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European Review of Agricultural Economics
, Volume 45 (2) – Apr 1, 2018

31 pages

/lp/ou_press/adaptive-local-parametric-estimation-of-crop-yields-implications-for-gZnDSjBWNU

- Publisher
- Oxford University Press
- Copyright
- © Oxford University Press and Foundation for the European Review of Agricultural Economics 2017; all rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0165-1587
- eISSN
- 1464-3618
- D.O.I.
- 10.1093/erae/jbx028
- Publisher site
- See Article on Publisher Site

Abstract The estimation model of crop yields is a prerequisite for deriving actuarially sound insurance premiums. A major challenge in estimating crop yield models arises from the non-stationarity of the data generating process due to technological change and climate change. In this paper, we introduce an adaptive local parametric approach to deal with the non-stationarity of crop yields. An empirical application to major crops in the USA indicates that yield risks for corn and cotton are decreasing, but are increasing for winter wheat. A rating analysis suggests that the proposed model shows potential to obtain more accurate rates than commonly used methodology. 1. Introduction Crop insurance is considered as an important risk management tool to protect farmers against financial shortfalls resulting from yield losses. In fact, the insurance industry has developed a broad spectrum of insurance products, including multiple peril crop insurance, index-based crop insurance and weather derivatives. The premium for crop insurance, for example, amounted to $13 billion worldwide in 2008 (Mahul and Stutley, 2010). The calculation of adequate insurance premiums is an important step when designing these insurance products. On the one hand, insurance demand is quite elastic, i.e. high premium rates will render crop insurance unattractive to potential buyers (Woodard, 2016; Smith and Glauber, 2012). On the other hand, from the viewpoint of insurers, premiums should be sufficiently large to cover expected indemnities, risk loadings, moral hazard and administrative costs (Coble and Barnett, 2012). Thus, it is not surprising, that considerable attention has been paid to develop accurate pricing models (cf. Ker et al., 2016 for an overview). Pricing insurance essentially means to apply a premium principle (e.g. mean value principle or equivalent utility principle) to a loss distribution function. In the case of crop insurance, the loss distribution is derived from a stochastic crop yield model (Ker and Goodwin, 2000; Norwood et al., 2004; Ozaki et al., 2008). In this paper, we discuss challenges in estimating crop yield models from real world data and propose a statistical procedure that supports this task. We then elaborate on the implications for crop insurance rate making. A specific challenge in estimating crop yield models arises from non-stationarity of the data generating process, i.e. the evolution of crop yield distributions over time cannot be adequately described by a simple model with constant parameters. For instance, agricultural crop yields usually show an upward trend over time and deviations from the trend (residuals) frequently exhibit heteroscedasticity. Major causes of non-stationarity are climate change and technological change. McCarl et al. (2008), for example, find that the mean and variance of key climate variables change over time. This, in turn, has a significant effect on average crop yields, yield variability and hence on the cost of insurance. Tolhurst and Ker (2015) investigate the impact of technological developments on crop yields and conclude that technological change not only shifts the mean and variance of yields, but also affects other moments of the yield distribution. These changes make it difficult to determine the data generating process and to accurately model yield distributions using observed time series data. Hence, historical crop yield distributions need to be regularly updated. Otherwise, insurance losses and insurance premiums derived from these yield models will likely be biased. A further question related to the non-stationarity of the data generating process concerns the appropriate length of the sample period to be used for the calculation of crop insurance rates. In case of time-varying parameters, a shorter interval of historical data might be appropriate for estimation purposes. In the USA, for example, the Risk Management Agency (RMA) uses less than 20 years of loss observations to calculate crop insurance premiums, arguing that yield losses from more than 20 years ago may not be representative for current agricultural risk despite data normalisation (Ker et al., 2016). On the other hand, researchers have cast doubt on the use of short samples of available data since it may not be sufficient to properly determine crop yield loss distributions from short historical data, such as 30 years (Coble et al., 2010; Smith and Goodwin, 2010). The lack of historical data may lead to incorrect estimations of the time trend and yield distribution and also introduces a new model risk into the insurer’s decision problem (Courbage and Liedtke, 2003). Thus, insurers face a trade-off when choosing the appropriate sample length: in general, shorter intervals will result in larger variations compared to longer intervals, while longer intervals lead to increases in the modelling bias. Despite the importance of this issue, studies that investigate the impacts of sample length selection in crop insurance pricing are rare. Woodard (2014) compares weather distributions based on 30 and 115 year data and finds little efficiency gains in rate making from using the longer period. On the other hand, Ker et al. (2016) report significant differences in the rating performance depending on the length of sample period of crop yield data. The objective of this study is to present a flexible and parsimonious local model to capture the non-stationary nature of crop yields and to adaptively estimate crop yields and insurance rates. We propose the adaptive local parametric approach (LPA) to model county-level crop yield data in the USA. We begin with a simple linear time trend model as the underlying local parametric model. The time-varying parameters of the local model are determined via adaptive data-driven statistical techniques. The idea is to find for each time point as an optimal longest past-time estimation interval for which the assumption of a (local) parametric model with constant parameters holds. The selection of the optimal longest interval is crucial to the procedure and is accomplished by a backward sequential testing procedure for each considered interval candidate. To examine the performance of the proposed model, we conduct an out-of-sample forecast and an insurance rating game that mimics the rating procedure applied by the RMA in the USA federal crop insurance programme. The contribution of this article to the existing literature is threefold. First, to the best of our knowledge, this is the first study to introduce a local adaptive procedure to deal with the non-stationarity of crop yields and to estimate the time-varying parameters of crop yield models. The results indicate that this approach has the potential to improve the quantification of crop yield risks and the estimation of crop insurance premiums. Second, the study contributes to the longstanding debate on the selection of the optimal sample period for crop yield data. Unlike previous ad hoc analyses, this study offers a novel data-driven perspective to adaptively determine the appropriate sample length via a backward sequential testing procedure. Third, our empirical results contribute to the ‘increasing-risk-hypothesis’ that has been raised in the climate change literature and which has immediate implications for the pricing of crop insurance (Carriquiry and Osgood, 2012; Wang et al., 2013). The rest of the article is structured as follows. Section 2 describes the theoretical framework of crop insurance rate making. First, we provide a brief overview of crop yield models that are applied in crop insurance pricing. Subsequently, we introduce the local parametric estimation procedure. In Section 3, this model is applied to crop yield data, such as winter wheat, corn, soybean and cotton in the USA. We present the empirical models for two representative counties for each crop, followed by a discussion of their estimation results. In Section 4, we assess the forecast performances of different models in terms of crop yields and insurance rates. The final section provides a conclusion and discussion on the potential of local adaptive crop yield models and offers suggestions for further research. 