This paper considers two-step e¢ cient GMM estimation and inference where the weighting matrix and asymptotic variance matrix are based on the series long run variance estimator. We propose a simple and easy-to-implement modi…cation to the trinity of test statistics in the two-step e¢ cient GMM setting and show that the modi…ed test statistics are all asymptotically F distributed under the so-called …xed-smoothing asymptotics. The modi…cation is multiplicative and involves the J statistic for testing over-identifying restrictions. This leads to convenient asymptotic F tests that use standard F critical values. Simulation shows that, in terms of both size and power, the asymptotic F tests perform as well as the nonstandard tests proposed recently by Sun (2014b) in …nite samples. But the F tests are more appealing as the critical values are readily available from standard statistical tables. Compared to the conventional chi-square tests, the F tests are as powerful, but are much more accurate in size.JEL Classi…cation: C12, C32

This paper develops a new asymptotic theory for two-step GMM estimation and inference in the presence of clustered dependence. The key feature of alternative asymptotics is the number of clusters G is regarded as small or …xed when the sample size increases. Under the small-G asymptotics, this paper shows the centered two-step GMM estimator and the two continuously-updating GMM estimators we consider have the same asymptotic mixed normal distribution. In addition, the J statistic, the trinity of two-step GMM statistics (QLR, LM and Wald), and the t statistic are all asymptotically pivotal, and each can be modi…ed to have an asymptotic standard F distribution or t distribution. We suggest a …nite sample variance correction to further improve the accuracy of the F and t approximations. Our proposed asymptotic F and t tests are very appealing to practitioners because our test statistics are simple modi…cations of the usual test statistics, and critical values are readily available from standard statistical tables. A Monte Carlo study shows that our proposed tests are more accurate than the conventional inferences under the large-G asymptotics.

No abstract

According to the conventional asymptotic theory, the two-step Generalized Method of Moments (GMM) estimator and test perform as least as well as the one-step estimator and test in large samples. The conventional asymptotic theory, as elegant and convenient as it is, completely ignores the estimation uncertainty in the weighting matrix, and as a result it may not re ‡ect …nite sample situations well. In this paper, we employ the …xed-smoothing asymptotic theory that accounts for the estimation uncertainty, and compare the performance of the one-step and two-step procedures in this more accurate asymptotic framework. We show the two-step procedure outperforms the one-step procedure only when the bene…t of using the optimal weighting matrix outweighs the cost of estimating it. This qualitative message applies to both the asymptotic variance comparison and power comparison of the associated tests. A Monte Carlo study lends support to our asymptotic results. JEL Classi…cation: C12, C32

We propose a new finite sample corrected variance estimator for the linear generalized method of moments (GMM) including the one-step, two-step, and iterated estimators. Our formula additionally corrects for the over-identification bias in variance estimation on top of the commonly used finite sample correction of Windmeijer ( 2005) which corrects for the bias from estimating the efficient weight matrix, so is doubly corrected. Formal stochastic expansions are derived to show the proposed double correction estimates the variance of some higher-order terms in the expansion. In addition, the proposed double correction provides robustness to misspecification of the moment condition. In contrast, the conventional variance estimator and the Windmeijer correction are inconsistent under misspecification. That is, the proposed double correction formula provides a convenient way to obtain improved inference under correct specification and robustness against misspecification at the same time.