The ‘wrong skewness’ problem: a re-specification of stochastic frontiers

Abstract

In this paper, we study the so-called "wrong skewness" anomaly in Stochastic Frontiers (SF), which consists in the observed difference between the expected and estimated sign of the asymmetry of the composite error. We propose a more general and flexible specification of the SF model, introducing dependence between the two error components and asymmetry (positive or negative) of the random error. This re-specification allows us to decompose the third moment of the composite error in three components, namely: i) the asymmetry of the inefficiency term; ii) the asymmetry of the random error; and iii) the structure of dependence between the error components. This decomposition suggests that the "wrong skewness"anomaly is an ill-posed problem, because we cannot establish ex ante the expected sign of the asymmetry of the composite error. We report a relevant special case that allows us to estimate the three components of the asymmetry of the composite error and, consequently, to interpret the estimated sign.We present two empirical applications. In the first dataset, where the classic SF displays wrong skewness, estimation of our model rejects the dependence hypothesis, but accepts the asymmetry of the random error, thus justifying the sign of the skewness of the composite error.In the second dataset, where the classic SF does not display any anomaly, estimation of our model provides evidence of the presence of both dependence between the error components and asymmetry of the random error.

Stochastic Frontier Analysis: Foundations and Advances I

“…In this case the null hypothesis of interest is H 0 : σ 2 u = 0 against H 1 : σ 2 u > 0. 20 The direct way to test the H 0 is through a likelihood ratio test, keeping in mind that the unrestricted model is the assumed SFM and the restricted model is the linear regression model (or more specifically the normal regression model). There is a problem with implementation of this test however.…”

Estimation of stochastic production functions: the state of the art

“…The common feature of all these strategies is that the phenomenon of wrong skewness was approached from an inefficiency error term point of view. According to Bonanno-De Giovanni-Domma [2017], this only partially addresses the problem because the wrong skewness anomaly is a direct consequence of all the assumptions underlying the stochastic frontier model specification. Therefore, Bonanno-De Giovanni-Domma [2017] describe a more general framework, where they relaxed the hypothesis of symmetry for i ν , of positive skewness for i u , and of independence between i u and i ν , and extended Greene's [1990] third central moment of the composite error as:…”

“…/18/ From Equation /18/, the sign of the asymmetry of i u , i ν , and the dependence between i u and i ν affect the expected sign of the asymmetry of the composite error. In Bonanno-De Giovanni-Domma's [2017] model, the dependence structure is modelled with a copula function that allows them to specify the joint distribution with different marginal probability density functions. The copula joint distribution function,   , , U V i i f u ν has the following standard representation:…”

“…Applying the copulas technique, Bonanno-De Giovanni-Domma's [2017] present a new specification for the composite error, which gives rise to a semi-closed expression for the composite error of a stochastic production frontier through an exponential random variable…”

Specification Testing of Production in a Stochastic Frontier Model

“…Another merit of the method of [16] is that only local distributional assumptions on U are needed, although symmetry is still necessary for V. Nonetheless, it should be realized that the methods discussed above would not be necessary if the hypothetical parametric model was satisfied. Studying the "wrong skewness phenomenon" in stochastic frontiers (SF), Bonanno et al [17] proposed a more general and flexible specification of the SF model by introducing dependences between the two error components and asymmetry of the random error.…”