2. NASS yield data, detrending methodology, and heteroscedasticity treatment

NASS provides 49 categories of field crops including beans, cotton, corn, grain, hay, peanuts, mint, rice, soybeans, and wheat. The data generally date back to the 1950s. We use county-level yield data for corn, soybeans, and winter wheat for the period 1951–2017 (67 years). Our corn and soybean analysis focuses on states that account for the majority of national corn and soybean production. We removed counties with one or more missing yield observations, as well as any state that does not have 25 or more counties. We also removed all states that reported more than 10% of their acreage as irrigated in the 2012 Census of Agriculture. After doing so, we were left with seven states for corn: Illinois (IL), Indiana (IN), Iowa (IA), Minnesota (MN), Ohio (OH), South Dakota (SD), and Wisconsin (WI). These states accounted for 57.8% of harvested acreage and 61.8% of national production in 2017. All corn states except South Dakota met the inclusion criteria for soybeans. These six states accounted for 50.5% and 53.9% of national harvested acreage and production, respectively, in 2017. For winter wheat, we considered the top 15 states that had less than 10% of their acreage irrigated in 2012 Census of Agriculture, only two of which met the inclusion criteria: Kansas (KS) and Michigan (MI). These two states accounted for 29.2% and 28.9% of national harvested acreage and production, respectively, in 2017. In total, our data comprise 414 corn, 373 soybean, and 64 winter wheat counties.

Premium rates are estimated using a two-step process in which a trend is first estimated and then residuals are adjusted for possible heteroscedasticity. A two-step process is by far the most common in the literature as noted by Zhu, Goodwin, and Ghosh (Reference Zhu, Goodwin and Ghosh2011). FCIC estimates the temporal process of yields, denoted $({y_t} = ({y_1},...,{y_T}))$, for each crop-county combination using a robust two-knot linear spline:

(1)$$({y_t} = {\theta _1} + {\theta _2}t + {\delta _1}{d_1}(t - {k_1}) + {\delta _2}{d_2}(t - {k_2}) + {\varepsilon _t},)$$

with $({d_1} = 1)$ if $(t \ge {k_1})$ and $({d_2} = 1)$ if $(t \ge {k_2})$ for knots $({k_1},{k_2} \in (1 + \bar k, \ldots ,T - \bar k))$ and $({k_2} - {k_1} \ge \underline k )$. The $(\underline k ,\bar k \ge 10)$ are imposed, which prevent the knots from locating too close together ($(\underline k )$) or too close to either end point ($(\bar k)$). Knot locations $({k_i})$ are selected using a grid search (least-squares criterion). The model is run with zero, one, and two knots and then the number of knots used is selected using the Akaike information criterion.Footnote ¹ Given the number of knots, two robustness procedures are performed; the spline is iterated to convergence with Huber weights and then twice through a bisquare function. Specifically, let $({\tilde \varepsilon _t})$ be the estimated residuals from the robust spline with the chosen number of knots and $({\tilde \eta _t} = {\tilde \varepsilon _t}/\sqrt {{T^{ - 1}}\sum \tilde \varepsilon _t^2} )$. The Huber function assigns weight one to observations if $(|{\tilde \eta _t}| \lt c)$ and weight $(c/|{\tilde \eta _t}|)$ otherwise with a default $(c = 1.345)$. Similarly, the bisquare function weights observations $({(1 - {({\tilde \eta _t}/c)^2})^2})$ if $(|{\tilde \eta _t}| \lt c)$ and zero otherwise with default $(c = 4.685)$.Footnote ²

Denote the residuals from the aforementioned detrending process as $({\hat \varepsilon _t})$ and the fitted values as $(\hat g(t) = {\hat y_t})$. The heteroscedasticity adjustment via the Harri et al. (Reference Harri, Coble, Ker and Goodwin2011) estimates

