Fixed-smoothing Asymptotics in a Two-step GMM Framework

The paper develops the …xed-smoothing asymptotics in a two-step GMM framework. Under this type of asymptotics, the weighting matrix in the second-step GMM criterion function converges weakly to a random matrix and the two-step GMM estimator is asymptotically mixed normal. Nevertheless, the Wald statistic, the GMM criterion function statistic and the LM statistic remain asymptotically pivotal. It is shown that critical values from the …xedsmoothing asymptotic distribution are high order correct under the conventional increasingsmoothing asymptotics. When an orthonormal series covariance estimator is used, the critical values can be approximated very well by the quantiles of a noncentral F distribution. A simulation study shows that the new statistical tests based on the …xed-smoothing critical values are much more accurate in size than the conventional chi-square test. JEL Classi…cation: C12, C32 Keywords: F distribution, Fixed-smoothing Asymptotics, Heteroskedasticity and Autocorrelation Robust, Increasing-smoothing Asymptotics, Noncentral F Test, Two-step GMM Estimation 1 Introduction Recent research on heteroskedasticity and autocorrelation robust (HAR) inference has been focusing on developing distributional approximations that are more accurate than the conventional chi-square approximation or the normal approximation. To a great extent and from a broad perspective, this development is in line with many other areas of research in econometrics where more accurate distributional approximations are the focus of interest. A common theme for coming up with a new approximation is to embed the …nite sample situation in a di¤erent limiting thought experiment. In the case of HAR inference, the conventional limiting thought experiment assumes that the amount of smoothing increases with the sample size but at a slower rate. The Email: [email protected]. For helpful comments, the author thanks Jungbin Hwang, David Kaplan, and Min Seong Kim, Elie Tamer, the coeditor, and anonymous referees. The author gratefully acknowledges partial research support from NSF under Grant No. SES-0752443. Correspondence to: Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508.