with the geographic distribution of county yield variance, county yield variance explains about 6 5% of systemic risk. Unit yield sensitivity to county yield explains about 28% while unit idiosyncratic yield variance explains the remaining 6%. For unit yield ™s sensitivity to count y yield, column (2) in Table 4 shows that GDD explains about 47% of the variability that can be explained by county growing conditions . SDD explains about 10% and wetness explains a bout 35%. Column (3) shows that land quality is the most important determinant of unit ™s idiosyncratic yield variance as it solely explains about 86% of explained variability unit ™s idiosyncratic yield variance. For county yield variance, column (4) shows that GDD explains the largest part of its growing -condition -explainable variability while SDD explains the second largest part. wetness variable also explains more than 16% of the explainable part of county yield variability. Overall, Column (5) shows that GDD is the most important growing condition variable in explaining systemic risk variability. SDD also explains a significant part of systemic risk ™s variability. This result is consistent with the fact that systemic risk is high in the southern and western parts of the Corn Belt, where both GDD and SDD are high. Thus, county heat conditions can be viewed as the most important factors affecting systemic risk. High heat accumul ation counties are more likely to have higher systemic risk, especially for those that are less likely to experience severe wetness. 35 4.3. Systemic Risk Estimate and Inadequate Investigating Time Horizon Our estimation of systemic risk is based entirely on Miranda ™s one factor capital market model, which implicitly assumes that the correlation between county yield and unit yield is constant within the sample period. However, empirical evidences have demons trated that the spatial correlation of crop yields tends to be higher in extreme weather years than in a typical year (Goodwin 2001; Okhrin et al. 2013). Thus, our model might over restimate a county ™s systemic risk in years that a county experienc es except ionally good weather and under estimate its systemic risk in years that the county experienc es exceptionally bad weather. To see which county might suffer more from such bias, we first investigate whether unit yield ™s sensitivity to county yield is higher w hen county yield is below its expectation than when county yield is above its expectation, and then check how this pattern correlates with systemic risk . By introducing an interaction term into equation (25), we get , (31) ,,=+,,+,,,+,, where ,=1 whenever ,<, and ,=0 whenever ,,. Now, there are three mutually exclusive groups that complete the value set of ,,, i.e., i) ,=,, ii) ,>,, and iii) ,<,. Parameter measures the mean value of ,, when ,=,, measures unit yield ™ sensitivity to county yield whenever ,>,, + measures unit yield ™s sensitivity to coun ty yield whenever ,<,, and captures the difference. A positive value of indicates that unit yield ™s sensitivity to county yield is higher when county yield is below its expectation than when it is above its expectation. Table 5 lists the sign and significance of . Among the 579 sample counties, 48.7% have a significantly positive and only 24.7% have a significantly negative . Moreover, about 13.6% counties have insignificantly positive and about 13% have insignificant ly negative . Thus, for most sample counties, unit yield is more sensitive to county yield when county 36 yield is below its expectation than when county yield is above its expectation. Since systemic risk is increasing in unit yield ™s sensitivity to coun ty yield, this finding confirms the conjecture that we might underestimate a county ™s systemic risk if its county yields were exceptionally bad over the sample period and overestimate the systemic risk if the county yields were exceptionally good. We the n plot the geographic distributions of to ascertain which area ™s systemic risk estimate is more likely to suffer from potential bias caused by misspecification. Panel A in Figure 4 plots the geographic distribution of ™s value and panel B plots th e geographic distribution of ™s sign and significance. There does not exist a clear pattern in ™s geographic distributions and the Moran ™s I statistics of Panel A is 0.009 and that of Panel B is -0.001, asserting that can be treated as indepen dently distributed among sample counties. The Pearson ™s correlation test coefficient between and is -0.0115 and is statistically insignificant. Given the short time interval at hand, our view is that we have insufficient information available to establish whether systemic risk estimate and the bias resulting from the short time period. Thus, there is no clue about whether counties with high systemic risk or low systemic risk are more likely to suffer from the bias. To obtain more accurate estimat e of typical systemic risk, a longer unit yield time series is required . 