On Testing Equality of Distributions of Technical Efficiency Scores

Abstract

The challenge of the econometric problem in production efficiency analysis is that the efficiency scores to be analyzed are unobserved. Statistical properties have recently been discovered for a type of estimator popular in the literature, known as data envelopment analysis (DEA). This opens up a wide range of possibilities for well-grounded statistical inference about the true efficiency scores from their DEA estimates. In this paper we investigate the possibility of using existing tests for the equality of two distributions in such a context. Considering the statistical complications pertinent to our context, we consider several approaches to adapting the Li test to the context and explore their performance in terms of the size and power of the test in various Monte Carlo experiments. One of these approaches shows good performance for both the size and the power of the test, thus encouraging its use in empirical studies. We also present an empirical illustration analyzing the efficiency distributions of countries in the world, following up a recent study by Kumar and Russell (2002), and report very interesting results.Bootstrap, DEA, Kernel density estimation and tests,

Does Farm Size and Specialization Matter for Productive Efficiency? Results from Kansas

“…Nonparametric kernel density estimation techniques have become common in graphically illustrating various results in nonparametric production efficiency analysis (Henderson and Zelenyuk, 2007;Simar and Zelenyuk, 2006). Compared with histograms, kernel densities have the advantage of providing smoother density estimates and do not depend on the width and number of bins (Wand and Jones, 1995).…”

“…This method is useful in this study because no distributional assumptions were imposed on the efficiency scores across farms. When using kernel density estimation, Simar and Zelenyuk (2006) note that one has to take care of at least three things: the random variable whose density is to be estimated must have a bounded support, only the consistent estimate of the efficiency scores are used, and there is no violation of the continuity assumption needed to ensure consistency of the density estimation. In this paper, the Silverman reflection method is used to correct for the bounded support, bootstrap DEA is used to compute the consistent efficiency scores, and a Gaussian kernel density is estimated using the bias corrected efficiency scores.…”

Output complexity, environmental conditions, and the efficiency of municipalities

“…However, as demonstrated by Simar and Zelenyuk (2006), although controlling for the boundary effect is important in density estimation, the statistic based on the reflection method is "essentially the same as the original Li (1996) test, with the difference being a factor of √ 2 and the fact that the bandwidth used in estimation of the statistic is obtained from the data with reflection rather than the original data". Therefore, since the independency issue is not negligible, we should then follow Simar and Zelenyuk (2006), who provide a way of adapting the Li (1996) test to the order-m context via bootstrapping techniques to improve its performance. These authors provide consistent bootstrap estimates of the p-values of the Li (1996) test…”

“…However, order-m efficiency estimates are dependent in the statistical sense, since perturbations of observations lying on the estimated frontier will in most cases cause changes in efficiencies estimated for other observations. As indicated by Simar and Zelenyuk (2006), under these circumstances the Li (1996) test has to be modified in several ways.…”

“…Simar and Zelenyuk (2006) consider somewhat ad hoc methods to solve the discontinuity problem generated by the spurious probability mass at the unity-recall that, by construction, at least one observation will always be on the frontier, and in most circumstances the number will be quite large. We adopt one of their proposed methods (Algorithm II; see Simar and Zelenyuk, 2006), based on computing and bootstrapping the Li (1996) statistic using the sample of order-m estimates where those equal to unity are "smoothed" away from the boundary. We add a small noise, within, say, 5% of the empirical distribution ofα m s , disregarding those equal to the unity, but with an order of magnitude smaller than the noise of the estimation.…”