2. Theoretical framework We consider an insurance company that aims to determine the actuarially fair insurance premium for an area yield insurance policy such that insurance premiums cover the expected indemnity payments. An actuarially fair premium π with a coverage level c of (unconditional) expected yield ye is given by1: π(c)=Pr(yt≤cye){cye−E[yt|yt<cye]}=∫0cye(cye−yt)f(yt|Ft)dyt, (1)where yt is the random crop yield at time t. The expectation is based on the conditional yield density f(yt|Ft) with the available information set Ft at time t. Given the non-stationarity of crop yields, the standard approach to pricing crop insurance is a two-stage estimation procedure. In the first step, the trend component and heteroscedasticity from the data are removed. In the second step, a parametric or non-parametric distribution is fitted to the detrended data (Harri et al., 2011; Woodard and Sherrick, 2011; Annan et al., 2014). With the estimated yield density f(yt|Ft) and predictions yt at hand, an actuarially fair premium can be derived by equation (1). We focus on an area yield insurance contract since county-level yield data are used in our study. Following usual practice, we select a 90 per cent coverage level, i.e. c=0.9 (e.g. Annan et al., 2014). 2.1. Crop yield modelling in crop insurance pricing The literature on crop yield modelling offers various approaches to deal with non-stationarity when estimating a yield density. The dynamics of average yields are captured by either a deterministic or a stochastic trend. Deterministic time trend models are dominant in the literature and consist of a simple linear trend, polynomial trend (Just and Weninger, 1999) and spline functions (Harri et al., 2011). Trends using stochastic approaches have been estimated by the Kalman filter (Kaylen and Koroma, 1991) or autoregressive integrated moving average (ARIMA) process (Goodwin and Ker, 1998). Harri et al. (2009) provide an empirical comparison of the deterministic and stochastic time trend models and find limited support for stochastic trends in crop yields. A time trend model only captures mean shifts of the crop yield distribution, but a growing body of empirical evidence has shown that higher-order moments also vary over time due to climatic change or technological change (Yu and Babcock, 2010; Edgerton et al., 2012). To account for the adjustment of heteroscedasticity in historical yields, Skees et al. (1997) assume proportional heteroscedasticity over time, i.e. that the standard deviations of the residuals increase proportionally with increasing yields. Harri et al. (2011) doubt the universal validity of the assumption and find that arbitrarily imposing a specific form of heteroscedasticity in insurance rate calculations limits actuarial soundness. To capture the non-stationarity of other higher moments, Zhu et al. (2011) propose a time-varying yield distribution model by allowing location, scale, skewness and kurtosis parameters to evolve over time, while Tolhurst and Ker (2015) suggest using a mixture of Normals with embedded trend functions to account for different rates of technological change in different components (e.g. mean, variance and skewness) of the yield distribution. After normalising crop yield data, the primary interest is the estimation of the conditional crop yield distribution. Although the probability distribution of crop yields has been extensively investigated over the last few decades, its characterisation remains an open issue. Many parametric and non-parametric distribution models have been introduced to fit crop yield distributions, including the Normal distribution (Botts and Boles, 1958), Gamma distribution (Gallagher, 1987), Beta distribution (Nelson and Preckel, 1980), Logistic distribution (Atwood et al., 2003), Weibull distribution (Sherrick et al., 2004), Inverse hyperbolic sine transformation (Moss and Shonkwiler, 1993), non-parametric/semi-parametric density (Goodwin and Ker, 1998; Ker and Coble, 2003), maximum entropy (Tack et al., 2012) and Normal mixtures (Woodard and Sherrick, 2011; Tolhurst and Ker, 2015). 2.1.1. A benchmark model In this paper, we consider one- and two-knot linear spline models as a benchmark. This choice is motivated by the fact that the USA RMA adopts this method to model the dynamics of crop yields when rating their crop insurance programmes (Harri et al., 2011). It is assumed that crop yields yt at time t=1,…,T follow: yt=α+βt+γ1I{t≥knot1}(t−knot1)+γ2I{t≥knot2}(t−knot2)+et (2)where α refers to the constant, parameters β, γ1 and γ2, knot1 and knot2 are trend coefficients and et is a random error term. For γ2 = 0 equation (2) includes a one-knot linear spline model ( γ2 = 0) and a simple linear trend model ( γ1=γ2=0) as special cases. To increase the stability of the estimated trend over time and across space, RMA imposes restrictions on the knots for the spline models (Harri et al., 2011; Annan et al., 2014). For example, the estimated knots have to be at least 10 years away from the time series endpoints. Instead of making assumptions on the distribution of et, insurance premiums are estimated based on the empirical distribution after adjusting for heteroscedasticity. Yield variance is modelled as a function of predicted yield: var(et)=σ2[E(yt)]δ=σ2yˆtδ. The estimate δˆ is obtained by regressing lneˆt2 on lnyˆt and a constant. Then, the adjusted historical yields yt̃ extrapolated at time T + 1 can be calculated according to: yt̃=yˆT+1+eˆtyˆT+1δˆyˆtδˆ, (3)where yˆT+1 and yˆt are the predicted yields at time T + 1, t=1,…,T based on the trend model (2), and eˆt is the residual yt−yˆt. The actuarially fair premium π corresponding to equation (1) is thus: π(c)=1T∑t=1Tmax[(cyˆT+1−yt̃),0]. (4) A feature of the aforementioned approaches dealing with non-stationarity of crop yields is that they allow for some or all of the model parameters to vary over time based on all of the observed historical data. An identification of these models either requires structural assumptions about the transition process over time or presumes that the parameters follow smooth functions of time. For instance, the linear spline models require knowledge about the number and time of structural breaks and then the adjustment of heteroscedasticity needs a function form of predicted yields over time. In the next section, we present a method that explicitly addresses these issues. 2.2. Adaptive LPA To capture the dynamics of crop yields, we consider a local perspective and develop an adaptive local parametric crop yield model, which is based on the local parametric assumption, i.e. an arbitrary non-stationary process can be well approximated by a simple time-homogeneous model within a given time interval (cf. Spokoiny, 2009). Such an approach makes sense, if the variability of yields is high compared with the variability of the underlying model parameters, so that the latter can be estimated from more recent data. The adaptive local estimation procedure identifies the largest data set (also known as the interval of homogeneity) for which a homogeneous parametric model can be assumed. In a way, this approach addresses the bias-variance trade-off in statistics: the longer the estimation period, the lower the variance of the parameter estimate, but increasing the data set entails the risk that the data no longer follow the same parametric model. This approach provides a well-defined solution for dealing with this trade-off. The theoretical properties distinguish this method from other heuristic procedures that also target the bias-variance trade-off, such as rolling window estimation or exponential smoothing. Recently, this LPA