4.4. Calibrating TRASR We now tu rn to calibrat ing TRASR and investigating the relationship between TRA SR and systemic risk. When calibrating TRASR from equ ation (13) we need to know the distribution s of both county yield and the unit ™s idiosyncratic yield. Studies investigatin g crop yield distributions mainly adopt two distinct methodologies , parametric and nonparametric . Parametric method s 37 often assume that crop yield follows a specific distribution, such as the normal, gamma or beta distribution (Botts and Boles 1958; Gallagher 1987; Nelson 1990; Sherrick et al. 2004; Harri et al. 2011). Nonparametric method s, on the other hand, do not assume that crop yield follows a specific distribution and thus offer flexibility in capturing local idiosyncrasies in yield distribution that may not be captured by parametric method s (Goodwin and Ker 1998). Since our study contains many counties and units while appropriate corn yield distribution specifications might differ by location , flexibility considerations lead us to employ nonparametric kernel density estimation. Since kernel density estimation requires a stationary data series, before estimating a distribution, we need to first retrend county yield data. The retrending model follows the detrending model and retrended yield for county c in year t is given by (32) ,=,,×, +, , where , and , ({1958,1959,–,2007}) are county yield predictio ns from equation (26), and the ,s are prediction error s. The kernel density function for county c is given by (33) ()=1, , ,, where is a specific point whose density is to be evaluated , ,s are retrended county yield s located within a pre -selected bandwidth centering at , is the bandwidth parameter, () is the kernel function, and ,™s are the associated sample weights. There is general consensus among researchers in the area that the kernel function choi ce is less importance than the bandwidth choice in kernel estimation (Parzen 1962; Tapia and Thompson 1978; Newton 1988). Thus, we choose the Epanechnikov kernel function because it is most efficient in minimizing the mean integrated squared error (MISE), which is the most 38 common optimality criterion used to select bandwidth . As for bandwidth choice, we follow Goodwin and Ker (1998) to adopt Silverman ™s modified rule -of-thumb method to select the bandwidth parameters. The bandwidth parameter ™s formula is gi ven by (34) =0.9×min ,1.349. , where is the standard deviation of , and is the interquartile range of possible realization of ,. Since we only have fifty -year observations for each county, kernel estimation of such a short time period might be imprecise. To extend the data pool, following Goodwin and Ker (1998) , we u se yield information from adjacent counties. To be qualified for th e calibration, the central county and its adjacent counties must have a total number of 200 yearly yield records, i.e., the central county must have at least three adjacent counties that have no missing yield records over 1958 -2007. This requirement leaves us a sample comprised of 208,549 units in 540 counties. Each yield record in adjacent counties is assigned with weight 1/[(2+1)×50] while each yield record in the central county is assigned with weight (+1)/[(2+1)×50], where is the number of adjacent counties. After getting the density estimate for each point we want to evaluate, (), the cumulative density at each point, (), is calculated by the Tra pezoid rule in the form (35) ()=(),+,2,, where ,=,, and is the total number of density -evaluation points below . The two integrals, ˜˚(˛˝˙)ˆ˛˝˙ˇ˘(˘) and ˜˚(˛˝˙)ˆ˛˝˙ˇˇ, are also calculated by the Tra pezoid rule in a similar way. 39 As for the estimation of each unit ™s idiosyncratic yield distribution, (), since we only have ten estimates of idiosyncratic yield for each unit and are unable to access unit adjacency information , we are unable to perform unit -level nonparametric kernel estimation. Rather, within each county we pool units with similar 2008 APH values together and then perform nonparametric kernel estimation on the APH group level. By doing so, we are assuming that units with simila r 2008 APH values also have similar idiosyncratic yield distribution. Since APH is determined by historical yield, which is in turn determined by land quality, it is reasonable to expect that units with similar APH values should have similar idiosyncratic yield distributions. Empirically, within each county, the 2008 APH values are grouped in tens. That is, units with the smallest to the tenth smallest 2008 APH values are assigned to group 1, units with the eleventh to twentieth smallest 2008 APH values ar e assigned to group 2, and so on. Whenever the last group contains fewer than ten APH values, these units are merged with the second last group. This grouping method yields 2,393 unique APH groups in total. As shown by Table 6, 37 counties have only one AP H group and 54 counties have only two AHP groups. Units in these counties are more likely to be assigned into APH groups where the idiosyncratic yield distributions of some units differ considerably from the idiosyncratic yield distributions from other uni