Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series

Nunzio Cappuccio,Diego Lubian

doi:10.3390/econometrics4020021

Abstract

In cointegration analysis, it is customary to test the hypothesis of unit roots separately for each single time series. In this note, we point out that this procedure may imply large size distortion of the unit root tests if the DGP is a VAR. It is well-known that univariate models implied by a VAR data generating process necessarily have a finite order MA component. This feature may explain why an MA component has often been found in univariate ARIMA models for economic time series. Thereby, it has important implications for unit root tests in univariate settings given the well-known size distortion of popular unit root test in the presence of a large negative coefficient in the MA component. In a small simulation experiment, considering several popular unit root tests and the ADF sieve bootstrap unit tests, we find that, besides the well known size distortion effect, there can be substantial differences in size distortion according to which univariate time series is tested for the presence of a unit root.

Highlights

It is well known that unit root tests may have large size distortion when the autoregressive parameter is close to unity and/or when there is a large MA component
We (a) provide a theoretical motivation for the finite sample size distortion observed in the presence of a large negative MA root; (b) give additional simulation evidence on its extent comparing standard and bootstrap unit root tests and (c) provide some suggestions for empirical researchers working with univariate time series implied by a VAR data generating process
It turns out that the univariate models implied by a VAR data generating process always have a finite order MA component

Summary

Introduction

It is well known that unit root tests may have large size distortion when the autoregressive parameter is close to unity and/or when there is a large MA component (see, for instance, [1]). In the bivariate case, unit root tests applied to the one component may have an effective size as large as 90% while the same unit root test applied to the other component may have an effective size close to the nominal one These finite sample results hold for DGPs characterized by quite different parameter values and for a wide range of roots of the AR component, and even in the case where the roots are 1 and 0. We (a) provide a theoretical motivation for the finite sample size distortion observed in the presence of a large negative MA root; (b) give additional simulation evidence on its extent comparing standard and bootstrap unit root tests and (c) provide some suggestions for empirical researchers working with univariate time series implied by a VAR data generating process.

The Model

A Simulation Experiment

Findings

Conclusions