Simultaneous bandwidths determination for DK-HAC estimators and long-run variance estimation in nonparametric settings

We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroscedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the tradeoff between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews, we establish how nonstationarity affects the bias-variance tradeoff. We use the plug-in approach to construct data-dependent bandwidths for the DK-HAC estimators and compare them with the DK-HAC estimators from Casini that use data-dependent bandwidths obtained from a sequential MSE criterion. The former performs better in terms of size control, especially with stationary and close to stationary data. Finally, we consider long-run variance (LRV) estimation under the assumption that the series is a function of a nonparametric estimator rather than of a semiparametric estimator that enjoys the usual T rate of convergence. Thus, we also establish the validity of consistent LRV estimation in nonparametric parameter estimation settings.