Inference in Dynamic, Nonparametric Models of Production: Central Limit Theorems for Malmquist Indices

The Malmquist index gives a measure of productivity in dynamic settings and has been widely applied in empirical work. The index is typically estimated using envel- opment estimators, particularly data envelopment analysis (DEA) estimators. Until now, inference about productivity change measured by Malmquist indices has been problematic, including both inference regarding productivity change experienced by particular firms as well as mean productivity change. This paper establishes properties of a DEA-type estimator of distance to the conical hull of a variable returns to scale production frontier. In addition, properties of DEA estimators of Malmquist indices for individual producers are derived as well as properties of geometric means of these estimators. The latter requires new central limit theorem results, extending the work of Kneip, Simar, and Wilson (2015, Econometric Theory 31, 394–422). Simulation results are provided to give applied researchers an idea of how well inference may work in practice in finite samples. Our results extend easily to other productivity indices, including the Luenberger and Hicks–Moorsteen indices.