has been successfully applied in many research fields, such as localising temperature risk (Härdle et al., 2016), yield curve term structure modelling (Chen and Niu, 2014), localised realised volatility forecasting (Chen et al., 2010) and time-varying GARCH modelling (Čížek et al., 2009). While this approach has been applied to many complex models in the above-mentioned financial literature, we incorporate the approach into the standard two-stage estimation framework for modelling crop yield risk. Details of the adaptive estimation procedure are presented in Appendix 1. In the first stage, a simple linear time trend model is adopted as the underlying local parametric model, which is reasonable for short time intervals, such as 10 years. We assume that the crop yield yt, at time t∈(1,…,T) is as follows: yt=αt+βtt+ϵt, (5)where αt,βt denote the time-varying local linear trend coefficient at time t and ϵt is a mean-zero random error term over a fixed interval I=[t−m,t] of (m+1) (m≤t) observations. To mitigate the effect of outliers on trend estimation, a robust iterative reweighted least square Huber M-estimator is employed (Finger, 2010; Annan et al., 2014). In the second step, the detrended yields ỹt are assumed to follow a normal distribution2 with mean μt and time-varying variance σt2. The distribution parameters θt=(μt,σt) are time dependent and can be estimated by a (quasi) maximum likelihood estimation over a fixed interval I of detrended yields ỹt. For time point t0 the (quasi) maximum likelihood estimator θ̃(t0) of (m+1) detrended yields observations is defined as θ̃(t0)=argmaxθ∈ΘL(ỹ;I,θ), (6)where Θ=R+×R+ denotes the parameter space and L(ỹ;Is,θ) is the local log-likelihood function. We refer to θ̃(t0) as the local maximum likelihood estimator. Note that the only difference with the conventional two-stage estimation is the fact that the LPA depends on optimally selected time intervals I. Since time-varying trend coefficients and distribution parameters are both determined by the same selected intervals, we refer to θt=(αt,βt,μt,σt) from now on. How are the optimal intervals, for which local homogeneity assumption of estimated parameters holds, optimally selected? For nested intervals I0⊂I1⊂…⊂IK of crop yield data, the estimation procedure embeds a multiple local change point test to detect the longest homogeneous interval. Since the true distribution of the test statistic of the local change point test is unknown, critical values have to be determined by simulation. Based on the estimated local parametric model from in-sample crop yield data, one can simulate homogeneous time series of crop yield data and corresponding homogeneous nested intervals. Next, the small modelling bias condition is imposed, which essentially means that the distance between the true and the estimated model parameters is bounded by a small constant with a high probability (see Appendix 1 for a formal definition of the small modelling bias condition). The risk bound derived from the small modelling bias condition allows us to define confidence sets that are used to determine critical values in the local change point test. Note that the derivations of risk bound and critical values depend on the specification of a few hyper-parameters. Appendix 2 details the specification of these hyper-parameters and presents a robustness analysis. With critical values at hand, we implement a backward sequential testing procedure that is based on the local change point test. For each time point, we start from the shortest past-time interval to estimate the (locally) constant parameters. Then, the interval is iteratively extended and tested for local homogeneity of the estimated parameters. The procedure ends when local homogeneity is rejected, i.e. a significant difference is detected in the values of the estimated parameters between the current and previous intervals. After the longest homogeneous interval has been determined, we estimate the linear trend coefficients and distribution parameters from which we can predict yˆT+1 and estimate the yield density f(yt|Ft). The insurance premiums π are then derived according to equation (1). 3. Application to USA crop yields Due to data availability and quality, we utilise annual USA county-level crop yield data from the National Agricultural Statistical Service (NASS) to investigate the performance of the local parametric crop yield model. The considered crops are winter wheat, corn, soybean and cotton. We choose two states for each crop according to their important role in the national production (see Table 1). County-level crop yield data cover the period from 1955 to 2014 with the exception of winter wheat in Texas, where data from 1968 to 2014 are available. After excluding counties without continuous yield records, we have data from 106, 178, 187 and 34 counties for winter wheat, corn, soybean and cotton, respectively. The broad coverage of the data set allows us to explore the importance of non-stationarity for different crops and different geographical locations. Practical relevance arises from the fact these data form the basis of area yield insurance programmes, such as the group risk plan implemented by the RMA. Summary statistics of yield data are provided in Table 1. Table 1. Summary statistics of yield data State Number of counties Min. Mean Median Max. Std Dev. Coef. of variation Winter wheat Kansas 61 5.0 32.7 32.8 80.0 10.3 0.31 Texas 45 6.4 26.5 25.5 64.6 9.5 0.36 Corn Illinois 79 19.0 114.5 114.0 236.0 39.2 0.34 Iowa 99 18.3 115.0 114.7 206.6 38.9 0.34 Soybean Illinois 89 9.5 36.5 36.0 69.3 9.7 0.27 Iowa 98 7.3 37.3 36.9 64.0 9.6 0.26 Cotton Georgia 20 127.0 589.1 564.0 1,264.0 212.3 0.36 Mississippi 14 237.0 726.9 711.0 1,482.0 208.8 0.29 State Number of counties Min. Mean Median Max. Std Dev. Coef. of variation Winter wheat Kansas 61 5.0 32.7 32.8 80.0 10.3 0.31 Texas 45 6.4 26.5 25.5 64.6 9.5 0.36 Corn Illinois 79 19.0 114.5 114.0 236.0 39.2 0.34 Iowa 99 18.3 115.0 114.7 206.6 38.9 0.34 Soybean Illinois 89 9.5 36.5 36.0 69.3 9.7 0.27 Iowa 98 7.3 37.3 36.9 64.0 9.6 0.26 Cotton Georgia 20 127.0 589.1 564.0 1,264.0 212.3 0.36 Mississippi 14 237.0 726.9 711.0 1,482.0 208.8 0.29 3.1. Model specification The proposed adaptive LPA depends on a set of parameters, namely, the considered interval candidates and hyper-parameters used for the calculation of the critical values. In the following, we justify the specification of these parameters. Motivated by empirical applications in the literature we select (K+1)=8 nested intervals from 5 to 40 historical observations, i.e. {5,10,15,20,25,30,35,40}. Here, we assume that the shortest interval (5 years) is always homogeneous and test if the homogeneity assumption applies to longer intervals via the local change point test. The longest interval includes 40 observations. This choice allows us to use the remaining data to evaluate the out-of-sample performance of the proposed method. Second, the hyper-parameters r and ρ are crucial to the calibration of risk bounds and critical values. We follow Chen and Niu (2014) and Härdle et al. (2013) and consider the hyper-parameters r=0.5 and ρ=0.5. To evaluate the performance of the proposed LPA, we compare it with three benchmark models: a one-knot linear spline model (hereafter, Spline1), a two-knot linear spline model (hereafter, Spline2) and a simple linear trend model with a 40-year rolling window (hereafter, RW40). Unlike the LPA where a potential structural break in the data is determined among all candidate intervals, the linear spline models first limit the number of structure changes as one or two and then estimate the knots with all available historical data. The RW40 can be considered as a special case of the LPA in which the homogeneity assumption of the parametric model holds for all historical crop yield data and the longest possible interval (40 years) is always selected. Figure 1a compares the three benchmark models for the case of wheat in Ellis County, Kansas in 2014. The graph of observed data suggests that one- or two-knot linear spline models might not sufficiently capture the non-stationarity of winter wheat yields and that further structural breaks might exist after 1985. Figure 1b demonstrate the effect of the length of the sample periods on the estimated parameters of the yield model. The next section explores how different estimates of the trend model translate into mean forecasts and risk assessments for crop yields. Fig. 1. View largeDownload slide (a) Different trend estimations and (b) coefficients of linear trend over selected data periods; winter wheat in Ellis County, Kansas. Fig. 1. View largeDownload slide (a) Different trend estimations and (b) coefficients of linear trend over selected data periods; winter wheat in Ellis County, Kansas. 