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Invariant tests based on M-estimators, estimating functions, and the generalized method of moments

Author

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  • Jean-Marie Dufour
  • Alain Trognon
  • Purevdorj Tuvaandorj

Abstract

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 (Godambe, 1960, Ann. Math. Stat.) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982, Econometrica), and models estimated by pseudo-likelihood (Gouri´eroux, Monfort and Trognon, 1984, Econometrica) 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()−type criteria. Extending the approach used in Dagenais and Dufour (1991, Econometrica), 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 GMM procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant scoretype and C()−type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987, Int. Econ. Rev.) 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.

Suggested Citation

  • Jean-Marie Dufour & Alain Trognon & Purevdorj Tuvaandorj, 2015. "Invariant tests based on M-estimators, estimating functions, and the generalized method of moments," CIRANO Working Papers 2015s-27, CIRANO.
  • Handle: RePEc:cir:cirwor:2015s-27
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    References listed on IDEAS

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    1. Gourieroux, Christian & Monfort, Alain & Trognon, Alain, 1984. "Pseudo Maximum Likelihood Methods: Theory," Econometrica, Econometric Society, vol. 52(3), pages 681-700, May.
    2. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    3. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models 2 volume set," Cambridge Books, Cambridge University Press, number 9780521478373, October.
    4. Dagenais, Marcel G & Dufour, Jean-Marie, 1991. "Invariance, Nonlinear Models, and Asymptotic Tests," Econometrica, Econometric Society, vol. 59(6), pages 1601-1615, November.
    5. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models," Cambridge Books, Cambridge University Press, number 9780521471626, October.
    6. Davidson, Russell & MacKinnon, James G., 1993. "Estimation and Inference in Econometrics," OUP Catalogue, Oxford University Press, number 9780195060119.
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    8. Newey, Whitney K & West, Kenneth D, 1987. "Hypothesis Testing with Efficient Method of Moments Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 28(3), pages 777-787, October.
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    Cited by:

    1. Pascale VALERY (HEC-Montreal) & Jean-Marie Dufour (University of Montreal), 2004. "A simple estimation method and finite-sample inference for a stochastic volatility model," Econometric Society 2004 North American Summer Meetings 153, Econometric Society.

    More about this item

    Keywords

    Testing; invariance; hypothesis reformulation; reparameterization; measurement unit; estimating function; generalized method of moment (GMM); pseudo-likelihood; M-estimator; Linear exponential model; Nonlinear model; Wald test; Likelihood ratio test; score test; lagrange multiplier test; C(∝) test.;

    JEL classification:

    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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