ts. However, as Figure 5 shows, counties with fewer APH groups also have smaller APH ranges. Moreover, since counties at the southern and western fringes of the Corn Belt, where systemic risk is high and which are counties we care most about, generally hav e four or more APH groups, bias caused by pooling units with different idiosyncratic yield distributions into the same APH group should be minor. Among the 2,393 APH groups, only 7.5% have less 40 than 200 unit -year observations. Thus, the nonparametric kerne l density estimation for most APH groups is free from imprecision concerns. As with county yield kernel density estimation, we also choose the Epanechnikov kernel function and adopt Silverman ™s modified rule -of-thumb method to select the bandwidth paramete rs. After obtaining the density function, the cumulative density function, (), and the integral ˜˜˚(˛˝˙)ˆ˛˝˙ˇ˘(˘)ˆ(), are again evaluated by the Tra pezoid rule. Table 7 presents descriptive statistics for coverage -level conditional unit -level TRASR . We set the protection factor, , to equal to 1.2 as it is the protection factor level that brings the highest expected AYP indemnity payments . Since increases in 0.05 i ncrements from 0.5 to 0.85 and increases in 0.05 increments from 0.70 to 0.9 we have 8×5=40 possible coverage level combination s. Consistent with Proposition 4, the mean and median of TRASR both increase with MPCI coverage level. Although Proposition 4 makes no predictions about the effect of AYP coverage level on TRASR, results in Table 7 show that TRASR decreases with increase in . This finding suggests that the negative effect of on unscaled AYP indemnity payments dominates the positive effect of on the AYP scaling factor. We focus on TRASRs where MPCI coverage level is no less than 75% because 75% is the minimum coverage level chosen by most produ cers in the Corn Belt (Schnitkey and Sherrick 2014). Since some units have extremely large TRASR 13 , we thus focus on medians other than 13 Extremely large TRASR values occur whenever unit ™s M()s are very large or M is in the low density area of (), or both apply. The first case occurs when a unit ™s idiosyncratic yield is negative and large in magnitude because M() is decreasing in . The second case occurs when the AYP coverage level is low. In the first case, ()() is large and in the second case, () is small. Table 5 also shows that some units have a zero -value TRASR. This is because kernel density estimation will assign zero probabilities to M()s whose values that are too far away from observed county retrended yields. Thus, zero values of TRASR occur when a unit ™s M() is either too large or too small . 41 means as medians are less affected by extreme values than means. Results in Table 5 show that the medians of TRASR are ge nerally higher than 60% when AYP coverage level is no greater than than 80%. The medians of TRASR are even higher than 100% for groups =80%,=70%, =85%,=70% and =85%,=75%. Thus, generally, a necessary condition to induce most produc ers to choose a low coverage level AYP contract over a high coverage level MPCI contract is to set the related AYP subsidy rate higher than the related MPCI subsidy rate. The medians of TRASR are generally smaller than 50% when AYP coverage level is higher than 80%. We then check whether the current AYP subsidy rate discourages producers from choosing AYP over MPCI. Here we define a new concept, threshold area subsidy rate, or TASR, which is the product of TRASR and MPCI subsidy rate. By equation (8), TAS R thus constitutes the lower bound on the AYP subsidy rate below which the expected AYP indemnities would be strictly less than the expected MPCI indemnities and risk averse growers would never choose AYP over MPCI. Since MPCI offers contracts at different unit levels and subsidy rates differ by unit levels, we choose the enterprise unit level subsidy rate to derive our TASR because enterprise unit level MPCI contract is the most similar to AYP and in recent years it has covered the largest share of insured corn acres (Coble 2017). Subsidy rate information for AYP and different unit -level MPCI contracts are listed in Table 8. Enterprise unit level subsidy rates are higher than AYP subsidy rates at all available coverage levels except for the level 85%, but t he coverage -level conditional subsidy rate gap between the two insurance contracts decrease with increase in coverage level. Panel A of Table 9 reports sample means of the coverage -level conditional percent of units whose TASR is higher than the current A YP subsidy rate. To save space, we only report results 42 corresponding to a MPCI coverage level no less than 75%. The complete results are listed in Table 12 in Appendix B . Consistent with results in Table 7, the percent of units whose TASR is higher than the current AYP subsidy rate is increasing in MPCI coverage level and is generally decreasing in AYP coverage level. When AYP coverage level is less than 80%, generally more than half of producers will find their TASR higher than the current AYP subsi dy rate. This result