3.2. Empirical results In this section, we illustrate the implementation of the proposed local parametric crop yield model for eight random representative combinations of crops and counties. We apply the adaptive estimation procedures to select the longest interval for which the parameter homogeneity assumption is not violated. The estimation results based on adaptively selected optimal intervals for two representative counties are shown in Figure 2. Since the considered maximum estimation interval is 40, we estimate the time-varying parameters θ via equations (5) and (6) for 1994 based on 40 observations from 1955 to 1994. Then, the proposed LPA will determine the optimal interval of homogeneity over which we can estimate the time-varying parameters for 1994 and then predict the crop yield for 1995. The estimation procedure is repeated for each time point from 1994 to 2014. The resulting optimal interval for wheat in Ellis in 1994, for example, is 20. The estimation of the trend model (5) should be based on observations from 1975 to 1994. This implies that a local change point (a structural break of the model parameters) has been detected between 1969 and 1974. In fact, this finding is in line with the visual inspection of observed wheat yields in Figure 1a. The structural break in this period might result from technological progress and favourable weather conditions. From the 1940s through the 1960s, winter wheat in Kansas experienced a significant gradual increase in yield due to higher yielding wheat varieties, increased fertiliser usage and increased irrigation. Meanwhile, unusually favourable weather from the mid-1960s to the early 1970s has further contributed to rich harvests of winter wheat (Hill et al., 1980; NASS, 2016). As the targeting year moves forward, optimal interval lengths for 1995, … , 2014 change, but mainly range from 20 to 30 years. In contrast to wheat, optimal interval lengths of corn in Story County, Iowa, for recent years are getting shorter, implying that certain structural change took place during those periods. Fig. 2. View largeDownload slide Estimated trend (left) and volatility σt (right) of different yield models for (a) wheat (Ellis, Kansas) and (b) corn (Story, Iowa). Note: Longest optimal time intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. Fig. 2. View largeDownload slide Estimated trend (left) and volatility σt (right) of different yield models for (a) wheat (Ellis, Kansas) and (b) corn (Story, Iowa). Note: Longest optimal time intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. Figure 2 also describes the estimated trend coefficient and standard deviations from all considered models.3 Based on the selected intervals of homogeneity, the estimated local parameters βt and σt from the LPA change considerably over time. The slope coefficient βt and standard deviation σt represent the time-varying annual change and dispersion of crop yields, respectively, resulting from technological change and climate change over time (McCarl et al., 2008; Tolhurst and Ker, 2015). Unlike in a global linear trend model, the effect of severe crop shortfalls has more influence on the estimated βt and σt in a local parametric model. This may reflect how quickly the advances in seed technology respond to changing climate conditions and could allow us to better capture current crop yield risks. For wheat in Ellis County, Kansas, the estimated slope coefficients of all models are similar, except for the RW40. Moreover, the two-knot spline linear model leads to a considerably biased trend coefficient for 1999, which highlights a potential flaw when a fixed number of structural breaks is assumed. The slope coefficients for the period from 1994 to 2014 are mostly positive, but exhibit a decreasing pattern in recent years. This indicates that the rate of crop yield growth is decreasing; however, the magnitudes of the changes are rather moderate and range between 0.1 and 0.4 (bushels per acre). On the other hand, the standard deviation σt is constantly increasing over time, suggesting that wheat yield risk has become much higher compared to 20 years ago. The estimated standard deviation σt of model Spline2 are higher and more volatile compared to the other models. For corn in Story County, Iowa, the trend coefficients estimated from the LPA and Spline2 in recent years are also decreasing similar to that of wheat yields in Ellis County, Kansas, but the magnitude of the trend coefficients is considerably higher for corn. In contrast, the RW40 and Spline1 result in more stable trend coefficients over years. Interestingly, in contrast to the increasing dispersions of wheat yields, the estimated standard deviations σt for corn yields exhibit different patterns after 2000. While the estimated σt of one- or two- knot spline models indicates an increasing corn yield risk, the LPA suggests the opposite. Reduced corn yield risks in recent years are also reported by Woodard et al. (2011) and may reflect a higher resistance of corn against weather stress, especially drought (Yu and Babcock, 2010). In this situation, a global trend model estimated with historical corn data overestimates standard deviations and corn yield risks. In turn, the corresponding crop insurance would be overpriced. The opposite finding, however, applies to estimated standard deviations over time of cotton in Dooly County, Georgia. The estimated σt of the LPA, though decreasing over time, are often larger than the benchmark models. For soybean, changes in the estimated trend coefficients and standard deviations over time appear to be moderate, suggesting that the assumption of homogeneous parameters likely holds over this time period. The results for soybean and cotton are presented in Figure 3. Fig. 3. View largeDownload slide Estimated trend (left) and volatility σt (right) under different models for (a) soybean (Henry, Illinois) and (b) cotton (Dooly, Georgia). Note: Longest optimal time intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. Fig. 3. View largeDownload slide Estimated trend (left) and volatility σt (right) under different models for (a) soybean (Henry, Illinois) and (b) cotton (Dooly, Georgia). Note: Longest optimal time intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. 4. Implications for crop yield insurance 4.1. Forecast of expected yield and estimation of insurance premium To recover an actuarially fair premium, one has to correctly determine the expected yield ye and conditional yield density f(yt|Ft) according to equation (1). In this section, we compare the performance of the LPA in terms of expected yields and implied insurance premiums. To analyse the mean forecasting performance, we calculate one-step point forecasts of crop yields for each time point in the out-of-sample period, i.e. from 1995 to 2014 (and from 2008 to 2014 for wheat in Texas). The forecast accuracy is measured by the root mean squared error (RMSE). Figure 4 depicts the observed and predicted crop yields, as well as the RMSE for eight counties. Overall, the predicted crop yields of the LPA captures the crop yield dynamics well, but they largely deviate when single erratic shortfalls occur. Comparing the RMSE of the LPA with the benchmark models shows a mixed picture of the forecast accuracy: the LPA shows the smallest RMSE for three out of eight representative crop-county combinations and ranks second best for three crop-county combinations. The Spline1 and Spline2 models are superior in three and two cases, respectively. Fig. 4. View largeDownload slide One-step ahead forecasts of crop