A COMPANION TO Theoretical Econometrics
Power and finite sample properties
A number of studies have examined the small sample properties of nonnested tests. For a limited number of cases it is possible to determine the exact form of the test statistic and the sampling distribution. For example, Godfrey (1983) shows that under Hf if X and Z are non-stochastic with normal errors, then the /A-test has an exact t (T - kf - 1) distribution.22 In the majority of cases the finite sample properties have been examined using Monte Carlo studies. A recurrent finding is that many Cox-type tests for nonnested regression models have a finite sample size which is significantly greater than the nominal level. Modifications based upon mean and variance adjustments have been proposed in Godfrey and Pesaran (1983), and are shown to affect a substantial improvement in finite sample performance. The authors demonstrate that in experimental designs allowing for nonnested models with either nonnormal errors, different number of regressors, or a lagged dependent variable, the adjusted Cox-test performs favorably relative to the /-test or F-test.23 In the case of nonnested linear regression models, Davidson and MacKinnon (1982) compared a number of variants of the Cox-test with F-, /A - and /-test.
An analysis of the power properties of non-tested tests has been undertaken using a number of approaches. In the case of nested models local alternatives are readily defined in terms of parameters that link the null to the alternative. Obviously in the case of models that are globally nonnested (i. e. the exponential and lognormal) this procedure is not possible. In the case of regression models Pesaran (1982a) is able to develop a asymptotic distribution of Cox-type tests under a sequence of local alternatives defined in terms of the degree of multicol - linearity of the regressors from the two rival models. Under this sequence of local alternatives he shows that the F-test based on the comprehensive model is less powerful than the Cox-type tests, unless the number of non-overlapping variables of the alternative over the null hypothesis is unity. An alternative approach to asymptotic power comparisons which does not require specification of local alternatives is advanced by Bahadur (1960) and Bahadur (1967) and holds the alternative hypothesis fixed but allows the size of the test to tend to zero as the sample size increases. Asymptotic power comparisons of nonnested tests by the Bahadur approach is considered in Gourieroux (1982) and Pesaran (1984).
