UC San Diego

In the widely used over-identified econometric model, the two-step Generalized Methods of Moments (GMM) estimator and inference, first suggested by Hansen (1982), require the estimation of optimal weighting matrix at the initial stages. For time series data and clustered dependent data, which is our focus here, the optimal weighting matrix is usually referred to as the long run variance (LRV) of the (scaled) sample moment conditions. To maintain generality and avoid misspecification, nowadays we do not model serial dependence and within-cluster dependence parametrically but use the heteroscedasticity and autocorrelation robust (HAR) variance estimator in standard practice. These estimators are nonparametric in nature with high variation in finite samples, but the conventional increasing smoothing asymptotics, so called small-bandwidth asymptotics, completely ignores the finite sample variation of the estimated GMM weighting matrix. As a consequence, empirical researchers are often in danger of making unreliable inferences and false assessments of the (efficient) two-step GMM methods. Motivated by this issue, my dissertation consists of three papers which explore the efficiency and approximation issues in the two-step GMM methods by developing new, more accurate, and easy-to-use approximations to the GMM weighting matrix.

The first chapter, "Simple and Trustworthy Cluster-Robust GMM Inference" explores new asymptotic theory for two-step GMM estimation and inference in the presence of clustered dependence. Clustering is a common phenomenon for many cross-sectional and panel data sets in applied economics, where individuals in the same cluster

will be interdependent while those from different clusters are more likely to be independent. The core of new approximation scheme here is that we treat the number of clusters G fixed as the sample size increases. Under the new fixed-G asymptotics, the centered two-step GMM estimator and two continuously-updating estimators have the same asymptotic mixed normal distribution. Also, the t statistic, J statistic, as well as the trinity of two-step GMM statistics (QLR, LM and Wald) are all asymptotically pivotal, and each can be modified to have an asymptotic standard F distribution or t distribution. We also suggest a finite sample variance correction further to improve the accuracy of the F or t approximation. Our proposed asymptotic F and t tests are very appealing to practitioners, as test statistics are simple modifications of the usual test statistics, and the F or t critical values are readily available from standard statistical tables. We also apply our methods to an empirical study on the causal effect of access to domestic and international markets on household consumption in rural China.

The second paper "Should we go one step further? An Accurate Comparison of One-step and Two-step procedures in a Generalized Method of Moments Framework$''$ (coauthored with Yixiao Sun) focuses on GMM procedure in time-series setting and provides an accurate comparison of one-step and two-step GMM procedures in a fixed-smoothing asymptotics framework. The theory developed in this paper shows that the two-step procedure outperforms the one-step method only when the benefit of using the optimal weighting matrix outweighs the cost of estimating it. We also provide clear guidance on how to choose a more efficient (or powerful) GMM estimator (or test) in practice.

While our fixed smoothing asymptotic theory accurately describes sampling distribution of two-step GMM test statistic, the limiting distribution of conventional GMM statistics is non-standard, and its critical values need to be simulated or approximated by standard distributions in practice. In the last chapter, "Asymptotic F and t Tests in an

Efficient GMM Setting" (coauthored with Yixiao Sun), we propose a simple and easy-to-implement modification to the trinity (QLM, LM, and Wald) of two-step GMM statistics and show that the modified test statistics are all asymptotically F distributed under the fixed-smoothing asymptotics. The modification is multiplicative and only involves the J statistic for testing over-identifying restrictions. In fact, what we propose can be regarded as the multiplicative variance correction for two-step GMM statistics that takes into account the additional asymptotic variance term under the fixed-smoothing asymptotics. The results in this paper can be immediately generalized to the GMM setting in the presence of clustered dependence.