Exact tests on returns to scale and comparisons of production frontiers in nonparametric models

When benchmarking production units by non-parametric methods like data envelopment analysis (DEA), an assumption has to be made about the returns to scale of the underlying technology. Moreover, it is often also relevant to compare the frontiers across samples of producers. Until now, no exact tests for examining returns to scale assumptions in DEA, or for test of equality of frontiers, have been available. The few existing tests are based on asymptotic theory relying on large sample sizes, whereas situations with relatively small samples are often encountered in practical applications. In this paper we propose three novel tests based on permutations. The tests are easily implementable from the algorithms provided, and give exact significance probabilities as they are not based on asymptotic properties. The first of the proposed tests is a test for the hypothesis of constant returns to scale in DEA. The others are tests for general frontier differences and whether the production possibility sets are, in fact, nested. The theoretical advantages of permutation tests are that they are appropriate for small samples and have the correct size. Simulation studies show that the proposed tests do, indeed, have the correct size and furthermore higher power than the existing alternative tests based on asymptotic theory.