Let's Fix it : Fixed-b Asymptotics Versus Small-b Asymptotics in Heteroscedasticity 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 asymptotics 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 F* test that employs a finite 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 finite samples