(2)$$(\ln (\hat \varepsilon _t^2) = \alpha + \gamma \ln \left( {{{\hat y}_t}} \right) + {v_t}.)$$

Note, constant and proportional variance in the underlying yield data correspond to $(\gamma = 0)$and $(\gamma = 2)$, respectively. Yields are adjusted based on a one-step ahead forecast ($({\hat y_{T + 1}})$) and the heteroscedasticity coefficient ($(\hat \gamma )$):Footnote ³

(3)$$(\hat y_t^* = {\hat y_{T + 1}} + {\hat \varepsilon _t}{\left( {\frac{{{{\hat y}_{T + 1}}}} \over {{{{\hat y}_t}}}} \right)^{\frac{{\hat \gamma }} \over {2}}}.)$$

The adjusted yields are then used to generate the empirical premium rate for period T + 1:

(4)$$(\pi _{T + 1}^* = {\rm{ }}{1 \over T}\sum\limits_{t = 1}^T {\max } \{ 0,\,\lambda \hat y_{T + 1}^* - \hat y_t^*{\rm{\} ,}}$$

where $(\lambda )$ is the coverage level such that $(\lambda \hat y_{T + 1}^*)$ is the yield guarantee.

3. Testing the identically distributed assumption

When testing for structural change, generally a Chow-type test is used; the sample is split into different subpopulations, and residuals from regression equations within the subpopulations are combined with the residuals from a regression equation spanning the two samples to form a Wald-type test statistic. The Bai-Perron test is a sup-type test of the Chow test in that it does not assume the break point is known or the number of break points. The Wilcoxon rank sum test is like a Chow test and primarily has power against changes in location. Overall, these tests have power only against changes in the conditional mean function (first moment) and thus, unsurprisingly, resulted in very few rejections on the adjusted yields across the crop-county combinations.Footnote ⁴ We are interested in structural changes in the higher moments of the data generating process, beyond the conditional mean or variance. A common choice is the Kolmogorov-Smirnov (KS) test, which considers the maximum difference between two empirical distribution functions and thus has power against differences in all moments. Note that the test is nonparametric in that the test statistic is a function of the two empirical distribution functions. Also, the KS test has been shown to have relatively low power in comparison with the Chow or Bai-Perron tests as these tests have an infinitely smaller space of alternatives (Wilcox, Reference Wilcox1997). Moreover, the difference between two empirical distribution functions is most pronounced for differences in the location, followed by differences in scale, and then higher moments in sequential order. Recall we will only be testing differences in the higher moments, and thus the power of the KS test is further weakened in that the two samples we are comparing have near identical first two moments. The KS test statistic is denoted $({D_{n,m}})$ and is defined as

(5)$$({D_{n,m}} = \mathop {\sup }\limits_x |{F_{1,n}}(x) - {F_{2,m}}(x)|,)$$

where $({F_{1,n}})$ and $({F_{2,m}})$ are the empirical distribution functions of the first and the second samples, respectively. Specifically, the entire yield series is detrended and corrected for heteroscedasticity and then split pre- and post-1991 corresponding to the different FCIC rating procedures; recall the area-based programs use all the yield data, whereas the newer supplemental loss programs only use yields from 1991 onward. The null of the KS test is rejected at level $(\alpha )$ when

(6)$$({D_{n,m}} \gt c(\alpha )\sqrt {\frac{{n + m}}\over{{nm}}} ,)$$

where $(c(\alpha ))$ is calculated from the Kolmogorov distribution.

