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Unformatted text preview: Let&s Fix It: Fixed-b Asymptotics versus Small-b Asymptotics in Heteroscedasticity and Autocorrelation Robust Inference Yixiao Sun & Department of Economics, University of California, San Diego July 20, 2010 Abstract In the presence of heteroscedasticity and autocorrelation of unknown forms, the covariance matrix of the parameter estimator is often estimated using a nonparametric kernel method that involves a lag truncation parameter. Depending on whether this lag truncation parameter is specied to grow at a slower rate than or the same rate as the sample size, we obtain two types of asymptotic approximations: the small- b asymptotics and the xed- b asymptotics. Using techniques for probability distribution approximation and high order expansions, this paper shows that the xed- b asymp- totics provides a higher order renement to the rst order small- b asymptotics. This result provides a theoretical justication on the use of the xed- b asymptotics in em- pirical applications. On the basis of the xed- b asymptotics and higher order small- b asymptotics, the paper introduces a new and easy-to-use F & test that employs a nite sample corrected Wald statistic and uses an F-distribution as the reference distribution. Finally, the paper develops a novel bandwidth selection rule that is testing-optimal in that the bandwidth minimizes the type II error of the F & test while controlling for its type I error. Monte Carlo simulations show that the F & test with the testing-optimal bandwidth works very well in nite samples. JEL Classi&cation: C13; C14; C32; C51 Keywords: Asymptotic expansion, F-distribution, Heteroscedasticity and Autocorrelation Robust, Hotelling&s T-squared distribution, long-run variance, robust standard error, testing- optimal smoothing parameter choice, type I and type II errors. & Email: firstname.lastname@example.org. 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. 1 Introduction In linear and nonlinear models with moment restrictions, it is standard practice to employ the generalized method of moments (GMM) to estimate the model parameters. Consis- tency of the GMM estimator in general does not depend on the dependence structure of the moment conditions. However, we often want not only point estimators of the model parameters, but also their covariance matrix in order to conduct inference. A popular co- variance estimator that allows for general forms of dependence is the nonparametric kernel estimator. The underlying smoothing parameter is the truncation lag (or bandwidth para- meter) or the ratio b of the truncation lag to the sample size. See Newey and West (1987) and Andrews (1991). In econometrics, this covariance estimator is often referred to as the heteroscedasticity and autocorrelation robust (HAR) estimator. The major di culty in using the HAR covariance estimator to perform hypothesis testing lies in how to select the...
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