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Abstract
The first essay, "On the Distribution of Estimated Technical Efficiency in Stochastic Frontier Models," considers a stochastic frontier model with error &egr; = ν – u, where ν is normal and u is half normal. We derive the distribution of the usual estimate of E(u | &egr;). We show that as the variance of ν approaches zero, E( u | &egr;) – u converges to zero, while as the variance of ν approaches infinity, E(u | &egr;) converges to E(u). We graph the density of E(u | &egr;) for intermediate cases. To show that E(u | &egr;) is a shrinkage of u towards its mean, we derive and graph the distribution of E(u | &egr;) conditional on u. We also consider the distribution of estimated inefficiency in the fixed-effects panel data setting.
The second essay, "Goodness of Fit Tests in Stochastic Frontier Models," discusses goodness of fit tests for the distribution of technical inefficiency in stochastic frontier models. If we maintain the hypothesis that the assumed normal distribution for statistical noise is correct, the assumed distribution for technical inefficiency is testable. We show that a goodness of fit test can be based on the distribution of estimated technical efficiency, or equivalently on the distribution of the composed error term. We consider both the Pearson chi-squared test and the Kolmogorov-Smirnov test. The bootstrap can be used to account for the effects of parameter estimation. Alternatively, for the Pearson test, we use existing results in the literature to account for the fact that estimated parameters are used to construct the actual and/or the expected cell counts. Finally, we provide simulation results to show the extent to which the tests are reliable in finite samples.
The third essay, "Testing Equality of Distribution for Two Correlated Variables," discusses how to test the null hypothesis that y 1, y2, ..., yn and x1, x2, ..., xn from a correlated paired sample of size n: (yi, xi), i = 1, 2, 3, ..., n, have the same distribution. We implement the Pearson chi-squared test, based on differences of frequencies in non-overlapping intervals (cells) that span the support of the variables, in a GMM setting. This procedure makes no assumption about the correlation between the two variables. We also suggest a novel bootstrapping procedure that enables us to generate asymptotically valid critical values for the Kolmogorov-Smirnov and Baumgartner-Weiss-Schindler tests.
