Conical FDH estimators for testing returns to scale and making inference about changes in productivity

Non parametric envelopment estimators are frequently used to estimate production sets, efficiency, and changes in productivity. In previous papers we provide asymptotic theory enabling inference about expected efficiency, extend these results to develop tests of (i) convexity of the production set and (ii) constant versus variable returns to scale when the production set is convex, and further extend the results to make inferences about expected changes in productivity measured by Malmquist productivity indices (MPIs), provided the production set is convex. However, convexity is a strong assumption and has been rejected in a number of empirical studies. This article extends our earlier work to fill a gap in the literature by developing asymptotic properties of a non parametric envelopment estimator of distance to the boundary of the cone spanned by a production set without requiring convexity. These new results are then further extended to make inferences about productivity change measured by MPIs and to test constant versus non constant returns to scale without requiring convexity of the production set. We revisit and extend the study of U.S. municipalities by O’Loughlin and Wilson, who rejected convexity and hence were unable to examine returns to scale or to estimate and make inferences about MPIs. Using the new methods, we (i) test and reject constant returns to scale and (ii) estimate and make inferences about MPIs without imposing convexity. We find evidence of significant increases in productivity among U.S. municipalities during 1997–2012.