Some fixed-b results for regressions with high frequency data over long spans

This paper develops fixed-$b$ asymptotic results for heteroskedasticity autocorrelation robust (HAR) Wald tests for high frequency data using the continuous time framework of Chang et al. (2023) (CLP). It is shown that the fixed-$b$ limit of HAR Wald tests for high frequency stationary regressions is the same as the standard fixed-$b$ limit in Kiefer and Vogelsang (2005). For the case of cointegrating regression the form of the fixed-$b$ limits are different from the stationary case and may or may not be pivotal but also have the same fixed-$b$ limits that have been obtained for tests based on ordinary least squares (OLS) (Bunzel, 2006) and integrated modified OLS (Vogelsang and Wagner, 2014). A simulation study shows that fixed-$b$ critical values provide rejection probabilities closer to nominal levels than traditional chi-square critical values when using data-dependent bandwidths. The Andrews (1991) data-dependent method works reasonably well for a wider range of persistence parameters than those considered by CLP. In contrast, the Newey and West (1994) data-dependent method is sensitive to the choice of pre-tuning parameters. The data-dependent method of Sun et al. (2008) give results similar to the Andrews (1991) method with slightly less over-rejection problems when used with fixed-$b$ critical values. Our results for bandwidth choice reinforce the importance of high frequency compatibility of bandwidths as emphasized by CLP. Regardless of the bandwidth method used in practice, it is clear that fixed-$b$ critical values can and should be used for high frequency data whenever HAR tests are based on kernel estimators of long run variances. Our results complement the analysis of Pellatt and Sun (2023) who focused on HAR tests based on orthonormal series estimators of long run variance estimator.