Inference in Second-Order Identified Models

First-order asymptotic analyses of the Generalized Method of Moments (GMM) estimator and its associated statistics are based on the assumption that the population moment condition identifies the parameter vector both globally and locally at first order. In linear models, global and first-order local identification are equivalent but in nonlinear models they are not. In certain econometric models of interest, parameters are globally identified but only identified locally at second order. In these scenarios the standard GMM inference techniques based on first-order asymptotics are invalid, see Dovonon & Renault (2013) and Dovonon & Hall (2016). In this paper, we explore how to perform inference in moment condition models that only identify the parameters locally to second order. For inference about the parameters, we consider inference based on conventional Wald and LM statistics, and also the Generalized Anderson Rubin (GAR) statistic (Anderson & Rubin, 1949; Dufour, 1997; Staiger & Stock, 1997; Stock & Wright, 2000) and the KLM statistic (Kleibergen, 2002, 2005). Both the GAR and KLM statistics have been proposed as methods of inference in the presence of weak identification and are known to be ``identification robust'' in the sense that their limiting distribution is the same under first-order and weak identification. For inference about the model specification, we consider the identification-robust J statistic (Kleibergen, 2005) and the GAR statistic. In each case, we derive the limiting distribution of statistics under both null and local alternative hypotheses. We show that under their respective null hypotheses the GAR, KLM and J statistics have the same limiting distribution as would apply under first-order or weak identification, thus showing their identification robustness extends to second-order identification. We explore the power properties in detail in two empirically relevant models with second-order identification. In the panel autoregressive (AR) model of order one, our analysis indicates that the Wald test of whether the AR parameter is one has superior power to the corresponding GAR test which, in turn, dominates the KLM and LM tests. For the conditionally heteroskedastic factor model, we compare Kleibergen's (2005) J and the GAR statistics to Hansen's (1982) overidentifying restrictions test (previously analyzed in this context by Dovonon & 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.