Measuring Firm Performance using Nonparametric Quantile-type Distances

When faced with multiple inputs X ∈ Rp + and outputs Y ∈ Rq +, traditional quantile regression of Y conditional on X = x for measuring economic efficiency in the output (input) direction is thwarted by the absence of a natural ordering of Euclidean space for dimensions q (p) greater than one. Daouia and Simar (2007) used nonstandard conditional quantiles to address this problem, conditioning on Y ≥ y (X ≤ x) in the output (input) orientation, but the resulting quantiles depend on the a priori chosen direction. This paper uses a dimensionless transformation of the (p + q)-dimensional production process to develop an alternative formulation of distance from a realization of (X, Y ) to the efficient support boundary, motivating a new, unconditional quantile frontier lying inside the joint support of (X, Y ), but near the full, efficient frontier. The interpretation is analogous to univariate quantiles and corrects some of the dis- appointing properties of the conditional quantile-based approach. By contrast with the latter, our approach determines a unique partial-quantile frontier independent of the chosen orientation (input, output, hyperbolic or directional distance). We prove that both the resulting efficiency score and its estimator share desirable monotonic- ity properties. Simple arguments from extreme-value theory are used to derive the asymptotic distributional properties of the corresponding empirical efficiency scores (both full and partial). The usefulness of the quantile-type estimator is shown from an infinitesimal and global robustness theory viewpoints via a comparison with the previous conditional quantile-based approach. A diagnostic tool is developed to find the appropriate quantile-order; in the literature to date, this trimming order has been fixed a priori. The methodology is used to analyze the performance of U.S. credit unions, where outliers are likely to affect traditional approaches.