GMM with Weak Identification

Abstract

This paper develops asymptotic distribution theory for GMM estimators and test statistics when some or all of the parameters are weakly identified. General results are obtained and are specialized to two important cases: linear instrumental variables regression and Euler equations estimation of the CCAPM. Numerical results for the CCAPM demonstrate that weak-identification asymptotics explains the breakdown of conventional GMM procedures documented in previous Monte Carlo studies. Confidence sets immune to weak identification are proposed. We use these results to inform an empirical investigation of various CCAPM specifications; the substantive conclusions reached differ from those obtained using conventional methods.

Evaluating the German (New Keynesian) Phillips curve

“…We start with the pure forward looking specification where we test the null hypothesis of whether β and θ are (0.99, 0.58) or (0.99, 0.18) which corresponds to the GMM estimates of Table (3) with A = 0.175. According to Stock and Wright (2000) S(β 0 , θ 0 ) D → χ 2 k , where S(β 0 , θ 0 ) is the objective as defined above evaluated at the true parameter values. Figure (4) reports the results of this test type.…”

“…We now check whether our baseline GMM results hold when we use S-sets as suggested by Stock and Wright (2000). First, we can examine whether our GMM point estimates are also included the S-sets.…”

“…So we reevaluate our GMM results with an identification robust method that is fully robust to problems induced by weak instruments and weak identification. Therefore, we stick to a non-linear variant of the Anderson-Rubin Statistic as suggested by Stock and Wright (2000). They show that identification robust confidence sets can be obtained from the continuous-updating GMM (CUE) objective function.…”

“…This should be the case when the model is correctly specified and there are no weak instrument problems present. According to Stock and Wright (2000) S(ϑ 0 ) D → χ 2 k , where S(ϑ 0 ) is the objective as defined above evaluated at the true parameter values (β 0 , θ 0 , ξ 0 ) or (β 0 , θ 0 , ω 0 ). Second, we can construct confidence intervals for the parameters of interest (so-called S-sets).…”

“…They extended this framework in Dufour, Khalaf and Kichian (2007b) and in Dufour, Khalaf and Kichian (2008) to allow for heteroskedastic and autocorrelated residuals. Kleibergen and Mavroeidis (2008) consider not only Stock and Wright's (2000) approach, but also a Lagrange Multiplier statistic (as discussed in Kleibergen, 2007) and a Likelihood Ratio statistic (see Kleibergen, 2005) to construct identification robust confidence intervals for the NKPC. Martins and Gabriel (2006) evaluate the NKPC with generalized empirical likelihood (GEL) methods.…”