Abstract
We explore the local power properties of different test statistics for conducting inference in moment condition models that only identify the parameters locally to second order. We consider the conventional Wald and LM statistics, and also the Generalized Anderson–Rubin (GAR) statistic (Anderson and Rubin, 1949; Dufour, 1997; Staiger and Stock, 1997; Stock and Wright, 2000), KLM statistic (Kleibergen, 2002; Kleibergen, 2005) and the GMM extension of Moreira (2003) (GMM-M) conditional likelihood ratio statistic. The GAR, KLM and GMM-M statistics are so-called “identification robust” since their (conditional) limiting distribution is the same under first-order, weak and therefore also second order identification. For inference about the model specification, we consider the identification-robust J statistic (Kleibergen, 2005), and the GAR statistic. Interestingly, we find that the limiting distribution of the Wald statistic under local alternatives not only depends on the distance to the null hypothesis but also on the convergence rate of the Jacobian. We specifically analyse two empirically relevant models with second order identification. In the panel autoregressive model of order one, our analysis indicates that the Wald test of a unit root value of the autoregressive parameter has better power compared to the corresponding GAR test which, in turn, dominates the KLM, GMM-M and LM tests. For the conditionally heteroskedastic factor model, we compare Kleibergen (2005) J and the GAR statistics to Hansen (1982) overidentifying restrictions test (previously analysed in this context by Dovonon and Renault, 2013) and find the power ranking depends on the sample size. Collectively, our results suggest that tests with meaningful power can be conducted in second-order identified models.
Highlights
The Generalized Method of Moments (GMM) is a popular method for estimating the parameters of econometric models based on the information in population moment conditions
We focus on the case where parameters are globally identified, identification fails locally at first order but holds at second order. This pattern of identification has been shown to arise in a number of situations in statistics and econometrics such as2: ML for skew-normal distributions, Azzalini (2005); ML for binary response models based on skew-normal distributions, Stingo et al (2011); ML for missing not at random (MNAR) models, Lee and Chesher (1986) and Jansen et al (2006); ML estimation of production function models, Lee and Chesher (1986) and Lee (1993); GMM estimation of conditionally heteroskedastic factor models, Dovonon and Renault (2009, 2013a); GMM estimation of panel data models using second moments, Madsen (2009) and Bun and Kleibergen (2016); modified-ML estimation of panel data models, (Kruiniger, 2014)
The Wald statistic is shown to have a non-standard distribution under both null and local alternatives, which depends on the convergence rate of the Jacobian, but the distribution under the null is simulated making inference practicable
Summary
The Generalized Method of Moments (GMM) is a popular method for estimating the parameters of econometric models based on the information in population moment conditions. We focus on the case where parameters are globally identified, identification fails locally at first order but holds at second order This pattern of identification has been shown to arise in a number of situations in statistics and econometrics such as: ML for skew-normal distributions, Azzalini (2005); ML for binary response models based on skew-normal distributions, Stingo et al (2011); ML for missing not at random (MNAR) models, Lee and Chesher (1986) and Jansen et al (2006); ML estimation of production function models, Lee and Chesher (1986) and Lee (1993); GMM estimation of conditionally heteroskedastic factor models, Dovonon and Renault (2009, 2013a); GMM estimation of panel data models using second moments, Madsen (2009) and Bun and Kleibergen (2016); modified-ML estimation of panel data models, (Kruiniger, 2014). The proofs of the main results are presented in a mathematical appendix, with some additional results relegated to an on-line appendix available at https://sites.google.com/site/prosperdovonon/cab/ DHK_180925_SupApp_200424.pdf?attredirects=0
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