Advances on axiomatic nonparametric approaches to productivity and efficiency analysis

This dissertation contains several important methodological results on nonparametric productivity and efficiency analysis. It spans both theoretically motivated and practically challenging topics in the planning, implementation, and interpretation of efficiency analysis. The dissertation consists of five articles, all of which contribute to the subject from closely related perspectives. Firstly, we study the connection between nonparametric nonconvex efficiency analysis and isotonic regression, and propose a mathematical programming approach for estimating multivariate isotonic regression. We also examine the estimation of non-convex frontier technologies in the presence and absence of stochastic noise. Secondly, we propose an enumeration technique for estimating the nonparametric nonconvex efficient frontiers under different returns to scale assumptions. Furthermore, we present an application of efficiency analysis to explore the changes in value added for a sample of Portuguese secondary schools over a four-year time period. A modified Malmquist index is proposed to measure how the value added evolves over time. We also study the linear relationship between the inputs and outputs of a nonparametric efficiency problem. We consider transformations that can be applied to inputs and outputs without disrupting the efficiency results. We use the concept of dominating cones to analyze the transformed problems. We focus on the selection of variables, and explain how this can impact the dominating cone. A discussion on using linear transformation for dealing with the curse of dimensionality is provided in the dissertation. Lastly, we build a transformation based on the Fourier-Motzkin elimination method, which permits the derivation of a single variable optimization problem from which efficiency scores could be estimated. We build the connection between the constraints of the transformed problem and the supporting hyperplanes of the production possibility set, and show that this transformation generates all the efficient and weakly supporting hyperplanes. Moreover, we show that the proposed transformation does not generate redundant constraints. Extensive numerical comparisons are done to show the performance of the proposed algorithm.