Theory of Low Frequency Contamination from Nonstationarity and Misspecification: Consequences for HAR Inference

We establish theoretical and analytical results about the low frequency contamination induced by general nonstationarity for estimates such as the sample autocovariance and the periodogram, and deduce consequences for heteroskedasticity and autocorrelation robust (HAR) inference. We show that for short memory nonstationarity data these estimates exhibit features akin to long memory. We present explicit expressions for the asymptotic bias of these estimates. This bias increases with the degree of heterogeneity. in the data and is responsible for generating low frequency contamination or simply making the time series exhibiting long memory features. The sample autocovariances display hyperbolic rather than exponential decay while the periodogram becomes unbounded near the origin. We distinguish cases where this contamination only occurs as a small-sample problem and cases where the contamination continues to hold asymptotically. We show theoretically that nonparametric smoothing over time is robust to low frequency contamination.in that the sample local autocovariance and the local periodogram are unlikely to exhibit long memory features. Simulations confirm that our theory provides useful approximations. Since the autocovariances and the periodogram are key elements for HAR inference, our results provide new insights on the debate between consistent versus inconsistent small versus long/fixed-b bandwidths for long-run variance (LRV) estimation-based inference.