Using the power kernels of Phillips, Sun and Jin (2006, 2007), we examine the large sample asymptotic properties of the t -test for diﬀerent choices of power parameter (τ). We show that the nonstandard ﬁxed-τ limit distributions of the t -statistic provide more accurate approximations to the ﬁnite sample distributions than the conventional large-τ limit distribution. We prove that the second-order corrected critical value based on an asymptotic expansion of the nonstandard limit distribution is also second-order correct under the large-τ asymptotics. As a further contribution, we propose a new practical procedure for selecting the test-optimal power parameter that addresses the central concern of hypothesis testing: the selected power parameter is test-optimal in the sense that it minimizes the type II error while controlling for the type I error. A plug-in procedure for implementing the test-optimal power parameter is suggested. Simulations indicate that the new test is as accurate in size as the nonstandard test of Kiefer and Vogelsang (2002a, 2002b; KV), and yet it does not incur the power loss that often hurts the performance of the latter test. The new test therefore combines the advantages of the KV test and the standard (MSE optimal) HAC test while avoiding their main disadvantages (power loss and size distortion, respectively). The results complement recent work by Sun, Phillips and Jin (2008) on conventional and bT HAC testing.
Sun, Yixiao; Phillips, Peter C.B.; and Jin, Sainan, "Power Maximization and Size Control in Heteroskedasticity and Autocorrelation Robust Tests with Exponentiated Kernels" (2010). Cowles Foundation Discussion Papers. 2077.