A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models

This paper derives an analytic closed-form formula for the cumulative distribution function (cdf) of the composite error of the stochastic frontier analysis (SFA) model. Since the presence of a cdf is frequently encountered in the likelihood-based analysis with limited-dependent and qualitative variables as elegantly shown in the classic book of Maddala (Limited-dependent and qualitative variables in econometrics. Cambridge University Press, Cambridge, 1983), the proposed methodology is useful in the framework of the stochastic frontier analysis. We apply the formula to the maximum likelihood estimation of the SFA models with a censored dependent variable. The simulations show that the finite sample performance of the maximum likelihood estimator of the censored SFA model is very promising. A simple empirical example on the modeling of reservation wage in Taiwan is illustrated as a potential application of the censored SFA.