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Prewhitened Long-Run Variance Estimation Robust to Nonstationarity

Alessandro Casini, Pierre Perron

Abstract

We introduce a nonparametric nonlinear VAR prewhitened long-run variance (LRV) estimator for the construction of standard errors robust to autocorrelation and heteroskedasticity that can be used for hypothesis testing in a variety of contexts including the linear regression model. Existing methods either are theoretically valid only under stationarity and have poor finite-sample properties under nonstationarity (i.e., fixed-b methods), or are theoretically valid under the null hypothesis but lead to tests that are not consistent under nonstationary alternative hypothesis (i.e., both fixed-b and traditional HAC estimators). The proposed estimator accounts explicitly for nonstationarity, unlike previous prewhitened procedures which are known to be unreliable, and leads to tests with accurate null rejection rates and good monotonic power. We also establish MSE bounds for LRV estimation that are sharper than previously established and use them to determine the data-dependent bandwidths.

Prewhitened Long-Run Variance Estimation Robust to Nonstationarity

Abstract

We introduce a nonparametric nonlinear VAR prewhitened long-run variance (LRV) estimator for the construction of standard errors robust to autocorrelation and heteroskedasticity that can be used for hypothesis testing in a variety of contexts including the linear regression model. Existing methods either are theoretically valid only under stationarity and have poor finite-sample properties under nonstationarity (i.e., fixed-b methods), or are theoretically valid under the null hypothesis but lead to tests that are not consistent under nonstationary alternative hypothesis (i.e., both fixed-b and traditional HAC estimators). The proposed estimator accounts explicitly for nonstationarity, unlike previous prewhitened procedures which are known to be unreliable, and leads to tests with accurate null rejection rates and good monotonic power. We also establish MSE bounds for LRV estimation that are sharper than previously established and use them to determine the data-dependent bandwidths.

Paper Structure

This paper contains 26 sections, 15 theorems, 209 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The Statistical Environment
  3. Prewhitening DK-HAC Estimator
  4. Data-Dependent Bandwidths
  5. Large-Sample Results When $\mathbb{E}\left(V_{t}\right)=0$
  6. Extension to General Nonstationary Random Variables
  7. Consistency, Rate of Convergence and MSE Bounds
  8. Optimal Bandwidths and Kernels
  9. Data-dependent DK-HAC Estimation
  10. Theoretical Results About the Power of HAR Tests Under General $\mathbb{E}\left(V_{t}\right)$
  11. Small-Sample Evaluations
  12. Conclusions
  13. Appendix
  14. Tables
  15. Preliminaries
  16. ...and 11 more sections

Key Result

Theorem 3.1

Suppose $K_{1}\left(\cdot\right)\in\boldsymbol{K}_{3}$, $q$ is as in $\boldsymbol{K}_{3}$, $K_{2}\left(\cdot\right)\in\boldsymbol{K}_{2}$, $||\int_{0}^{1}f_{D}^{*\left(q\right)}\left(u,\,0\right)||<\infty$. Then, we have: (i) If Assumption Assumption Smothness f(u,w), Assumption A - Dependence-Assum

Theorems & Definitions (16)

  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.1
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Theorem 5.1
  • Definition S.A.1
  • Lemma S.B.1
  • ...and 6 more