Let’s fix it: Fixed- asymptotics versus small- asymptotics in heteroskedasticity and autocorrelation robust inference

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 specified 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 fixed- b asymptotics. Using techniques for probability distribution approximation and high order expansions, this paper shows that the fixed- b asymptotic approximation provides a higher order refinement to the first order small- b asymptotics. This result provides a theoretical justification on the use of the fixed- b asymptotics in empirical applications. On the basis of the fixed- b asymptotics and higher order small- b asymptotics, the paper introduces a new and easy-to-use asymptotic F test that employs a finite 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 finite samples.