Is Newey–West optimal among first-order kernels?

Newey–West (1987) standard errors are the dominant standard errors used for heteroskedasticity and autocorrelation robust (HAR) inference in time series regression. The Newey–West estimator uses the Bartlett kernel, which is a first-order kernel, meaning that its characteristic exponent, $q$, is equal to 1, where $q$ is defined as the largest value of $r$ for which the quantity $k^{\left[r\right]}\left(0\right)={lim}_{t\to 0}{\left|t\right|}^{-r}(k\left(0\right)-k\left(t\right))$ is defined and finite. This raises the apparently uninvestigated question of whether the Bartlett kernel is optimal among first-order kernels. We demonstrate that, for $q<2$, there is no optimal $q$th-order kernel for HAR testing in the Gaussian location model or for minimizing the MSE in spectral density estimation. In fact, for any $q<2$, the space of $q$th-order positive-semidefinite kernels is not closed and, moreover, all continuous $q$th-order kernels can be decomposed into a weighted sum of $q$th and second-order kernels, which suggests that there is no meaningful notion of ‘pure’ $q$th-order kernels for $q<2$. Nevertheless, it is possible to rank any given collection of $q$th-order kernels using the functional $I_q\left[k\right]={\left(k^{\left[q\right]}\left(0\right)\right)}^{1/q}\int k^2\left(t\right)dt$ with smaller values corresponding to better asymptotic performance. We examine the value of $I_q\left[k\right]$ for a wide variety of first-order estimators and find that none improve upon the Bartlett kernel. These comparisons provide additional justification for the continued use of the Newey–West estimator with testing-optimal smoothing parameters and fixed-$b$ critical values despite the lack of optimality of Bartlett among first-order kernels.