Estimating production uncertainty in stochastic frontier production function models

One of the main purposes of the frontier literature is to estimate inefficiency. Given this objective, it is unfortunate that the issue of estimating "firm-specific'' inefficiency in cross sectional context has not received much attention. To estimate firm-specific (technical) inefficiency, the standard procedure is to use the mean of the inefficiency term conditional on the entire composed error as suggested by Jondrow, Lovell, Materov and Schmidt (1982). This conditional mean could be viewed as the average loss of output (return). It is also quite natural to consider the conditional variance which could provide a measure of production uncertainty or risk. Once we have the conditional mean and variance, we can report standard errors and construct confidence intervals for firm level technical inefficiency. Moreover, we can also perform hypothesis tests. We postulate that when a firm attempts to move towards the frontier it not only increases its efficiency, but it also reduces its production uncertainty and this will lead to shorter confidence intervals. Analytical expressions for production uncertainty under different distributional assumptions are provided, and it is shown that the technical inefficiency as defined by Jondrow et al. (1982) and the production uncertainty are monotonic functions of the entire composed error term. It is very interesting to note that this monotonicity result is valid under different distributional assumptions of the inefficiency term. Furthermore, some alternative measures of production uncertainty are also proposed, and the concept of production uncertainty is generalized to the panel data models. Finally, our theoretical results are illustrated with an empirical example.