Abstract
Several tests of model structure developed by Kneip et al. (J Bus Econ Stat 34:435–456, 2016) and Daraio et al. (Econ J 21:170–191, 2018) rely on comparing sample means of two different efficiency estimators, one appropriate under the conditions of the null hypothesis and the other appropriate under the conditions of the alternative hypothesis. These tests rely on central limit theorems developed by Kneip et al. (Econ Theory 31:394–422, 2015) and Daraio et al. (Econ J 21:170–191, 2018), but require that the original sample be split randomly into two independent subsamples. This introduces some ambiguity surrounding the sample-split, which may be determined by choice of a seed for a random number generator. We develop a method that eliminates much of this ambiguity by repeating the random splits a large number of times. We use a bootstrap algorithm to exploit the information from the multiple sample-splits. Our simulation results show that in many cases, eliminating this ambiguity results in tests with better size and power than tests that employ a single sample-split.
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Notes
For two vectors a and b of length n with ith elements ai and bi, c = a ∘ b is a vector of length n with ith element ci = aibi.
For an estimator \(\widehat{\theta }(x,y)\) of θ(x, y) converging at rate nκ, \(\widehat{\theta }(x,y)-\theta (x,y)={O}_{p}({n}^{-\kappa })\). In other words, the estimation error of \(\widehat{\theta }(x,y)\) is of order in probability n−κ. In such cases, estimation error becomes less in probabilistic terms as the sample size n increases, but how fast this happens depends on the magnitude of κ.
In addition, the results on consistency, limiting distributions and rates of convergence have been extended to hyperbolic versions of the FDH and VRS-DEA estimators by Wheelock and Wilson (2008) and Wilson (2011), and to directional-distance versions by Simar and Vanhems (2012) and Simar et al. (2012). For each type of estimator—FDH, VRS-DEA or CRS-DEA—the value of κ remains the same across the different orientations.
These results extend trivially to the output-oriented estimators. Wilson (2019) extends the results to the hyperbolic orientation.
Simar and Zelenyuk (2020) propose adding a bias correction to the sample variances in (3.3) and (3.4) to improve performance in small samples. We have not used this idea here, in order to facilitate comparison with previous simulation results appearing in Kneip et al. (2016) and Daraio et al. (2018). Our focus here is on the impact of multiple sample-splits, but in applications one can use the improved variance estimator without increasing computational burden.
In the test of convexity versus non-convexity, one applies the FDH estimator to the observations in \({{\mathcal{X}}}_{1,{n}_{1}}\) and the VRS-DEA estimator to the observations in \({{\mathcal{X}}}_{2,{n}_{2}}\). In the test of separability, one applies the conditional VRS-DEA (or conditional FDH) estimator to observations in \({{\mathcal{X}}}_{1,{n}_{1}}\) and the unconditional VRS-DEA (or unconditional FDH) estimator to the observations in \({{\mathcal{X}}}_{2,{n}_{2}}\). In the test of CRS to VRS outlined here, both estimators used to construct the test statistic have the same rate of convergence, and so κ = 2/(p + q). In the tests of convexity and separability, the two estimators used for each test have different rates of convergence under the null, and so the variances in the denominators of the statistics for these tests are divided by different powers of n, reflecting the different convergence rates of the estimators. See Kneip et al. (2016) and Daraio et al. (2018) for details.
Daraio et al. (2018) report results from experiments with n = 1000 in addition to n = 100 and 200. However, with 10 sample splits, and 1000 bootstrap replications, the computational burden for each experiment here is 10,010 times that of the experiments in Daraio et al. (2018). Moreover, with the separability test, a bandwidth parameter must be selected by cross-validation, which requires time of order O(n2), and this must be done 10,010 times. Consequently, we consider only n = 100, 200 for the separability test.
Note that while the statistic in (3.8) is a Kolmogorov–Smirnov statistic, the usual tables cannot be used to assess significance due to the dependence problem here.
Computation times for the convexity test are faster than those given here due to the fact that the FDH estimator involves lower computational burden than the VRS-DEA estimator. On the other hand, times for the separability test are slower than those for the RTS test due to the necessity of cross-validation to optimize bandwidths used by the conditional efficiency estimators.
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Acknowledgements
We are grateful to the Cyber Infrastructure Technology Integration group at Clemson University for operating the Palmetto Cluster used for simulations in this paper.
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Simar, L., Wilson, P.W. Hypothesis testing in nonparametric models of production using multiple sample splits. J Prod Anal 53, 287–303 (2020). https://doi.org/10.1007/s11123-020-00574-w
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DOI: https://doi.org/10.1007/s11123-020-00574-w