Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Abstract

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.

Key Words:

• Bootstrap
• Nonnested hypotheses
• Nonnormality

JEL Classification:

• C12
• C15

Acknowledgments

We are indebted to Essie Maasoumi, an associate editor, and two referees for their constructive and detailed comments. We are also grateful to B.W. Brorsen for providing information about simulation experiments, and to K. Abadir, F. Bravo, G. Forchini, J. Hinde, P.N.W. Marsh, N. Shephard and A. Wood for helpful advice and discussions. Santos Silva gratefully acknowledges the partial financial support from Fundação para a Cieˆncia e Tecnologia, program POCTI, partially funded by FEDER.

Notes

^aGodfrey et al. (1988) recommend RESET on the basis of simulation evidence derived from simple regression models. It appears that their results on the relative merits of RESET do not generalize to situations in which multiple regression models are the data processes.

^bThe notation plim₀ denotes a probability limit taken under H ₀. This probability limit, in general, depends upon the error distribution specified by H ₀.

^cThe bootstrap could be used with standardized statistics calculated using the simulation methods of Pesaran and Pesaran (1995). Coulibaly and Brorsen (1999) use a different simulation method to obtain such statistics and then apply a bootstrap to derive a p-value.

^dThe probability limit of λˆ₀₁ does not depend on the error distribution when the competing nonnested models are linear regression equations with the same dependent variable; see Pesaran (1974, p. 157, Eq. 3.7).

^eHinde (1992) has suggested replacing the potentially invalid parametric bootstrap scheme by a nonparametric bootstrap scheme when obtaining a reference distribution for λˆ₀₁. However, it has not been possible to find a proof of the asymptotic validity of his method and the use of λˆ₀₁ (which, in practice, would probably often be derived under normality assumptions) would require some justification other than maximum likelihood considerations.

^fThis group of distributions represents a fairly standard set in simulation studies involving nonnormality. If the evidence suggests that a test has good performance for all these distributions, it seems likely that it will be well behaved in many cases in practice.

^gDetails of the definitions of these test variables are given by Godfrey et al. (1988).

^hIt would always be possible to use such a large number of replications that the point null hypothesis would be rejected with probability close to unity, even when the actual difference between the true value and 5% was of no practical relevance to applied workers.

ⁱIt seems reasonable to assume that estimators of significance levels are very close to being normally distributed when R = 25,000. Also, for true rejection rates of 4, 5 and 6%, the implied standard errors of the sample proportion of rejections are (approximately) 0.12, 0.14 and 0.15%, respectively.

^jExperiments were carried out using Hinde's nonparametric bootstrap (Hinde, 1992). This approach, for which no formal justification has been found, did not give good control of finite sample significance levels.

^kMacKinnon (2002) remarks that many bootstrap tests seem to perform very well when applied to single-equation models with exogenous or predetermined regressors and iid errors.

^lThe full set of results can be obtained from the authors on request.

^mDavidson and MacKinnon (1985, pp. 509–512) report cases in which the Andrews test lacks power. The use of parameter values in the Monte Carlo data processes of this paper that give low power estimates for Andrews reveals that the estimates for Andrews, Andrew s*, P _E and P _E * are again very similar.