We also consider a second test forwarded by Li (Reference Li1996) and further developed by Li, Maasoumi, and Racine (Reference Li, Maasoumi and Racine2009) (denoted the LMR test). This LMR test is similar to a Cramer–von Mises test in that rather than based on the supremum difference, it is based on the integrated squared difference. Specifically, the LMR test smooths the data using kernel methods and calculates the integrated squared difference. Moreover, Li, Maasoumi, and Racine (Reference Li, Maasoumi and Racine2009) find that power is increased if one bootstraps the null, using randomization methods, rather than use an asymptotic expansion for the distribution of the test statistic. Specifically, the entire yield series is detrended, corrected for heteroscedasticity, and then divided into two subsets pre- and post-1991. The test statistic is defined as follows:

(7)$$(LMR = \int_{ - \infty }^\infty {\left( {{{\hat f}_1}(x) - {{\hat f}_2}(x)} \right)^2}dx,)$$

where $({\hat f_1})$ and $({\hat f_2})$ are kernel estimates based on the two subsets of data. Li, Maasoumi, and Racine (Reference Li, Maasoumi and Racine2009) suggest using least-squares cross validation for bandwidth selection. Moreover, the kernel estimates under the bootstrap samples to recover the distribution of LMR under the null use the same two bandwidths in each bootstrap. In our application, 500 bootstrap samples were used to construct the null.

The test results are presented in Table 1. As expected, the LMR test has noticeably more power than the KS test given the null for the LMR test is calculated using randomization methods.Footnote ⁵ Second, the results reject that the data pre- and post-1991 come from the same distribution in many of the crop-state combinations despite the small number of observations. In corn, 30% of the counties reject at the 5% significance level, while 41% reject at the 10% significance level. The results are similar across the seven states. Soybeans exhibit somewhat less significance as compared with corn; 14% of the counties reject at the 5% significance level, while 24% reject at the 10% significance level. Winter wheat exhibits very little statistical significance, just 6% of the counties reject at the 5% significance level, while 14% reject at the 10% significance level, barely above the size of the test. Interestingly, these results correspond to the level of research expenditures in the three crops over the past half-century. Corn has seen the most innovation, whereas wheat has seen very little. The LMR test results (P value) for corn, soybeans, and winter wheat are graphically illustrated by county-crop combinations in Figure 1. There does appear to be geographical clustering. For example, with respect to corn the majority of the central counties—the high production counties—reject the null of identically distributed. With respect to soybeans, the clustering rejections are in the more eastern counties. With respect to winter wheat, the clustering in Michigan is to the west, while in Kansas it is to the southwest.

Table 1.

Identically distributed test results

Figure 1.

Maps of P values: LMR (Li, Maasoumi, and Racine) test. (a) Corn counties. (b) Soybean counties. (c) Wheat counties.

4. Trimming and estimating crop insurance rates

Results from the previous section call into question the identically distributed assumption from a statistical perspective but provide little information regarding economic importance. As previously mentioned, the loss functions for estimating a distribution versus a premium rate are over different subsets of the density space. We consider the effect of trimming in rating crop insurance contracts by using an out-of-sample retain-cede rating game consistent with the literature. Specifically, the game allows two players using different methodologies to estimate premium rates and adversely select against one another. The game was first proposed by Ker and McGowan (Reference Ker and McGowan2000) and has since been employed by Racine and Ker (Reference Racine and Ker2006), Harri et al. (Reference Harri, Coble, Ker and Goodwin2011), Annan et al. (Reference Annan, Tack, Harri and Coble2013), Tack and Ubilava (Reference Tack and Ubilava2015), Zhang (Reference Zhang2017), and Shen, Odening, and Okhrin (Reference Shen, Odening and Okhrin2018) to justify alternative rating methodologies. The game was modified with an additional test of rating efficiency by Ker, Tolhurst, and Liu (Reference Ker, Tolhurst and Liu2016). Park, Brorsen, and Harri (Reference Park, Brorsen and Harri2019) utilized both tests in proposing an alternative rating methodology that exploits spatial closeness. The game was inspired by the retain-cede decision of private insurers in regard to the crop insurance contracts they sell. Some salient features of the U.S. crop insurance program are relevant to the game. First, FCIC rather than private insurers sets the rates for all policies. Second, the private insurer must sell all policies in a state in which it operates (even if it deems the policy to be underpriced). Third, the private insurer shares, asymmetrically, in the underwriting gains and losses of all policies it sells. Fourth, there is a mechanism by which private insurers can significantly reduce their exposure on policies they deem unwanted.Footnote ⁶ Given these salient features, private insurers determine which policies to retain and which to cede. That is, private insurers retain policies that they believe are overpriced and for which they expect an underwriting gain and cede policies that they believe are underpriced and for which they expect an underwriting loss. As a result, private insurers necessarily develop their own rates in the attempt to strategically adversely select against FCIC and recover excess rents. Mimicking this allows one to hypothetically compare two sets of premium rates: one based on the full yield series and one based on the trimmed yield series. This is in contrast to the past literature that employs the retain-cede game to evaluate alternative rating methodologies using the same data.

