Testing the "separability" condition in two-stage nonparametric models of production

Simar and Wilson (J. Econometrics, 2007) provided a statistical model that can rationalize two-stage estimation of technical efficiency in nonparametric settings. Two-stage estimation has been widely used, but requires a strong assumption: the second-stage environmental variables cannot affect the support of the input and output variables in the first stage. In this paper, we provide a fully nonparametric test of this assumption. The test relies on new central limit theorem (CLT) results for unconditional efficiency estimators developed by Kneip et al. (Econometric Theory, 2015a) and new CLTs for conditional efficiency estimators developed in this paper. The test can be implemented relying on either asymptotic normality of the test statistics or using bootstrap methods to obtain critical values. Our simulation results indicate that our tests perform well both in terms of size and power. We present a real-world empirical example by updating the analysis performed by Aly et al. (R. E. Stat., 1990) on U.S. commercial banks; our tests easily reject the assumption required for two-stage estimation, calling into question results that appear in hundreds of papers that have been published in recent years.