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Invariant tests based on \(M\)-estimators, estimating functions, and the generalized method of moments. (English) Zbl 1524.62093

Summary: We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions [V. P. Godambe, Ann. Math. Stat. 31, 1208–1211 (1961; Zbl 0118.34301)] or moment conditions and are estimated by generalized method of moments (GMM) procedures [L. P. Hansen, Econometrica 50, 1029–1054 (1982; Zbl 0502.62098)], and models estimated by pseudo-likelihood [C. Gourieroux et al., Econometrica 52, 681–700 (1984; Zbl 0575.62031); ibid. 52, 701–720 (1984; Zbl 0575.62032)] and \(M\)-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and \(C(\alpha)\)-type criteria. Extending the approach used in [M. G. Dagenais and J.-M. Dufour, Econometrica 59, No. 6, 1601–1615 (1991; Zbl 0745.62103)], we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and \(C(\alpha)\)-type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by W. K. Newey and K. D. West [Int. Econ. Rev. 28, No. 3, 777–787 (1987; Zbl 0676.62029)] to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.

MSC:

62F03 Parametric hypothesis testing
62F05 Asymptotic properties of parametric tests
62P20 Applications of statistics to economics

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