suggests that the low coverage level AYP contracts which are currently available do not provide the minimum required subsidy rate to compete with the subsidized MPCI contracts as subsidy transfer instruments. Raising the related subsid y rate is a necessary condition if more low coverage level AYP demand is to be induced. When AYP coverage level is no less than 80%, generally only less than 30% of producers will find their TASR higher than the current AYP subsidy rate. Thus, for most pro ducers, the currently available high coverage level AYP contracts have met the minimum subsidy rate requirement to compete with MPCI contracts whenever producers are risk -neutral. This finding suggests that the current AYP subsidy rate might not be a major deterrence for producers to choose high coverage level AYP contracts over MPCI contracts and raising AYP subsidy rate might not be able to induce more high coverage level AYP demand. However, since risk aversion will increase the minimum required AYP subs idy rate, the percent of risk aversion producers whose TASR is higher than the current AYP subsidy rate will surely exceed the percent listed in Table 7 . Raising subsidy rate might still be a viable option to induce more demand even for high coverage level AYP contracts. Besides comparing with the current AYP subsidy rate, we also compare TASR to the 100% level to see whether some producers will not choose AYP over MPCI even when AYP is offered for free. Panel B of Table 9 presents the results. As expected , because the MPCI subsidy rate is 43 smaller than 100%, the percent of units with TASR below 100% is much smaller than the percent with TASR below the current AYP subsidy rate. For all coverage level combinations, at most 41% of producers will find their TAS R higher than 100%, so only a small proportion of producers will find that freely offered AYP contracts are not value for money when MPCI contracts are available. Thus, raising AYP subsidy rate to a level no greater than 100% might be a feasible option to induce greater AYP demand. High coverage level AYP contracts, at 80% or higher, might benefit more from this option because less than 10% producers will find their TASR higher than 100%. 4.5. TRASR, Systemic Risk, and Area Insurance Demand We now turn to study the correlation between TRASR and systemic risk and its implication for area insurance demand. Our conceptual framework suggests that TRASR should be negatively correlated with systemic risk. To test whether this conjecture holds in da ta, we then conduct a series of correlation test s between county systemic risk and the two county -level TRASR variables: county -level percent of units whose TASR is higher than the current AYP subsidy rate and county -level percent of units whose TASR is hi gher than 100%. Table 10 reports the Pearson ™s correlation coefficients between systemic risk and the two county -level TRASR variables. Panel A shows that generally, systemic risk is significantly positively correlated with the percent of units whose TASR is higher than the current AYP subsidy rate at low AYP coverage levels and is significantly negatively correlated with it at high AYP coverage levels. Thus, for low coverage level AYP contracts, producers in high systemic risk counties are more likely to find their TASR to exceed the current AYP subsidy rate than producers in low systemic risk counties, while for high coverage level AYP contracts, producers in high systemic risk counties are less likely to find their TASR higher than the AYP subsidy 44 rate than producers in low systemic risk counties. This finding indicates that the current AYP subsidy rate is more likely to be a major deterrence for producers in high systemic risk counties to choose low coverage level AYP contracts while it is more likely to be a major deterrence for producers in low systemic risk counties to choose high coverage level AYP contracts. Panel B of Table 10 show that systemic risk is significantly negatively correlated with the percent of units whose TASR is higher than 100% at m ost coverage level combinations, suggesting that producers in high systemic risk counties are less likely to have TASR exceeding 100% than are producers in low systemic risk counties. Thus, high systemic risk counties generally have larger room to raise AYP subsidy rate to induce more AYP demand. Since the correlation test coefficients only capture the overall relationships between systemic risk and TRASR variables among all sample counties and we care more about whether high systemic risk counties are truly associated with lower TASR, we then plot the geographic distributions for county -level TRASR variables. Figures 6 through 8 plot the geographic distributions for county -level percent of units whose TASR exceeds the current AYP subsidy rate. These f igures again show that the percent of units whose TASR is higher than the current AYP subsidy rate is increasing in MPCI coverage level and is decreasing in AYP coverage level. For all coverage level combinations, counties at the Corn Belt ™s northeastern f