A test statistic equation for obtaining alternative Wald and score statistics in the generalized method of moments framework

Abstract

In this article we derive a test statistic generating equation (TSGE) from which several statistics suitable for testing parametric restrictions in models estimated by the generalized method of moments (GMM) may be obtained. This statistic presents three main features: it is valid for producing statistics appropriate for testing constraints expressed in very general forms, which may involve not only the parameters of interest but also a vector of nuisance parameters; some of the tests proposed in the GMM literature are special cases of the TSGE; it can be evaluated at any consistent and asymptotically normal estimator.

Acknowledgements

I am grateful to Richard J. Smith for helpful comments. Financial support from Fundação para a Ciência e a Tecnologia through program POCTI, partially funded by FEDER, is also gratefully acknowledged.

Notes

¹ For some recently proposed alternatives, see inter alia Newey and Smith (2004).

² Throughout this article we assume that all efficient GMM estimators use the same estimator V_n . Otherwise, the numerical equivalence between some tests discussed later will not hold, although their asymptotic equivalence remains unaffected.

³ In this article we use the same definitions adopted by Smith (1987). Other authors prefer the terms mixed, implicit and explicit restrictions, respectively.

⁴ Another alternative would be to estimate α₀ under the null hypothesis, that is, we could minimize (2) subject to (5) using a Lagrangean function. See Gourieroux and Monfort (1989).

⁵ Note that and M_SM _Ω = M_S . Hence, .

⁶ Note that choosing S = Ω implies efficient GMM estimation of α₀ since , see expression (24) in the Appendix, and a generalized inverse for is Ω. In this case, an initial consistent estimator of α is needed to evaluate S_n  = Ω _n in (6), which can be obtained by solving also (6) but considering a matrix S_n , such as the identity, not dependent on unknown parameters.

⁷ Note that as α₀ is not identified only under the alternative hypothesis and the C(α) statistic is evaluated under H₀ , since we do not need to use (6) to estimate α₀. The same happens with the other score-type test presented next.