Let's Fix it: Fixed-b Asymptotics Versus Small-b Asymptotics in Heteroscedasticity and Autocorrelation Robust Inference

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 speci…ed 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 asymptotic approximation provides a higher order re…nement to the …rst order small-b asymptotics. This result provides a theoretical justi…cation on the use of the …xed-b asymptotics in empirical applications. On the basis of the …xed-b asymptotics and higher order small-b asymptotics, the paper introduces a new and easy-to-use asymptotic F test that employs a …nite sample corrected Wald statistic and uses an F-distribution as the reference distribution. Finally, the paper develops a bandwidth selection rule that is testing-optimal in that the bandwidth minimizes the type II error of the asymptotic F test while controlling for its type I error. Monte Carlo simulations show that the asymptotic F test with the testing-optimal bandwidth works very well in …nite samples.

A New Asymptotic Theory for Vector Autoregressive Long-Run Variance Estimation and Autocorrelation Robust Testing

“…For the Parzen and QS kernels, the smoothing parameters are determined by a testing-optimal rule developed by Sun (2010). Results for our recommended F*-VAR-Par and F*-VAR-QS tests are in bold fonts.…”

“…We consider two commonly used second-order kernels: the Parzen and QS kernels. For each kernel, the bandwidth is determined via the testing-oriented criterion (Sun 2010). The bandwidth is selected to solve the constrained minimization problem in (15).…”

“…Following Sun (2010), we show that, after some modi…cation, the nonstandard …xed-smoothing limiting distribution can be approximated by a standard F distribution. The F approximation can also be derived from a high-order Edgeworth expansion under the conventional asymptotics where p !…”

SIMPLE, ROBUST, AND ACCURATEFANDtTESTS IN COINTEGRATED SYSTEMS

“…For the model without a time trend, it follows from (22) and Theorem 6 that the asymptotic variances of^ T AOLS and^ T OLS conditioning on ( ; ) are…”

“…The …xed-smoothing asymptotics includes the …xed-b asymptotics of Kiefer and Vogelsang (2005) as a special case. For more discussions on these two types of asymptotics, see Sun (2014aSun ( , 2014b. There is a growing number of papers on …xed-b asymptotic theory for stationary data starting with Kiefer and Vogelsang (2005).…”