ringe, where systemic risk is low, generally have the highest percent of units whose TASR surpasses the current AYP subsidy rate. Counties in Central Corn Belt generally have the lowest percent of units with TASR higher than the current AYP subsidy rate. W estern and Southern Corn Belt counties, where systemic risk is high, also have relative low percent of units whose TASR is higher than the current AYP subsidy rate, especially when AYP coverage level is no less than 80%. The geographic distribution pattern s remain for the 45 percent of units whose TASR is higher than 100% and Western and Southern Corn Belt counties also have low percent of units whose TASR is higher than 100%, as plotted by figures 9 through 11. These results assert that producers in Western a nd Southern Corn Belt counties not only enjoy better risk protection from AYP, but also require less subsidies to possibly choose AYP over MPCI than counties in other areas. Subsidy rates also have larger room to increase to induce more AYP demand in these counties. Although our evidence suggests that counties in the Corn Belt ™s southern and western fringe s have the best potential to grow area insurance demand , this conjecture does not match the reality of area insurance demand. Working with data from RMA Summary of Business reports we draw the geographic distribution for share of acres insured by area insurance in Figure 12. Clearly, although the overall demand f or area insurance is low except in year 2006, counties with larger share of acres insured by area insurance are mainly located in the e astern and northeastern parts of the Corn Belt, where systemic risk is low while TRASR is high. This fact suggest s that area insurance demand might not be m ainly driven by producers ™ risk attitude and the contract designs, but by some factors unobserved in our data, such as the marketing strategy of insurance companies (Skees et al. 1997) . 46 CHAPTER 5. CONCLUSION AND DISCUSSION The Federal Crop Insurance Program has long been afflicted by high operation costs where asymmetric information and systemic risk play major roles. Area -based insurance programs have been widely proposed as viable options to deal with these problems . Howev er, program tak e-up rate s remain low and knowledge of determinants of the low demand is lacking in literature. In this paper, we investigated two factors that determine the demand of area insurance programs. First, we directly measured systemic risk in corn production across the Midwest. The magnitude of systemic risk determines the effectiveness of risk protection for area insurance programs. The greater the systemic risk, the more workable is area -based insurance (Skees et al. 1997). We find that , in general, systemic risk explains slightly more than one -third of total unit yield variability . It is high est in Sout hern and Western Corn Belt counties but low in the Central and Northern Corn Belt. Further investigations show that the geographic distribution of systemic risk is most likely to be driven by the geographic distribution of county yield variance where syste mic risk significantly increases with county heat conditions and the appearance of severe drought . In addition to systemic risk, we also investigate the role of premium subsidy in producer ™s choice between area and individual insurance contracts. By calibr ating TRASR, the relative subsidy rate at which AYP expected indemnity payments equals th ose from MPCI indemnity payments , we find that most producers would require the AYP subsidy rate to be higher than the enterprise unit level MPCI subsidy rate to possi bly choose AYP over MPCI. The percent of producers holding this requirement is increasing in MPCI coverage level and is decreasing in AYP coverage level. We also find that the current AYP subsidy rate might be the major deterrence for producers to not choo se AYP at low coverage levels, but not the major deterrence 47 for them to not choose AYP at high coverage levels. Since only a small fraction of producers would require the subsidy rate of high coverage level AYP contracts to be higher than 100% to possibly choose these AYP contracts over MPCI, raising the subsidy rate of high coverage level AYP contracts might be a feasible option to induce more demand for AYP contracts. We also find a negative correlation between systemic risk and TRASR and unit ™s idiosync ratic yield variance seems to be the best predictor of TRASR. P roducers in Southern and Western Corn Belt counties , where systemic risk is high, generally have low TRASR . These counties are ideal areas to implement area insurance contracts as produce rs the re would enjoy better risk protection and require lower subsidies to compensate for their risk exposure. However, this conjecture is at variance with the fact that counties with relatively high area insurance take -up rate are mainly located in the e astern and northeastern portions of the Corn Belt. Thus, some unobserved factors, such as insurance companies ™ marketing strategy, might play more important role than producers ™ risk attitude and area insurance ™s contract design in determining producers ™ demand o f area insurance. Our study largely extends current literature studying demand of area insurance contract by providing some basic facts about systemic risk and pecuniary motivation s for potential area insurance buyers. The correlation between these two fac tors are also explored. Our study also reveals that only AYP contracts at high coverage levels has the potential to induce producers to choose it over MPCI contracts by raising its subsidy rate to a still reasonable level. But we also find that the current AYP subsidy rate might not be the major deterrence for producers to not choose high coverage level AYP contracts. Our study does not explicitly include the risk attitude of producers. This helps us free from choosing among various utility function forms and risk aversion parameters, but it also leads us 48 to underestimate the lower bound of the relative subsidy rate that is required by risk averse producers to choose AYP over MPCI. Our study also does not explore the case for r evenue crop insurances, which has a much larger market share than yield insurance nowadays. Exploring the effects of these aspects asks for more work, and our study provides an analysis framework for future studies to work on. 49 APPENDICES 50 APPENDIX A Major Tables and Figures 51 Table 1. Descriptive statistics for yield variables and county growing condition variables Variable N Mean St. Dev Min Max Unit yield ( bu./ac) 2,133,320 149 39.4 0 374 County yield ( bu./ac) 5,790 137 28.9 27 204 579 1,290 152 360 1,612 579 14.8 14.1 0.49 92.4 579 4.72 1.89 1 13 579 5.53 2.04 0 13 579 47.3 22.5 2.20 93.5 Note s: Mean unit yield is the average of all unit -year yields over the 213,332 units and over the period 1998 -2007. Mean county yield is the average of all county -year yields over the 579 counties and over 1998 -2007. Table 2. Descriptiv e statistics for county systemic risk and its components Variable N Mean St. Dev Min Max Systemic risk, 579 0.37 0.15 0.02 0.78 County -level unit yield sensitivity, 579 1.05 0.20 0.19 2.05 Unit idiosyncratic yield standard deviation, 579 23.4 4.66 9.32 43.0 County yield standard deviation, 579 19.4 5.90 7.51 43.5 52 Table 3. Regression results for equation (30) (1) (2) (3) (4) (5) 2ln () 2ln () 2ln () 2ln () 2ln () ˙/100 -0.068*** -0.050*** 0.126*** 0.086** (0.017) (0.015) (0.022) (0.0 26) ˙/10 0.040** 0.002 0.081*** 0.121*** (0.019) (0.015) (0.023) (0.03 2) ˙ 0.008 0.006 0.044*** 0.048*** (0.010) (0.008) (0.012) (0.016) ˙ -0.032*** -0.008 -0.039*** -0.068*** (0.011) (0.009) (0.012) (0.0 15) ˙ (10 %) -0.005 -0.083*** -0.020* 0.027** (0.008) (0.006) (0.010) (0.014) 2ln () 1.048*** (0.071) 2ln () -0.625*** (0.043) 2ln () 1.004*** (0.028) constant 1.047*** 7.324*** 4.203*** -2.636*** (0.193) (0.206) (0.299) (0.356) Obs. 579 579 579 579 R-squared 0.057 0.255 0.308 0.831 Note s: Robust standard errors are in paretheses; *, **, and *** denote p<0.1, p<0.05, p<0.01, respectively. 53 Table 4. Shapley values for each growing condition variables on systemic risk, % (1) (2) (3) (4) (5) 2ln () 2ln () 2ln () 2ln () 2ln () 2ln () 28.35 2ln () 6.27 2ln () 65.38 ˙ 46.92 9.50 42.97 41.99 ˙ 10.29 2.28 36.00 26.59 ˙ 5.53 1.95 6.42 5.89 ˙ 35.53 0.30 9.60 16.37 ˙ 1.74 85.98 5.01 9.16 Total 100 100 100 100 100 Table 5. Sign and significance of among 5 79 Corn Belt counties Sign and P -value of N Percent >0 & P-value 282 48.7 >0 & P-value >0.1 79 13.6 0 & P-value 143 24.7 0 & P-value >0.1 75 13.0 Table 6. Number of counties with different number of APH groups Number of APH groups Number of counties Percent of counties with associated APH groups 1 37 6.9 2 54 10.0 3 60 11.1 4 87 16.0 5 137 25.4 6 123 22.8 7 40 7.4 8 1 0.2 9 1 0.2 54 Table 7. Descriptive statistics for unit -level TRASR, conditional on MPCI coverage level and AYP coverage level N Mean St.Dev Min Median Max =50%,=70% 208,549 28.3 70.5 0.00 12.4 3,421 =50%,=75% 208,549 19.0 37.5 0.00 9.2 1,084 =50%,=80% 208,549 13.6 23.3 0.00 6.9 482 =50%,=85% 208,549 10.0 15.8 0.00 5.1 280 =50%,=90% 208,549 7.5 11.3 0.00 3.8 179 =55%,=70% 208,549 38.1 84.5 0.00 19.4 4,022 =55%,=75% 208,549 25.7 44.7 0.00 14.2 1,275 =55%,=80% 208,549 18.3 27.6 0.00 10.6 566 =55%,=85% 208,549 13.5 18.6 0.00 8.0 328 =55%,=90% 208,549 10.1 13.3 0.00 5.9 199 =60%,=70% 208,549 51.4 101.0 0.01 29.0 4,711 =60%,=75% 208,549 34.5 53.1 0.00 21.3 1,493 =60%,=80% 208,549 24.6 32.6 0.00 15.9 659 =60%,=85% 208,549 18.1 21.9 0.00 11.9 382 =60%,=90% 208,549 13.5 15.6 0.00 8.9 232 =65%,=70% 208,549 69.2 120.8 0.05 42.1 5,500 =65%,=75% 208,549 46.4 62.9 0.04 30.9 1,743 =65%,=80% 208,549 33.0 38.3 0.03 23.0 761 =65%,=85% 208,549 24.2 25.5 0.02 17.3 442 =65%,=90% 208,549 18.0 18.1 0.02 12.9 268 =70%,=70% 208,549 93.0 144.4 0.26 60.1 6,397 =70%,=75% 208,549 62.2 74.4 0.19 43.9 2,028 =70%,=80% 208,549 44.1 44.9 0.15 32.7 873 =70%,=85% 208,549 32.2 29.6 0.11 24.5 507 =70%,=90% 208,549 23.9 20.8 0.09 18.3 307 =75%,=70% 208,549 125.1 173.1 1.07 85.1 7,408 =75%,=75% 208,549 83.3 88.0 0.80 61.7 2,348 =75%,=80% 208,549 58.9 52.4 