Permutation tests on returns to scale and common production frontiers in nonparametric models

Permutation techniques, where one recompute the test statistic over permutations of data, have a long history in statistics and have become increasingly useful as the availability of computational power has increased. Until now, no permutation tests for examining returns to scale assumptions, nor for test of common production possibility sets, when analysing productivity have been available. We develop three novel tests based on permutations of the observations. The first is a test for constant returns to scale. The other two are, respectively, tests for frontier differences and for whether the production possibility sets are nested. All tests are based on data envelopment analysis (DEA) estimates of efficiencies and are easily implementable. We show that our suggested permutations of the observations satisfy the necessary randomisation assumptions, and hereby that the sizes of the proposed tests are controlled. The advantages of permutation tests are that they are reliable even for relatively small samples and their size can generally be controlled upwards. We further add a lower bound showing that the proposed tests are very close to being exact. Finally, we show that our tests are consistent and illustrate the rate of convergence in simulation studies.