GLS-Based Unit Root Tests With Multiple Structural Breaks Under Both the Null and the Alternative Hypotheses

Abstract

Perron (1989, Econometrica 57, 1361–1401) introduced unit root tests valid when a break at a known date in the trend function of a time series is present. In particular, they allow a break under both the null and alternative hypotheses and are invariant to the magnitude of the shift in level and/or slope. The subsequent literature devised procedures valid in the case of an unknown break date. However, in doing so most research, in particular the commonly used test of Zivot and Andrews (1992, Journal of Business & Economic Statistics 10, 251–270), assumed that if a break occurs it does so only under the alternative hypothesis of stationarity. This is undesirable for several reasons. Kim and Perron (2009, Journal of Econometrics 148, 1–13) developed a methodology that allows a break at an unknown time under both the null and alternative hypotheses. When a break is present, the limit distribution of the test is the same as in the case of a known break date, allowing increased power while maintaining the correct size. We extend their work in several directions: (1) we allow for an arbitrary number of changes in both the level and slope of the trend function; (2) we adopt the quasi–generalized least squares detrending method advocated by Elliott, Rothenberg, and Stock (1996, Econometrica 64, 813–836) that permits tests that have local asymptotic power functions close to the local asymptotic Gaussian power envelope; (3) we consider a variety of tests, in particular the class of M-tests introduced in Stock (1999, Cointegration, Causality, and Forecasting: A Festschrift for Clive W.J. Granger) and analyzed in Ng and Perron (2001, Econometrica 69, 1519–1554).

Testing for a Deterministic Trend When There is Evidence of Unit Root

“…This work is in line with Kim and Perron (2006) and Carrion-i Silvestre, Kim, and Perron (2007), who recognize that structural breaks in the deterministic trend occur whether the noise component is stationary or has a unit root.…”

Empirical Evidence on Inflation and Unemployment in the Long Run

“…2001). 9 Using instead the annualized quarter-on-quarter changes, {ln(P t ) − ln(P t−1 )} 400, leads to very similar an alternative measure of inflation, we calculated the percentage changes for the GDP-deflator. The implied GDP-deflator was calculated based on the ratio of the nominal GDP (series GDP) and the real GDP (series GDPC1) in chained 2005 dollars.…”

Trends of U.S. Emissions of Nitrogen Oxides and Volatile Organic Compounds

“…To deal with these issues, we use a two-step method advocated by Perron (2007), Kim and Perron (2007), Harris et al (2007) and Carrion-i- Silvestre et al (2008). In the first step, we pretest the pollution series of interest for structural change using Exp-W RFS which is valid whether the series is DS or TS.…”

“…If the pre-test does not reject, we proceed as if we know a break does not occur and use the ADF-GLS test for a unit root that does not allow for a break. Harris et al (2007), Kim and Perron (2007) and Carrion-i- Silvestre et al (2008) showed that such a two step procedure leads to unit root tests with good size and power properties whether a break occurs or not.…”