0.60 45.7 996 =75%,=85% 208,549 42.8 34.1 0.47 34.2 578 =75%,=90% 208,549 31.8 23.8 0.37 25.6 351 =80%,=70% 208,549 168.2 208.5 3.62 119.6 8,538 =80%,=75% 208,549 111.5 104.2 2.69 86.3 2,706 =80%,=80% 208,549 78.6 61.0 2.03 63.6 1,129 =80%,=85% 208,549 57.0 39.1 1.60 47.4 655 =80%,=90% 208,549 42.1 26.9 1.25 35.5 397 =85%,=70% 208,549 226.1 252.8 9.68 165.5 9,791 =85%,=75% 208,549 149.4 123.9 7.19 120.1 3,103 =85%,=80% 208,549 104.9 71.1 5.44 88.1 1,273 =85%,=85% 208,549 75.8 44.6 4.28 65.3 739 =85%,=85% 208,549 55.8 30.1 3.34 48.8 448 55 Table 8. Premium subsidy rates for individual and area insurance contracts, conditional on coverage level Insurance Plan Coverage Level (%) CAT 50 55 60 65 70 75 80 85 90 Basic and Optional Units 100 67 64 64 59 59 55 48 38 n/a Enterprise Units n/a 80 80 80 80 80 77 68 53 n/a Area Yield Plans n/a n/a n/a n/a n/a 59 59 55 55 51 Whole Farm Units n/a 80 80 80 80 80 80 71 56 n/a Note s: Source: Shields, D. 2015. fiFederal Crop Insurance: Background. fl CRS Report for Congress, Congressional Research Service, 7 -5700, R40532. Washington, DC. Table 9. Coverage -level conditional percent of units whose TASR is higher than the current AYP subsidy rate and percent of units whose TASR is higher than 100% MPCI coverage level AYP coverage level 70% 75% 80% 85% 90% Panel A: Percent of units whose TASR>current AYP subsidy rate 75% 57.2 35.1 21.4 10.9 6.7 80% 71.8 49.5 30.8 14.2 8.2 85% 76.6 56.4 34.9 14.1 7.2 Panel B: Percent of units whose TASR>100% 75% 25.5 11.8 5.8 2.6 1.0 80% 36.3 16.7 7.1 3.1 1.1 85% 40.9 18.4 6.9 2.8 0.8 Table 10. Coveragel -level conditional Pearson ™s correlation test coefficients between county systemic risk and county -level TRASR variables MPCI coverage level AYP coverage level 70% 75% 80% 85% 90% Panel A: Pearson ™s correlation test coefficients between systemic risk and percent of units whose TASR>current AYP subsidy rate 75% 0.1397* 0.0483 -0.0367 -0.1118* -0.1112* 80% 0.1417* 0.0785* -0.0041 -0.1106* -0.1302* 85% 0.0541 -0.0031 -0.0568 -0.1716* -0.1908* Panel B: Pearson ™s correlation test coefficients between systemic risk and percent of units whose TASR>100% 75% -0.0973* -0.1741* -0.2111* -0.2110* -0.1684* 80% -0.0885* -0.1624* -0.2184* -0.2230* -0.1767* 85% -0.1343* -0.1812* -0.2244* -0.2162* -0.1804* Note: * denotes p<0.1. 56 Figure 1. Share of acres insured by area insurance contracts for all crops, 1993 -2018 Note s: Source: Summary of Business, 1993 -2018, Risk Management Agency (RMA). This figure plots acres insured by area insurance contracts and acres insur ed by any kind of crop insurance for all crops over 1993 -2018 (left y -axis), and plots the share of all insured acres that are insured by area insurance contracts for all crops over 1993 -2018 (right y -axis). Area insurance before year 2014 are Group Risk P lan (GRP), Group Risk Income Protection (GRIP), Group Risk Income Protection with Harvest Revenue Option (GRIP -HRO). Area insurance starting in 2014 are under Area Yield Protection (AYP), Area Revenue Protection (ARP), and Area Revenue Protection with Harv est Price Exclusion (ARP -HPE). 0.00 0.05 0.10 0.15 0.20 Share 0246810203040Acres: 10 Million 199319961999200220052008201120142017Year Acres insured by area contracts Total insured acres Share of acres insured by area contracts 57 Figure 2. Geographic distributions of county growing conditions Note s: This figure plots geographic distributions of county GDD, county SDD, frequency of severe drought, frequency of severe wetness, and the proportion of county land that is in land capability classes I or II. Numbers in legends are quartile ranges. No data Panel A: GDD, Gc No data Panel B: SDD, Sc No data Panel C: Drought, Dc No data Panel D: Wetness, Wc No data Panel E: Land Quality, Lc 58 Figure 3. Geographic distribution s of county systemic risk and its components Note: This figure plots geographic distributions of county systemic risk, county yield standard deviation, unit ™s idiosyncratic yield standard deviation , and unit yield ™s sensitivity to county yield. Numbers in legends are quartile ranges. No data Panel A: County systemic risk, Rc2 No data Panel B: County yield st.dev, c No data Panel C: Idiosyncratic yield st.dev, i No data Panel D: Unit yield sensitivity, c 59 Figure 4. Geographic distributions of ™s value , and sign /significance category . Note: This figure plots geographic distributions of . Panel A plots the geographic distribution of ™ values and numbers in the legend are quartile ranges. Panel B plots the geographic distribution of ™s sign and significance . Counties labeled 4 are significantly positive , counties labeled 3 are insignificantly positive , counties labeled 2 are significantly negative and counties labeled 1 are insignificantly negative . Figure 5. Geographic distributions of 2008 APH range and number of APH groups Note s: The 2008 APH range equals the maximum 2008 county APH value minus the minimum 2008 county APH value. Numbers in the legend of Panel A are quintile ranges. No data Panel A: Value of c4321No data Panel B: Sign & Significance of c No data