yields under different models with RMSE in parentheses. Fig. 4. View largeDownload slide One-step ahead forecasts of crop yields under different models with RMSE in parentheses. To better understand under what conditions the LPA outperforms the benchmark models, a closer look at the results for different crops is helpful. Apparently, yield data follow different patterns. Corn, wheat and cotton exhibit structural breaks either in their trend, yield, or dispersion (see Figures 1 and A4), implying that data from 20 or 30 years ago are not useful for predicting current yield levels and volatility and are likely introduce a parameter bias for recent years. Thus, shorter data intervals are appropriate in this case. The LPA is able to detect these structural breaks and selects appropriate intervals of parameter homogeneity. However, performance of the LPA is not superior if crop yields are characterised by erratic outliers rather than structural changes. This holds, for example, for corn yields in Mercer and in Story that show a shortfall in 2010 followed by an abrupt rise in 2014. A similar pattern applies to soybean yields (Figure A4(b)). Thus, if one considers these shortfalls as outliers, the assumption of homogeneous parameters likely holds over the entire time period. The local change point test recognises these single yield shortfalls as an indicator of a parameter change and rejects the homogeneity assumption for the longer time intervals. This results in higher variance of the parameter estimates and higher forecast errors. Since the differences in RMSE between all models are moderate, we test their significance (Table 2). A Diebold-Mariano (DM) test with the LPA as a benchmark shows that most of the aforementioned differences in the out-of-sample forecasts are not significant, which is likely due to the rather short sample period. A Mincer-Zarnowitz (MZ) test4 rejects the null hypothesis of an efficient forecast of the LPA for wheat in Coryell and for corn in Mercer and Story. For the two spline models, this hypothesis is rejected in only two out of eight cases. This suggests that the LPA cannot significantly improve the forecast accuracy of standard yield models. However, mean forecast errors do not provide full insight into economic implications and the value of the LPA for crop insurance pricing since insurance premium rates reflect crop yield risks and rely on variance estimation. Given the estimated crop yield density and predicted yields, insurance premiums are calculated according to equation (1) for the LPA and benchmark models. Figure 5 depicts the insurance premiums in the out-of-sample period. As expected, the patterns of estimated insurance premiums mimic those of estimated variances (Figures 2 and 3) and the same relationships among the considered models apply here as well. Particularly, for corn and soybean, the insurance premiums estimated with the LPA are systemically lower than those from global models, such as Spline1 and Spline2. However, the question of which model determines insurance premiums most accurately remains unanswered since true variances and ‘true’ premiums are latent and unobservable. We turn to this issue in the next section. Table 2. Forecast accuracy evaluation Model Wheat Corn Soybean Cotton Ellis Coryell Mercer Story Henry Cedar Dooly Coahoma Diebold-Mariano test LPA Benchmark Spline1 −1.674 0.735 0.000 −0.367 −0.989 1.278 −0.278 0.873 Spline2 −1.219 −1.419 0.827 0.291 −1.006 −1.227 −0.715 0.211 RW40 −0.673 −0.472 −0.013 −0.067 −0.376 1.499 −0.588 0.073 Mincer-Zarnowitz test LPA 1.505 15.844*** 2.761* 7.456*** 0.011 0.395 0.050 2.004 Spline1 2.344 4.748* 1.854 5.112** 0.230 0.514 1.755 2.186 Spline2 2.214 6.935** 0.913 6.850** 0.179 1.287 0.283 1.198 RW40 3.281* 3.693* 1.109 2.678* 0.228 0.678 3.508* 3.986** Model Wheat Corn Soybean Cotton Ellis Coryell Mercer Story Henry Cedar Dooly Coahoma Diebold-Mariano test LPA Benchmark Spline1 −1.674 0.735 0.000 −0.367 −0.989 1.278 −0.278 0.873 Spline2 −1.219 −1.419 0.827 0.291 −1.006 −1.227 −0.715 0.211 RW40 −0.673 −0.472 −0.013 −0.067 −0.376 1.499 −0.588 0.073 Mincer-Zarnowitz test LPA 1.505 15.844*** 2.761* 7.456*** 0.011 0.395 0.050 2.004 Spline1 2.344 4.748* 1.854 5.112** 0.230 0.514 1.755 2.186 Spline2 2.214 6.935** 0.913 6.850** 0.179 1.287 0.283 1.198 RW40 3.281* 3.693* 1.109 2.678* 0.228 0.678 3.508* 3.986** Notes: In a Diebold-Mariano test, the negative sign implies that the benchmark’s loss is lower than that implied by other models. A higher Mincer-Zarnowitz test statistic indicates that the null hypothesis is more likely to be rejected, and hence the considered model is less favourable. ***, ** and * represent the significance at the 1, 5 and 10 per cent level, respectively. Fig. 5. View largeDownload slide Fair insurance premiums under different models. Fig. 5. View largeDownload slide Fair insurance premiums under different models. 4.2. Out-of-sample rating game To cope with the non-observability of correct insurance premiums and to explore the potential of the LPA for improving the accuracy of estimated crop insurance rates, we conduct an out-of-sample rating game which has been commonly used in the crop insurance literature (Harri et al., 2011; Annan et al., 2014; Tolhurst and Ker, 2015). The rating game mimics features of the USA crop insurance programme, that is private insurance companies can choose to participate in the insurance programmes and sell insurance contracts to farmers with premium rates set by the RMA of the government. Therefore, private insurance companies will likely re-estimate the premiums and compare their premiums with those set by the RMA. If their premiums are higher than the RMA’s premiums, they cede the policies as they believe the RMA’s premiums to be underpriced. Conversely, if their premiums are lower than the RMA’s premiums, they retain the policies as they believe the RMA’s premiums to be overpriced (Tolhurst and Ker, 2015). Here, we compare the RMA premiums with those derived from the LPA. Then, given actual realised crop yields, we calculate the loss ratios of retained policies and ceded policies for each county and period combination. In the simulated rating game, the yield guarantee cye for an insurance policy is determined by the RMA. Thus, the difference between private insurance premiums and the RMA’s premiums depends solely on their estimations of the conditional yield density f(yt|Ft). For the out-of-sample period from 1995 to 2014, we repeat the rating game to derive premiums and loss ratios for each period and county combination using data from 1955 to 1994, … , 1955 to 2013. Note that for the LPA, only the selected longest data interval of homogeneity will be used for insurance rating. According to Harri et al. (2011) and Annan et al. (2014), the RMA models crop yields with a one- or two-knot linear spline and then adjusts the residuals for heteroscedasticity. Therefore, we compare our proposed LPA against the one-knot and two-knot linear spline models, respectively. Based on the detrended crop yields, the empirical premium is estimated as the RMA’s premium ( πg). In other words, this technique uses a burn analysis rating, whereas the LPA premium ( πp) is based on simulated predicted crop yields given the time-varying parameter estimates. The cede-retain decisions for each insurance policy are then made according to the comparison between πg and πp. Following usual convention, loss ratios are aggregated on a state basis using planted acreage for the final year 2014 as weights. If loss ratios for retained polices are smaller than those for ceded polices, we conclude that the LPA is better at estimating the premium rate than the RMA method. Statistical significance of differences in loss ratios is tested by a randomisation test.5 The results of the out-of-sample insurance rating game between the LPA model and RMA methods are presented in Table 3. The percentages of retained policies are quite high for all corn and soybean policies, suggesting that most of the LPA premium rates are lower than the RMA rates. This observation is in line with our finding in the analysis in Section 3.2 that the one- or two-knot spline model based on all historical