Improved HAR Inference

Employing power kernels suggested in earlier work by the authors (2003), this paper shows how to re.ne methods of robust inference on the mean in a time series that rely on families of untruncated kernel estimates of the long-run parameters. The new methods improve the size properties of heteroskedastic and autocorrelation robust (HAR) tests in comparison with conventional methods that employ consistent HAC estimates, and they raise test power in comparison with other tests that are based on untruncated kernel estimates. Large power parameter (rho) asymptotic expansions of the nonstandard limit theory are developed in terms of the usual limiting chi-squared distribution, and corresponding large sample size and large rho asymptotic expansions of the finite sample distribution of Wald tests are developed to justify the new approach. Exact finite sample distributions are given using operational techniques. The paper further shows that the optimal rho that minimizes a weighted sum of type I and II errors has an expansion rate of at most O(T^{1/2}) and can even be O(1) for certain loss functions, and is therefore slower than the O(T^{2/3}) rate which minimizes the asymptotic mean squared error of the corresponding long run variance estimator. A new plug-in procedure for implementing the optimal rho is suggested. Simulations show that the new plug-in procedure works well in finite samples.