Estimation of the threshold stochastic frontier model in the presence of an endogenous sample split variable

Abstract

Heterogeneity among firms has been an important issue in studying firms’ technical efficiencies. If firms do not randomly fall into different groups with different technologies but by self-selection, statistically it implies the data are subject to the sample selection bias. In this paper, we generalize the stochastic frontier (SF) model to accommodate heterogeneous technologies among firms by considering the threshold SF model with an endogenous threshold variable. We discuss the econometric techniques appropriate for the threshold SF model with panel data. To determine the optimal number of regimes, we use modified the model selection criteria of Gonzalo and Pitarakis (J Econom 110(2):319–352, 2002) and investigate their finite sample performance by some Monte Carlo experiments. Finally, we also demonstrate our approach by an empirical example.

Appendix

Given that υ _it and e _it follows a bivariate normal distribution with correlation coefficient ρ _j under regime j, we may written υ _it as

$$ \begin{aligned} \upsilon_{it} & = \kappa_{j} e_{it} + \xi_{it} , \\ & = \kappa_{j} {\text{E}}\left( {e_{it} |\gamma_{j - 1} - \pi^{\text{T}} z_{it} < e_{it} \le \gamma_{j} - \pi^{\text{T}} z_{it} } \right) + \varepsilon_{it} , \\ \end{aligned} $$

where $\varepsilon_{it} = \kappa_{j} \left[ {e_{it} - {\text{E}}\left( {e_{it} |\gamma_{j - 1} - \pi^{\text{T}} z_{it} < e_{it} \le \gamma_{j} - \pi^{\text{T}} z_{it} } \right)} \right] + \xi {}_{it}$ and j = 1, …, m + 1. Then it follows from p. 156 of Johnson et al. (1994) that

$$ {\text{E}}\left( {e_{it} |\gamma_{j - 1} - \pi^{\text{T}} z_{it} < e_{it} \le \gamma_{j} - \pi^{\text{T}} z_{it} } \right) = \left\{ {\begin{array}{*{20}l} { - \frac{{\phi (\gamma_{1} - \pi^{\text{T}} z_{it} )}}{{\Upphi (\gamma_{1} - \pi^{\text{T}} z_{it} )}},\quad {\text{if}}\;j = 1;} \\ {\frac{{\phi (\gamma_{j - 1} - \pi^{\text{T}} z_{it} ) - \phi (\gamma_{j} - \pi^{T} z_{it} )}}{{\Upphi (\gamma_{j} - \pi^{\text{T}} z_{it} ) - \Upphi (\gamma_{j - 1} - \pi^{T} z_{it} )}},\quad {\text{if}}\;j = 2, \ldots ,m;} \\ { - \frac{{\phi (\gamma_{m} - \pi^{\text{T}} z_{it} )}}{{\Upphi (\gamma_{m} - \pi^{\text{T}} z_{it} )}},\quad {\text{if}}\;j = m + 1.} \\ \end{array} } \right. \, $$