Specifically, we assume FCIC uses the full historical yield data from 1951 to 1997 on a county-crop basis to estimate the FCIC premium rates for 1998. Conversely, private insurers estimates their rates using a 25-year trimmed data set (i.e., yields from 1973 to 1997).Footnote ⁷ Both the FCIC and the private insurers use the FCIC rating methodology outlined previously, and as such, the only difference in the two sets of rates is the result of trimming the historical yield data. Based on the two sets of rates, the private insurer identifies which contracts to retain and which to cede. The underwriting gains or losses for the set of retained and ceded contracts are calculated using the actual yields in 1998. This process is repeated for 20 years, and the loss ratios (defined as the ratio of total underwriting losses to total premiums) for both the retained and ceded sets of contracts are calculated. We conduct the game for each crop-county combination at the 90% coverage level where the very large majority of area-based contracts are purchased.

As in the previously cited literature, we undertake two hypothesis tests. The first tests whether the loss ratio from the retained contracts is less than the loss ratio from retaining contracts randomly (choosing which contracts to retain randomly is equivalent to the private insurer being indifferent between the two sets of competing rates). Like Li, Maasoumi, and Racine (Reference Li, Maasoumi and Racine2009), randomization methods are used to recover the P value. Game 1 mimics the current reality of the U.S. crop insurance program. However, private insurers have an advantage because they react to the FCIC premium rates. As such, whichever of the two competing rates the private insurers use has an inherent competitive advantage in game 1. This advantage is nullified in game 2 by contrasting the changes in loss ratios under both sets of the competing rates (for details, see Ker, Tolhurst, and Liu, Reference Ker, Tolhurst and Liu2016). The number of contracts considered is the number of counties multiplied by 20 years: 8,280 contracts for corn, 7,460 contracts for soybeans, and 1,280 contracts for winter wheat. The results, which include percent retained by the private insurer, the government or ceded contracts loss ratio, the insurer or retained loss ratio, P value of game 1, and P value of game 2, are presented in Table 2.

Table 2.

Out-of-sample retain-cede rating game

Under a 25-year trimming decision rule, we find the private insurer’s loss ratio is less than the FCIC loss ratio for all 14 of the 15 state-crop combinations (Michigan wheat is higher only in the third decimal place). For corn, the private insurer loss ratio ranges from 66% to 89% of the FCIC loss ratio. For soybeans, the ratio ranges from 53% to 87%. Given only two states for winter wheat, the ratio is 53% for Kansas and 101% for Michigan. With respect to the first game, P value 1 is significant in all state-crop combinations but Michigan wheat, suggesting that economically and statistically significant rents can be recovered by private insurers by trimming the yield data. With respect to the second game, P value 2 is significant at the 10% level for 7 of the 15 state-crop combinations, suggesting that trimming leads to statistically significantly more accurate premium rates. In no cases did not trimming lead to statistically significantly more accurate premium rates. Specifically, with respect to corn, four of the seven states were significant. With respect to soybeans, two of the six states were significant. Finally, with respect to winter wheat, one of the two states was significant. In summary, the out-of-sample retain-cede rating game provides strong evidence for trimming, consistent with the results from the last section.