GC 16/01: The impact of the initial condition on covariate augmented unit root tests

Summary

Conventional testing for a unit root in a time series is typically carried out using the OLS-demeaning/detrending procedure of Dickey and Fuller (1979), or the GLS-demeaning/detrending procedure of Elliott, Rothenberg and Stock (1996). When the series under consideration covaries with an available stationary variable, Hansen (1995) showed that it is possible to substantially increase the power of the OLS-based unit root tests by augmenting the underlying OLS regression model with that stationary covariate. Elliott and Jansson (2003) and Westerlund (2013) show that incorporating covariates in a GLS-demeaning/detrending setting also improves the power of GLS- based unit root tests. The powers of covariance augmented OLS-based and GLS-based unit root tests are sensitive to the magnitude of the unobserved initial condition of a time series. For a small initial condition, GLS-based tests can have substantially more power than their OLS-based counterparts, while the reverse is true for a large initial condition. Typically, the power of OLS-based tests is an increasing function of this magnitude, whereas GLS-based tests demonstrate the opposite behaviour. In any practical testing situation, the magnitude of the initial condition is not known (nor can it be consistently estimated) and it is therefore unclear whether it is best to apply an OLS- or GLS-based unit root test in order to extract the most information about the presence, or otherwise, of a unit root.

In this Nottingham School of Economics working paper, Chrystalleni Aristidou, David I. Harvey and Stephen J. Leybourne propose to adopt a simple union of rejections strategy whereby (in its simplest guise) the unit root null hypothesis is rejected whenever either of the individual OLS- or GLS-based unit root tests rejects. The proposed a union of rejections based procedure, detects evidence in favour of the alternative hypothesis taken from both OLS- and GLS-based demeaned/detrended variants, and is shown to work very well, retaining attractive power levels across zero, small and large initial condition magnitudes. The findings mirror those found in the standard non-covariate augmented unit root testing environment, and our recommended procedure adds to the suite of available unit root testing procedures a covariate augmented approach that offers reliable power levels across the range of possible (unknown) initial conditions.

GC Discussion Paper 16/01, The impact of the initial condition on covariate augmented unit root tests by Chrystalleni Aristidou, David I. Harvey and Stephen J. Leybourne

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Authors

Chrystalleni Aristidou, David Harvey and Stephen Leybourne

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