Tests \(H_0\): no technical inefficiency, i.e. \(\gamma = 0\) in the Battese-Corra parameterization \(\gamma = \sigma_u^2/(\sigma_u^2 + \sigma_v^2)\), following Section 3 of Coelli (1995).
inefficiency_test(object, test = c("lr_1sided", "lr", "wald", "m3t"),
level = 0.05)A data frame with one row per test: test, statistic,
null (the reference distribution), p.value and reject.
The log-likelihoods under both hypotheses are attached as attributes.
An "sfareg" fit from sfm, with
model_name = "NHN" or "NTN".
Which statistics to compute; any subset, all by default.
Significance level used for the reject column.
The two tests most often reported are the two with the wrong size. \(H_0\) puts \(\gamma\) on the boundary of the parameter space, so the usual asymptotics fail. The likelihood ratio statistic is not \(\chi^2(1)\) but the Gourieroux, Holly and Monfort (1982) mixture \(\tfrac12\chi^2(0) + \tfrac12\chi^2(1)\), whose 5% critical value is 2.71 rather than 3.84; using the \(\chi^2(1)\) value makes the naive LR test conservative. The Wald ratio \(\hat\gamma/s_{\hat\gamma}\) is worse: it is not asymptotically standard normal at the boundary at all.
Measured in this package on 400 replications per cell, half-normal data with \(\sigma_u = 0\) and a nominal 5% level:
| \(n\) | LR one-sided | LR naive | Wald | M3T |
| 100 | 0.060 | 0.030 | 0.230 | 0.055 |
| 200 | 0.060 | 0.033 | 0.203 | 0.050 |
| 400 | 0.055 | 0.033 | 0.217 | 0.045 |
| 800 | 0.060 | 0.035 | 0.182 | 0.062 |
The one-sided LR test and the third-moment test hold their size; the naive LR test rejects about half as often as it should; the Wald test rejects three to four times too often and does not improve with \(n\). Under the alternative (\(\sigma_u = \sigma_v = 1\)) the one-sided LR test has the better power of the two valid tests at every sample size, which is Coelli's recommendation. The Wald column's apparently high power is not comparable, because its size is wrong.
For model_name = "NTN" the null restricts both \(\gamma\) and
\(\mu\), and the mixture becomes
\(\tfrac14\chi^2(0) + \tfrac12\chi^2(1) + \tfrac14\chi^2(2)\).
The "m3t" statistic is the same one skewness_test
reports as test = "coelli"; it is repeated here so the four tests can
be compared in a single table.
Coelli, T. (1995). Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis. Journal of Productivity Analysis 6, 247--268.
Gourieroux, C., Holly, A. and Monfort, A. (1982). Likelihood ratio test, Wald test, and Kuhn-Tucker test in linear models with inequality constraints on the regression parameters. Econometrica 50, 63--80.
skewness_test, spec_test, sfm
d <- data_gen_cs(N = 300, rand = 7, sig_u = 1, sig_v = 0.5, cons = 0.5,
beta1 = 0.5, beta2 = 0.5, a = 5, mu = 0.1)
f <- sfm(y_pcs ~ x1 + x2, model_name = "NHN", data = as.data.frame(d))
inefficiency_test(f)
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