data tends to overestimate current yield volatility. The fact that πg frequently exceeds πp results in a high number of retained policies, which, in turn, renders the randomisation test less meaningful. In other words, the out-of-sample rating assessment may fail if the considered rating methodology leads to systematically lower or higher premiums than the RMA method. Nevertheless, we observe that in seven of the 16 state-crop combinations, insurance companies could gain a significant economic rent using the LPA rating method. Particularly, the results for winter wheat and cotton are in favour of the LPA method since it captures tail yield risks better than the RMA approach. Given its high ratio of policy payouts, potential economic rent of private insurance companies using the LPA could be much more considerable. For the other nine state-crop combinations, the results show that either the RMA methodology leads to lower loss ratios or that the differences are not statistically significant. Table 3. Out-of-sample insurance rating game Crop-state Method Number of counties Retained policies (%) Loss ratio of retained oolicies Loss ratio of ceded polices p-value Payout policies (%) Loss ratio of all policies ( πg) Loss ratio of all policies ( πp) Winter Wheat Kansas One-knot 61 61.9 1.490 1.968 0.014 0.355 1.671 1.469 Texas 45 66.7 1.893 1.903 0.489 0.444 1.896 1.836 Kansas Two-knot 61 57.7 1.369 2.012 0.005 0.356 1.610 1.469 Texas 45 53.7 1.924 2.394 0.128 0.435 2.131 1.836 Corn Illinois One-knot 79 97.5 0.577 1.020 0.131 0.235 0.586 1.435 Iowa 99 92.5 0.303 0.179 0.857 0.145 0.296 0.845 Illinois Two-knot 79 96.6 0.628 0.973 0.179 0.249 0.636 1.435 Iowa 99 89.6 0.404 0.250 0.896 0.174 0.392 0.845 Soybean Illinois One-knot 89 88.9 0.648 0.806 0.268 0.199 0.659 1.413 Iowa 98 84.6 0.818 0.851 0.432 0.201 0.822 1.507 Illinois Two-knot 89 84.8 0.717 1.690 0.001 0.221 0.802 1.413 Iowa 98 82.3 0.961 1.004 0.439 0.232 0.966 1.507 Cotton Georgia One-knot 20 76.5 0.610 0.946 0.100 0.395 0.651 0.782 Mississippi 14 69.3 0.476 0.925 0.032 0.225 0.612 1.017 Georgia Two-knot 20 71.8 0.467 1.042 0.005 0.332 0.580 0.782 Mississippi 14 49.6 0.583 2.281 0.000 0.279 1.397 1.017 Crop-state Method Number of counties Retained policies (%) Loss ratio of retained oolicies Loss ratio of ceded polices p-value Payout policies (%) Loss ratio of all policies ( πg) Loss ratio of all policies ( πp) Winter Wheat Kansas One-knot 61 61.9 1.490 1.968 0.014 0.355 1.671 1.469 Texas 45 66.7 1.893 1.903 0.489 0.444 1.896 1.836 Kansas Two-knot 61 57.7 1.369 2.012 0.005 0.356 1.610 1.469 Texas 45 53.7 1.924 2.394 0.128 0.435 2.131 1.836 Corn Illinois One-knot 79 97.5 0.577 1.020 0.131 0.235 0.586 1.435 Iowa 99 92.5 0.303 0.179 0.857 0.145 0.296 0.845 Illinois Two-knot 79 96.6 0.628 0.973 0.179 0.249 0.636 1.435 Iowa 99 89.6 0.404 0.250 0.896 0.174 0.392 0.845 Soybean Illinois One-knot 89 88.9 0.648 0.806 0.268 0.199 0.659 1.413 Iowa 98 84.6 0.818 0.851 0.432 0.201 0.822 1.507 Illinois Two-knot 89 84.8 0.717 1.690 0.001 0.221 0.802 1.413 Iowa 98 82.3 0.961 1.004 0.439 0.232 0.966 1.507 Cotton Georgia One-knot 20 76.5 0.610 0.946 0.100 0.395 0.651 0.782 Mississippi 14 69.3 0.476 0.925 0.032 0.225 0.612 1.017 Georgia Two-knot 20 71.8 0.467 1.042 0.005 0.332 0.580 0.782 Mississippi 14 49.6 0.583 2.281 0.000 0.279 1.397 1.017 *Note: A p-value close to 0.00 indicates that the proposed method outperforms the RMA method. On the other hand, if a p-value is close to 1.00, the RMA method outperforms the proposed method. The loss ratio of all policies in the last column is calculated based on the LPA’s premiums. Though the high number of retained policies renders the results of the out-of-sample rating assessment useless, it does not imply that a private insurance company will not be able to make a profit or that the premiums derived from the proposed LPA method are underpriced. If the RMA premiums exceed actual average losses, profits can be made by retaining all policies. To capture this aspect, we also display the realised loss ratios of all policies in the last two columns of Table 3. If the realised loss ratio is smaller (larger) than one, insurance premiums are supposed to be overpriced (underpriced). For winter wheat and cotton, the loss ratios based on the LPA premiums are closer to the expected long run value of one than for the RMA’s premiums. Indeed, the RMA’s premiums are underpriced for winter wheat and overpriced for cotton. Interestingly, the loss ratio using the LPA’s premiums is also closer to one for corn, though the results for corn in Iowa in the out-of-sample rating assessment are not in favour of the LPA method. For soybean, the LPA underestimates the insurance premiums. This can be explained by the difficulty to distinguish between unusual shortfalls and structural changes in mean-forecasting of soybean yields. However, soybean insurance premiums estimated from the 40-year rolling window are similar and even smaller than that from the LPA (e.g. Cedar County in Figure 5). Thus, the disappointing performance of the LPA in this case is probably not rooted in the local change point test, but in the assumption of normally distributed yields. 5. Discussion and conclusions This article was motivated by the challenge of considering non-stationarity in the estimation of crop yield models, which is a building block for the pricing of crop insurance. Non-stationarity can be a result of technological change and/or climate change. To deal with non-stationarity, various approaches have been proposed to allow some or all model parameters to vary over time. An identification of these models in the current literature requires either structural assumptions about the transition process over time or presumes that the parameters follow smooth functions of time. In this article, we develop an alternative data-driven approach that is based on the local parametric assumption. To be specific, the idea of the LPA is to find an optimal interval of homogeneity over which one can fit a local parametric model with constant parameters. The selection of an optimal interval of homogeneity is determined in a backward sequential testing procedure with an embedded local change point test. The advantage of adaptively and promptly detecting structural change makes the proposed LPA more flexible and less restrictive for modelling time-varying parameters compared to previous approaches. In addition, the proposed approach enables us to contribute to the longstanding debate over the selection of the sample period for crop yields from a sound statistical perspective. The backward selected sample period allows us to more accurately determine the current rate of technological change and the current risk of crop yields, and therefore to mitigate the potential bias caused by historical crop yield data from many decades ago. We apply the proposed LPA to county-level winter wheat, corn, soybean and cotton yields in a large number of counties in the USA. Our empirical results demonstrate that the proposed LPA selects reasonable intervals of parameter homogeneity, mainly ranging from 20 to 30 years before the current period. In contrast to earlier work on this issue, we relax the assumption of a fixed sample period over the entire data set. In fact, a change of estimated local parameters, such as βt and σt, over time allows us to capture the deceleration of crop yield growth, the decrease in corn and cotton yield variability, and the increase in wheat yield risk that has been found in the literature (e.g. Yu and Babcock, 2010). In terms of the forecasting accuracy of crop yields, the results show that the LPA, in general, has the potential to reduce forecast errors of crop yields for several crops compared to traditional alternatives. This is particularly true when yield data exhibit several structural breaks or regime switches. A simulation exercise documents that the LPA has the potential to improve the pricing of insurance contracts because it allows for changes in yield variability. A shortcoming of the proposed LPA, however, is the difficulty of the local change point test to distinguish between singular shortfalls and persistent structural changes. This