Panel A: 2008 APH range >44321No data Panel B: Number of APH groups 60 Figure 6. Geographic distribution s of county -level percent of units whose TASR is higher than the current AYP subsidy rate, conditional on MPCI coverage level = 75% Figure 7. Geographic distribution s of county -level percent of units whose TASR exceeds the current AYP subsidy rate, conditional on MPCI coverage level = 80% No data Panel A: i=75%, c=70% No data Panel B: i=75%, c=75% No data Panel C: i=75%, c=80% No data Panel D: i=75%, c=85% No data Panel E: i=75%, c=90% No data Panel A: i=80%, c=70% No data Panel B: i=80%, c=75% No data Panel C: i=80%, c=80% No data Panel D: i=80%, c=85% No data Panel E: i=80%, c=90% 61 Figure 8. Geographic distribution s of county -level percent of units whose TASR exceeds the current AYP subsidy ra te, conditional on MPCI coverage level = 85% Figure 9. Geographic distribution s of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 75% No data Panel A: i=85%, c=70% No data Panel B: i=85%, c=75% No data Panel C: i=85%, c=80% No data Panel D: i=85%, c=85% No data Panel E: i=85%, c=90% No data Panel A: i=75%, c=70% No data Panel B: i=75%, c=75% No data Panel C: i=75%, c=80% No data Panel D: i=75%, c=85% No data Panel E: i=75%, c=90% 62 Figure 10. Geographic distribution s of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 80% Figure 11. Geographic distribution s of county -level percent of units whose TASR exceeds 100%, conditional on MPCI coverage level = 85% No data Panel A: i=80%, c=70% No data Panel B: i=80%, c=75% No data Panel C: i=80%, c=80% No data Panel D: i=80%, c=85% No data Panel E: i=80%, c=90% No data Panel A: i=85%, c=70% No data Panel B: i=85%, c=75% No data Panel C: i=85%, c=80% No data Panel D: i=85%, c=85% No data Panel E: i=85%, c=90% 63 Figure 12. Geographic distribution of share of corn acres insured by area insurance contracts Note s: Data Source: Summary of Business, 2000, 2006, 2012, and 2018, RMA. This figure plots geographic distributions of share of corn acres insured by area insurance contracts for year 2000, 2006, 2012, and 2018. For each county, the share is calculated as the rate of all corn acres insured by area insurance contracts over all corn acres insured by any kind of insurance contract. No data Panel A: Crop Year = 2000 No data Panel B: Crop Year = 2006 No data Panel C: Crop Year = 2012 No data Panel D: Crop Year = 2018 64 APPENDIX B Supplementary Tables 65 Table 11. Observation loss after each data screening and merging step Number of counties left Number of units left Initial observations 1,031 14,700,150 Drop units with any non -actual historical yield types 1,003 5,455,480 Drop units with historical year before 1998 980 2,669,560 Drop irrigated units and counties with irrigation rate higher than 20% 847 2,267,540 Drop counties with less than 50 -year data records between 1958 and 2007 712 2,234,220 Drop counties with less than 30 units 618 2,220,260 Merge with weather and land quality data 579 2,133,320 Table 12. Coverage -level conditional percent of units whose TASR is higher than the current AYP subsidy rate and percent of units whose TASR is higher than 100% MPCI coverage level AYP coverage level 70% 75% 80% 85% 90% Panel A: Percent of units whose TASR>current AYP subsidy rate 50% 7.1 4.3 2.7 1.3 0.7 55% 10.0 6.0 3.9 2.0 1.1 60% 14.6 8.6 5.7 3.1 1.8 65% 23.2 13.1 8.5 4.7 2.9 70% 38.0 21.5 13.4 7.3 4.6 75% 57.2 35.1 21.4 10.9 6.7 80% 71.8 49.5 30.8 14.2 8.2 85% 76.6 56.4 34.9 14.1 7.2 Panel B: Percent of units whose TASR>100% 50% 3.4 1.7 0.8 0.3 0.0 55% 4.7 2.5 1.2 0.5 0.1 60% 6.6 3.6 1.7 0.7 0.2 65% 9.8 5.3 2.6 1.1 0.4 70% 15.8 7.9 4.0 1.8 0.7 75% 25.5 11.8 5.8 2.6 1.0 80% 36.3 16.7 7.1 3.1 1.1 85% 40.9 18.4 6.9 2.8 0.8 66 Table 13. Coveragel -level conditional Pearson ™s correlation test coefficients between county systemic risk and county -level TRASR variables MPCI coverage level AYP coverage level 70% 75% 80% 85% 90% Panel A: Pearson ™s correlation test coefficients between systemic risk and percent of units whose TASR>current AYP subsidy rate 50% -0.1360* -0.1543* -0.1504* -0.1391* -0.1080* 55% -0.1238* -0.1515* -0.1514* -0.1467* -0.1201* 60% -0.1024* -0.1291* -0.1407* -0.1495* -0.1256* 65% -0.0273 -0.1026* -0.1194* -0.1391* -0.1360* 70% 0.0665 -0.0382 -0.0897* -0.1292* -0.1167* 75% 0.1397* 0.0483 -0.0367 -0.1118* -0.1112* 80% 0.1417* 0.0785* -0.0041 -0.1106* -0.1302* 85% 0.0541 -0.0031 -0.0568 -0.1716* -0.1908* Panel B: Pearson ™s correlation test coefficients between systemic risk and percent of units whose TASR>100% 50% -0.1866* -0.1841* -0.1549* -0.0939* -0.0380 55% -0.1846* -0.1915* -0.1715* -0.1209* -0.0770* 60% -0.1803* -0.1962* -0.1871* -0.1527* -0.0935* 65% -0.1701* -0.1971* -0.2060* -0.1720* -0.1137* 70% -0.1387* -0.1855* -0.2039* -0.1963* -0.1453* 75% -0.0973* -0.1741* -0.2111* -0.2110* -0.1684* 80% -0.0885* -0.1624* -0.2184* -0.2230* -0.1767* 85% -0.1343* -0.1812* -0.2244* -0.2162* -0.1804* Note: * denotes p<0 .1 67 REFERENCES 68 REFERENCES Babcock, B.A. and Hennessy, D.A., 1996. 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