The Skewness Issue in Stochastic Frontiers Models: Fact or Fiction?
Skewness plays an important role in the stochastic frontier model. Since the model was introduced by Aigner et al. (J. Econometric 6:21–37, 1977), Meeusen and van den Broeck (Int. Econ. Rev. 18:435–444, 1997), and Battese and Cora (Aust. J. Agr. Econ. 21:169–179, 1977), researchers have often found that the residuals estimated from these models displayed skewness in the wrong direction. In such cases applied researchers were faced with two main and often overlapping alternatives, either respecifying the model or obtaining a new sample, neither of which are particularly appealing due to inferential problems introduced by such data-mining approaches. Recently, Simar and Wilson (Econometric Rev. 29:62–98, 2010) developed a bootstrap procedure to address the skewness problem in finite samples. Their findings point to the latter alternative as potentially the more appropriate-increase the sample size. That is, the skewness problem is a finite sample one and it often arises in finite samples from a data generating process based on the correct skewness. Thus the researcher should first attempt to increase the sample size instead of changing the model specification if she finds the “wrong” skewness in her empirical analyses. In this chapter we consider an alternative explanation to the “wrong” skewness problem and offer a new solution in cases where this is not a mere finite sample fiction but also a fact. We utilize the Qian and Sickles (Stochastic Frontiers with Bounded Inefficiency, Rice University, Mimeo, 2008) model in which an upper bound to inefficiencies or a lower bound to efficiencies is specified based on a number of alternative one-sided bounded inefficiency distributions. We consider one of the set of specifications considered by Qian and Sickles (Stochastic Frontiers with Bounded Inefficiency, Rice University, Mimeo, 2008) wherein inefficiencies are assumed to be doubly-truncated normal. This allows the least square residuals to display skewness in both directions and nests the standard half-normal and truncated-normal inefficiency models. We show and formally prove that finding incorrect skewness does not necessarily indicate that the stochastic frontier model is misspecified in general. Misspecification instead may arise when the researcher considers the wrong distribution for the bounded inefficiency process. Of course if the canonical stochastic frontier model is the proper specification the residuals still may have the incorrect skew in finite samples but this problem goes away as sample size increases. This point was originally made in Waldman (Estimation in Economic Frontier Functions, Unpublished manuscript, University of North Carolina, Chapel Hill, 1977) and Olson et al. (J. Econometric. 13:67–82, 1980). We also conduct a limited set of Monte Carlo experiments that confirm our general findings. We show that “wrong” skewness can be a large sample issue. There is nothing inherently misspecified about the model were this to be found in large samples if one were to consider the bounded inefficiency approach. In this way the “wrong” skewness, while problematic in standard models, can become a property of samples drawn from distributions of bounded inefficiencies.
KeywordsFinite Sample Stochastic Frontier Technical Inefficiency Stochastic Frontier Model Negative Skewness
The authors would like to thank the participants of the Workshop on Exploring Research Frontiers in Contemporary Statistics and Econometrics in Honor of Leopold Simar, Institute of Statistics of the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, May 14–15, 2009, the 2009 XI European Workshop on Efficiency and Productivity Analysis, Pisa, Italy, and the 2009 Rice University Econometrics Workshop, as well as two anonymous referees and Co-Editor Paul Wilson for comments and criticisms that substantially strengthened this chapter. The usual caveat applies.
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