weakness leads to the rejection of a longer interval of homogeneity and to more volatile parameter estimates. This problem could be mitigated by the exclusion or devaluation of outliers, which is also recommended in the average loss cost ratio approach in the RMA rating methodology (Coble et al., 2010). How can these insights into the comparative advantage of different yield and pricing methods be translated into practical recommendations for insurance companies? Overall, the margin for improving rate making of crop insurance is rather small given the sophisticated methods that are currently in place. Nevertheless, if graphical inspection of yield series, sensitivity of insurance prices with respect to the length of a rolling window, or a priori considerations suggest the existence of structural breaks in the level or the volatility of yield data, it might be useful to apply the LPA in addition to standard spline models. This study provides the first empirical application of an adaptive local parametric model to crop yields. However, a number of potential extensions may further improve the performance of this model and are suggested for future research. First, one may apply alternative homogeneity tests that can detect structural change, but are less sensitive to occasional catastrophes. A simple likelihood ratio test is an example (Härdle et al., 2016). Second, the underlying crop yield model, which is a simple linear trend mode in our case, could be refined. More sophisticated crop yield models include a mixture Normals with embedded trend functions (Tolhurst and Ker, 2015) or a model that takes into account extreme weather events through exogenous weather variables. Finally, the incorporation of heavy-tailed distributions, such as the Weilbull distribution (Woodard, 2014) or Beta distribution (Zhu et al., 2011) might further improve the model results since empirical evidence suggest that area yields are not normally distributed. Supplementary data Supplementary data are available at ERAE online. Acknowledgements The authors would like to thank the editor Iain Fraser and two anonymous reviewers for their helpful comments. Financial support from the German Research Foundation (DFG) is gratefully acknowledged. Footnotes 1 Prices of crop yields are not taken into account and thus fair premiums are measured by a physical unit, e.g. bushel/acre. 2 The assumption of normal distributions for crop yields has been criticised when fitting a global model. Previous studies on local parametric models, however, document that normality nevertheless can be plausible in a local window (Andriyashin et al., 2006; Wang et al., 2013; Härdle et al., 2016). 3 The trend coefficients are calculated for each endpoint and refer to βt (the LPA and RW40), ( β+γ1) (Spline1) and ( β+γ1+γ2) (Spline2). The standard deviations for Spline1 and Spline2 are empirical standard deviations from the heteroscedasticity-adjusted yields. 4 The MZ test is based on the idea that the error of an efficient forecast has to be unbiased and uncorrelated with the forecast itself according to the Mincer-Zarnowitz regression: yt=a+bytf+ϵt, where yt and ytf are observed and forecast values. 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At a fixed time point t0, we use historical observed data Yt,t≤t0 to estimate the unknown parameters θ(t0) and repeat the procedure for each newly included time point t0. The objective is to select the longest interval of homogeneity of Yt over which the homogeneity assumption of the parametric model holds. Since the number of possible interval candidates can be large, we consider only a finite set of intervals, e.g. K + 1 increasingly nested intervals I0⊂I1⊂…⊂IK. For each interval, the corresponding quasi-ML estimators θ̃I0(t0),θ̃I1(t0),…,θ̃IK(t0) for the local parametric model can be determined for a fixed time point t0. From now on, we ignore the index t0 and describe the procedure for an arbitrary fixed time point. Let θˆIk refer to the accepted adaptive estimator in the interval Ik, and let θˆ denote the optimal estimator based on the longest homogeneous interval Iˆ. The selection algorithm is built on a sequential testing procedure. It starts from the shortest interval I0 over which local homogeneity holds by the assumption and the maximum likelihood estimator θ̃I0is accepted, i.e. θˆI0 = θ̃I0. Then, we iteratively extend to next longer intervals Ik over which the local change point detection test is conducted to test the hypothesis of local homogeneity provided that the null hypothesis has not been rejected over Ik−1. The selected interval Iˆ corresponds to the longest accepted interval Ikˆ, such that: Tk≤ξk,k≤kˆandTkˆ+1>ξkˆ+1, (A.1)where Tk is the test statistic of the local change point test at step k, ξk is its corresponding critical value (the sections below describe the local change point test and the derivation of its critical values in greater detail). Equation (A.1) indicates that a change point is detected over interval Ikˆ+1, i.e. parameter homogeneity of the interval Ikˆ+1 is rejected and extending the interval to Ikˆ+1 will introduce significant bias. As a last step, the longest accepted interval is Iˆ=Ikˆ, resulting in the adaptive optimal estimator θˆ=θˆIkˆ for the fixed time point t0. In summary, the procedure for a fixed time point t0 is provided as follows: Start with the smallest interval, Iˆ=I0, θˆI0=θ̃I0. For k=1, we test the interval I1 for the local homogeneity assumption. Select intervals I2,I1, and J1=I1/I0. If T1≤ξ1, θ̃I1 is accepted then θˆI1=θ̃I1. Otherwise, if θˆI1=θˆI0, we accept the parameter estimator from the smallest interval as the optimal estimator for t0. For k≥2, select intervals Ik+1,Ik, and Jk=Ik/Ik−1. θ̃Ik is accepted and θˆIk=θ̃Ik if Tk≤ξk and θ̃Ik−1 has not been rejected. Otherwise, θˆIk=θˆIk−1, where θˆIk is accepted after k steps. The final estimate is Iˆ=Ikˆ and θˆ=θˆIkˆ. Local change point detection test The local parametric approach (LPA) crucially relies on the sequencing test of local time-homogeneity to search for an interval of homogeneity among the considered intervals Ik(k=0,1,…,K) at a fixed time point t0. Here, we follow Härdle et al. (2015) and Čížek et al. (2009) and adopt the local change point detection test, in which the null hypothesis of parameter homogeneity for the intervals up to Ik is tested. The alternative hypothesis is that a change point within interval Ik exists. Assuming that the homogeneity assumption of interval Ik−1 has not been rejected, the test statistic for testing possible change points in interval Ik is defined via the corresponding fitted log-likelihood L(y;I,θ) by: Tk=supτ∈Jk{LAk,τ(y,Ak,τ,θ̃Ak,τ)+LBk,τ(y,Bk,τ,θ̃Bk,τ)−LIk+1(y,Ik+1,θ̃Ik+1)}, (A.2)where Jk=Ik/Ik−1, Ak,τ=[t0−nk+1,τ], and Bk,τ=(τ,t0] represent two parts of observations in interval Ik+1. As you can see from equation (A.2), the test statistic is defined as the supremum of the log-likelihood ratio statistics over τ∈Jk because we do not know where the change-point location τ is. Figure A1 visualises the construction of the test statistic. Suppose that at a fixed time point t0, the parameter homogeneity within Ik−1 has not been rejected. To test if the homogeneity of the interval should be extended to Ik, we examine all possible change-point τ within the data extension Jk=Ik/Ik−1 by calculating the supremum of two log-likelihood values based on the intervals Ak,τ and Bk,τ for any τ∈Jk and then compare it to the log-likelihood value estimated over Ik+1. We compare the test statistic Tk with the corresponding critical values ξk and reject the null hypothesis of parameter homogeneity if Tk>ξk. Fig. A1. View largeDownload slide Graphical illustration of the construction of test statistics Tk. Note: The dotted [t0−nk+1,τ] and solid line (τ,t0] represent Ak,τ and Bk,τ, respectively. Fig. A1. View largeDownload slide Graphical illustration of the construction of test statistics Tk. Note: The dotted [t0−nk+1,τ] and solid line (τ,t0] represent Ak,τ and Bk,τ, respectively. Calculation of critical values ξk Since the true distribution of the test statistic is unknown, the critical values have to be determined by simulation using the general approach of testing