The Conditional Mode in Parametric Frontier Models

We survey formulations of the conditional mode estimator for technical inefficiency in parametric stochastic frontier models with normal errors and introduce new formulations for models with Laplace errors. We prove the conditional mode estimator converges pointwise to the true inefficiency value as the noise variance goes to zero. We also prove that the conditional mode estimator in the normal-exponential model achieves near-minimax optimality. Our minimax theorem implies that the worst-case risk occurs when many firms are nearly efficient, and the conditional mode estimator minimizes estimation risk in this case by estimating these small inefficiency firms as efficient. Unlike the conditional expectation estimator, the conditional mode estimator produces multiple firms with inefficiency estimates exactly equal to zero, suggesting a rule for selecting a subset of maximally efficient firms. Our simulation results show that this “zero-mode subset” has reasonably high probability of containing the most efficient firm, particularly when inefficiency is exponentially distributed. The rule is easy to apply and interpret for practitioners. We include an empirical example demonstrating the merits of the conditional mode estimator.