The Fixed-b Limiting Distribution and the ERP of HAR Tests Under Nonstationarity

TL;DR

This paper demonstrates that fixed-b asymptotic distributions for HAR tests are non-pivotal under nonstationarity, leading to larger errors in rejection probability and questioning the validity of existing inference methods.

Contribution

It reveals the non-pivotal nature of fixed-b distributions under nonstationarity and quantifies the increased error in rejection probability, challenging current fixed-b inference methods.

Findings

Fixed-b limiting distribution depends on nonstationary data characteristics.

Error in rejection probability is significantly larger under nonstationarity.

Existing fixed-b methods are not valid for nonstationary data.

Abstract

We show that the nonstandard limiting distribution of HAR test statistics under fixed-b asymptotics is not pivotal (even after studentization) when the data are nonstationarity. It takes the form of a complicated function of Gaussian processes and depends on the integrated local long-run variance and on on the second moments of the relevant series (e.g., of the regressors and errors for the case of the linear regression model). Hence, existing fixed-b inference methods based on stationarity are not theoretically valid in general. The nuisance parameters entering the fixed-b limiting distribution can be consistently estimated under small-b asymptotics but only with nonparametric rate of convergence. Hence, We show that the error in rejection probability (ERP) is an order of magnitude larger than that under stationarity and is also larger than that of HAR tests based on HAC estimators under conventional asymptotics. These theoretical results reconcile with recent finite-sample evidence in Casini (2021) and Casini, Deng and Perron (2021) who showing that fixed-b HAR tests can perform poorly when the data are nonstationary. They can be conservative under the null hypothesis and have non-monotonic power under the alternative hypothesis irrespective of how large the sample size is.