theory: to provide a prescribed performance of the procedure under the null hypothesis (Čížek et al., 2009; Chen and Niu, 2014; Härdle et al., 2015). To be specific, we simulate 1,000 global homogeneous processes, i.e. a linear time trend model with constant parameters θ⁎ in equation (1). The simulated data ensure homogeneity for all of the considered intervals. To assess the quality of approximating the true (unknown) process over an interval I, Čížek et al. (2009) introduce the small modelling bias (SMB) condition, i.e. for some θ∈Θ, the distance between the true and the estimated model parameter is bounded by a small constant with a high probability. In the case of a quasi-MLE estimation with loss functions ( L(I,θ̃I,θ)=|L(I,θ̃I)−L(I,θ)|), the risk in an estimated local constant model (under SMB) differs from the risk in the true constant model satisfying the following condition: Eθ⁎|L(I,θ̃I)−L(I,θ⁎)|r≤Nr(θ⁎), (A.3)where r refers to the power of the loss and Nr(θ⁎) is the parametric risk bound. Given the assumption of parameter homogeneity, the largest considered interval IK should be achieved, over which the estimation loss of the ML estimator fulfills the risk bound equation (A.3). However, if the procedure stops earlier at Ik with k<K, instead of θ̃IK we select the adaptive estimator θˆIk=θ̃Ik, which can be regarded as a ‘false alarm’ (Čížek et al., 2009). The resulting loss due to such a false alarm is defined by LIK=L(IK,θ̃IK)−L(IK,θˆIk) and the corresponding risk bound given by equation (A.3) due to the adaptive estimation changes to: Eθ⁎|L(IK,θ̃IK)−L(IK,θˆIk)|r≤ρNr(θ⁎), (A.4)where ρ corresponds to the significance level. This condition in equation (A.4) ensures that the loss resulting from a ‘false alarm’ is the ρ-fraction of the loss of the estimator θ̃IK when the largest considered interval IK would have been achieved. To ensure a small probability of such a false alarm, Čížek et al. (2009) suggest the selection of minimal critical values. Similarly, at each step of the adaptive procedure, the estimate θˆIk after k steps should satisfy: Eθ⁎|L(Ik,θ̃Ik)−L(Ik,θˆIk)|r≤ρkNr(θ⁎),k=1,…,K, (A.5)where ρk=ρk/K and Nr(θ⁎)=maxk|L(I,θ̃Ik)−L(I,θ⁎)|r. Čížek et al. (2009) show that large values of ρ lead to smaller critical values. Given that the sequential testing procedure is used in a local change point test, the critical values are computed through the following steps (Härdle et al., 2016): Step 1. Consider ξ1 and let ξ2=ξ3=…=ξK=∞. We determine the estimates θˆIk(ξ1) and then the value ξ1 is selected as the minimal one for which supEθ⁎|L(Ik,θ̃Ik)−L(Ik,θˆIk(ξ1))|r≤ρNr(θ⁎)K,k=2,…,K. (A.6) Step k. Given that ξ1,ξ2,…,ξl−1 is fixed from the previous steps, and let ξl=…=ξK=∞. With estimate θˆIk(ξ1,…,ξl) for k=l+1,…,K, we choose ξl as the minimal value satisfying supEθ⁎|L(Ik,θ̃Ik)−L(Ik,θˆIk(ξ1,…,ξl))|r≤ρkNr(θ⁎)K,k=l+1,…,K. (A.7) Appendix 2. Hyper-parameters specification and robustness analysis According to Appendix 1, the calibration of risk bounds and critical values ξk depends on the specification of the hyper-parameters r and ρ. It has been shown that higher values of r lead to the acceptance of longer intervals of homogeneity and thus a higher modelling bias. An increase in ρ, in general, leads to a decrease in critical values ξk. The robustness and sensitivity of the empirical results to different hype-parameters will be discussed below. Second, critical values also depend on the ‘true’ parameters θ⁎used in the Monte Carlo simulation. Fortunately, previous studies document that the results are, in general, robust with respect to the selection of hypothetical parameters θ⁎ (Chen and Niu, 2014; Härdle et al., 2016). In our application, θ⁎ is specified as parameter estimates over the longest interval (i.e. 40 years starting in 1955). In the empirical application, the simulated risk bounds Nr(θ⁎) for eight representative crop-county combinations are presented in Table A1. We find risk bounds for different crop yields to be rather similar. This similarity might be due to the fact that the estimated θ⁎ based on a 40-year sample do not vary much across crops and counties, which is in line with the finding in Tolhurst and Ker (2015). As expected, larger values of the risk power parameter r lead to larger values for risk bounds. Table A1. Simulated risk bound Nr(θ⁎) Crop State County r=0.5 (baseline) r=1 Wheat Kansas Ellis 1.214 1.771 Texas Coryell 1.201 1.706 Corn Illinois Mercer 1.238 1.819 Iowa Story 1.313 2.030 Soybean Illinois Henry 1.275 1.918 Iowa Cedar 1.224 1.796 Cotton Georgia Dooly 1.217 1.751 Mississippi Coahoma 1.221 1.760 Crop State County r=0.5 (baseline) r=1 Wheat Kansas Ellis 1.214 1.771 Texas Coryell 1.201 1.706 Corn Illinois Mercer 1.238 1.819 Iowa Story 1.313 2.030 Soybean Illinois Henry 1.275 1.918 Iowa Cedar 1.224 1.796 Cotton Georgia Dooly 1.217 1.751 Mississippi Coahoma 1.221 1.760 With simulated risk bounds at hand, critical values can be derived from equations (A.6) and (A.7). These critical values for selected representative counties are displayed in Figure A2 for different significance levels ρ. For example, consider the two representative examples: wheat in Ellis County, Kansas and corn in Story County, Iowa. As expected, critical values decrease with the length of data intervals because the variance of the parameter estimate for short intervals is larger than that for longer intervals. The effect of parameter r on the critical values is the same as it is on the risk bounds, i.e. an increase in r leads to an increase in critical values, particularly for shorter intervals. For longer intervals, differences in critical values with respect to the choice of r are moderate. Figure A2 further shows that decreasing ρ generally results in an increase in critical values, especially for longer intervals.6 Again, the increase is moderate. That is, critical values are relatively robust to the choice of hyper-parameters ρ and r. We conjecture that this also holds for the final parameter estimates of the crop yield model. This is consistent with the results reported in financial or meteorological applications (Chen et al., 2010; Härdle et al., 2016). Similar findings apply to all combinations of crops and counties (see Figure A2). Fig. A2. View largeDownload slide Simulated critical values for different values of parameters r and ρ. Fig. A2. View largeDownload slide Simulated critical values for different values of parameters r and ρ. Fig. A3. View largeDownload slide Adaptively estimated trend βt (left) and volatility σt (right) and time intervals with alternative hyper-parameters. Note: Optimal intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. Fig. A3. View largeDownload slide Adaptively estimated trend βt (left) and volatility σt (right) and time intervals with alternative hyper-parameters. Note: Optimal intervals from the LPA are shown in the grey bars and correspond to the Y-axis on the right side. Fig. A4. View largeDownload slide Fig. A4. View largeDownload slide Since optimal intervals rely on the calibration of critical values based on the assumed hype-parameters, we investigate the robustness of the estimation results with respect to alternative hype-parameters. Figure A3 presents the adaptive estimation results based on alternative hyper-parameters for wheat in Ellis County, Kansas. The results in Figure A3, in general, confirm the robustness of the adaptive technique and low sensitivity to the choice of hype-parameters. The results suggest that in the data-driven adaptive procedure, changes in the simulated critical values derived from different choices of hype-parameters are rather moderate compared to changes in the test statistics due to the break point or structural change in empirical observations. Thus, similar optimal intervals are determined by the local change point test. Author notes Review coordinated by Iain Fraser © Oxford University Press and Foundation for the European Review of Agricultural Economics 2017; all rights reserved. For permissions, please e-mail: journals.permissions@oup.com

European Review of Agricultural Economics – Oxford University Press

**Published: ** Apr 1, 2018

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