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A Flexible Parametric Family for the Modeling and Simulation of Yield Distributions

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A Flexible Parametric Family for the Modeling and Simulation of Yield Distributions Journal of Agricultural and Applied Economics, 42,2(May 2010):303–319 Ó 2010 Southern Agricultural Economics Association A Flexible Parametric Family for the Modeling and Simulation of Yield Distributions Octavio A. Ramirez, Tanya U. McDonald, and Carlos E. Carpio The distributions currently used to model and simulate crop yields are unable to accom- modate a substantial subset of the theoretically feasible mean-variance-skewness-kurtosis (MVSK) hyperspace. Because these first four central moments are key determinants of shape, the available distributions might not be capable of adequately modeling all yield distributions that could be encountered in practice. This study introduces a system of distributions that can span the entire MVSK space and assesses its potential to serve as a more comprehensive parametric crop yield model, improving the breadth of distributional choices available to researchers and the likelihood of formulating proper parametric models. Key Words: risk analysis, parametric methods, yield distributions, yield modeling and simulation, yield nonnormality JEL Classifications: C15, C16, C46, C63 Agricultural economists have long recognized Shonkwiler (1993), Ramirez, Moss, and Boggess that the choice of an appropriate probability (1994), Coble et al. (1996), and Ramirez (1997). distribution to represent crop yields is critical These authors have provided strong statistical for an accurate measurement of the risks as- evidence of nonnormality and heteroskedasticity sociated with crop production. Anderson in crop yield distributions—specifically, the ex- (1974) first emphasized the importance of ac- istence of kurtosis and negative skewness in counting for nonnormality in crop yield distri- a variety of cases. The possibility of positive butions for the purpose of economic risk anal- skewness has been documented as well (Ramı´rez, ysis. Since then, numerous authors have Misra, and Field, 2003). focused on this issue, including Gallagher The three general statistical procedures that (1987), Nelson and Preckel (1989), Moss and have been used for the modeling and simulating of crop yield distributions are the parametric, the nonparametric, and semiparametric. All have Octavio A. Ramirez is professor and head, Department of Agricultural and Applied Economics, University of distinct advantages and disadvantages. The para- Georgia, Athens, GA. Tanya U. McDonald is research metric procedures assume that the data-generat- specialist, Department of Agricultural Economics and ing process can be adequately represented by a Agricultural Business, New Mexico State University, particular parametric probability distribution Las Cruces, NM. Carlos E. Carpio is assistant pro- fessor, Department of Applied Economics and Statis- function. For this reason, the main disadvantage tics, Clemson University, Clemson, SC. of this method is the potential error resulting This study was supported by the National Research from assuming a probability distribution that is Initiative of the Cooperative State Research, Education and not flexible enough to properly represent the Extension Service, USDA, Grant #2004-35400-14194, and by the Agricultural Experiment Stations of the Uni- data. The main advantage of this method is that, versity of Georgia and New Mexico State University. if the assumed distribution can adequately 304 Journal of Agricultural and Applied Economics, May 2010 represent the data-generating process, it performs Coble (2003) proposed a semiparametric model relatively well even in small sample applications. based on the Normal and Beta densities. Em- This is important because crop yield datasets do pirical comparisons of leading parametric not often span long periods. Distributions that models have been recently attempted (Norwood, have been used as a basis for parametric pro- Roberts, and Lusk, 2004). Despite such ad- cedures include the Normal, the Log-normal, the vances, the potential of different probability Logistic, the Weibull, the Beta, the Gamma, and density functions (pdfs) to serve as suitable yield the Inverse Hyperbolic Sine. and price distribution models has not been The nonparametric approaches also have assessed in the context of a rigorous theoretical strengths and weaknesses. Because these framework, and this critical research methods methods are free of a functional form assump- issue remains unsettled. tion, they are generally more flexible. However, According to basic statistical theory (Mood, they can be inefficient relative to parametric Graybill, and Boes, 1974), the first four central procedures under certain conditions. Specifi- moments of a pdf are the main descriptors of its cally, according to Ker and Coble (2003), ‘‘it is shape. Although there are other means for possible, perhaps likely, for very small samples characterizing and comparing distributions, this such as those corresponding to farm-level yield suggests that the flexibility of a pdf to accom- data, that an incorrect parametric form—say modate a variety of empirical shapes (i.e., data- Normal—is more efficient than the standard generating processes) should be closely related to nonparametric kernel estimator.’’ Other authors the Mean-Variance-Skewness-Kurtosis (MVSK) cite theoretical complexity and intensive com- combinations that are allowed by it. putational requirements as another disadvantage Unfortunately, in this regard, all of the pdfs of nonparametric procedures (Yatchew, 1998). that have been used as a basis for parametric Semiparametric methods show significant po- methods suffer from two significant range re- tential because they encapsulate the advantages strictions: (1) they can only accommodate lim- of the parametric and nonparametric approaches ited subsets of all of the theoretically feasible while mitigating their disadvantages (Ker and Skewness-Kurtosis (SK) combinations and, Coble, 2003; Norwood, Roberts, and Lusk, therefore, might only be capable of adequately 2004). Because the semiparametric approach is modeling underlying data distributions where based on nonparametrically ‘‘correcting’’ a par- third and fourth central moments are within that ticular parametric estimate, the availability of subset; and (2) their variance, skewness, and more flexible distributions such as the ones ad- kurtosis are controlled by only two parameters vanced in this study should improve the potential and, therefore, are arbitrarily interrelated (i.e., efficiency of semiparametric procedures as well. only two of these three key moments are free to Extensive efforts have been devoted to the vary independently). Although the expanded issue of the most appropriate probability dis- form of the SU advanced by Ramı´rez, Misra, and tribution to be used as a basis for parametric or Field (2003) allows for any mean and variance semiparametric methods. Gallagher (1987) used to be freely associated with the SK values per- the well-known Gamma density as a parametric mitted by the original parameterization of this model for soybean yields. Nelson and Preckel family, it is far from encompassing all theoret- (1989) proposed a conditional Beta distribution ically feasible SK combinations. to model corn yields. Taylor (1990) estimated This research contributes to the yield and multivariate nonnormal densities through a con- price distribution modeling literature through ditional distribution approach based on the the following measures: Hyperbolic Tangent transformation. Ramirez 1. Introduce a family of distributions (SB) that (1997) introduced a modified Inverse Hyper- perfectly complements the SU in its coverage bolic Sine transformation (also known in the of the SK space, and that spans significant statistics literature as the SU Distribution) as regions of the space not covered by any other a possible multivariate nonnormal and hetero- distribution that has been used for the mod- skedastic crop yield distribution model. Ker and eling and simulation of crop yields. Ramirez, McDonald, and Carpio: Flexible Yield Distribution Models 305 2. Derive expanded forms of this SB family and outline a reparameterization technique that of the Beta distribution, which are analogous expands any probability distribution by two to Ramı´rez, Misra, and Field’s (2003) repar- parameters that specifically and uniquely con- ameterization of the SU, so that they, too, can trol the mean and variance without affecting the allow for any mean and variance to be freely range of skewness and kurtosis values that can associated with their possible SK values. be accommodated. The expanded distribution 3. Assess the importance of MVSK coverage in obtained through this reparameterization can determining a model’s flexibility to adequately therefore model any conceivable mean and represent a wide range of distributional shapes. variance in conjunction with the set of SK combinations allowed by the original distribu- The SU-SB System tion. For the purposes of this study, the tech- nique is first applied to the SU and SB families. Johnson (1949) introduced the SU and the SB This yields a system that can model all theo- families of distributions and showed that, to- retically feasible MVSK combinations. The gether, they span the entire SK space. Figure 1 reparameterization begins with the original is constructed on the basis of the formulas for two-parameter families (Johnson, 1949): the skewness and kurtosis of the SU and SB À1 distributions, which were also first derived by (1) Z 5 g 1 d  sinh Y for the SU distribution Johnson (1949). Specifically, SK pairs are (2) Z 5 g
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