Regression-Based Seasonal Unit Root Tests

Abstract

It is well known that (seasonal) unit root tests can be seriously affected by the presence of weak dependence in the driving shocks when this is not accounted for. In the non-seasonal case both parametric (based around augmentation of the test regression with lagged dependent variables) and semi-parametric (based around an estimator of the long run variance of the shocks) unit root tests have been proposed. Of these, the M class of unit root tests introduced by Stock (1999), Perron and Ng (1996) and Ng and Perron (2001), appear to be particularly successful, showing good finite sample size control even in the most problematic (near-cancellation) case where the shocks contain a strong negative moving average component. The aim of this paper is threefold. First we show the implications that neglected weak dependence in the shocks has on lag un-augmented versions of the well known regression-based seasonal unit root tests of Hylleberg et al. (1990). Second, in order to complement extant parametrically augmented versions of the tests of Hylleberg et al. (1990), we develop semi-parametric seasonal unit root test procedures, generalising the methods developed in the non-seasonal case to our setting. Third, we compare the finite sample size and power properties of the parametric and semi-parametric seasonal unit root tests considered. Our results suggest that the superior size/power trade-off offered by the M approach in the non-seasonal case carries over to the seasonal case.

Semi-Parametric Seasonal Unit Root Tests

“…Relaxing this assumption will not alter the limiting null distributions of the test statistics we outline in this paper because tests based on data which have been de-trended under Cases 1, 2 or 3 will be exact similar with respect to the initial conditions; see Smith et al (2009).…”

“…Following HEGY (1990) and Smith et al (2009), among others, the regression-based approach to testing for seasonal unit roots in α(L) consists of two steps. In the first step one de-trends the data in order to yield tests which will be exact invariant (assuming µ Sn+s is not under-specified) to the elements of δ which characterise the deterministic component µ Sn+s in (2.1a).…”

“…This procedure allows the practitioner to test for unit root behaviour at each of the zero and seasonal frequency components of the data, either separately or via a joint test. HEGY outline their procedure for quarterly data; this has been extended to the case of monthly data by Beaulieu and Miron (1993) and to an arbitrary seasonal periodicity, say S, by Smith and Taylor (1999) and Smith et al (2009).…”

“…Focusing on the case where the shocks follow a finite-order autoregressive process of order p [AR(p)], Burridge and Taylor (2001) and Smith et al (2009) show that such lag augmentation can provide only a partial solution with the limiting null distributions of certain of the harmonic frequency unit root tests still depending, in general, on the parameters of the AR(p) polynomial with the consequence that not all of the HEGY-type tests can be reliably used in practice. However, it has been known since the seminal work of Box and Jenkins (1976) that seasonally observed time series tend to display significant moving average behaviour.…”

SPECIAL ISSUE OF ECONOMETRIC THEORY IN HONOR OF PROFESSOR RICHARD J. SMITH: GUEST EDITORS’ INTRODUCTION

“…Building on the foundations of earlier work of Richard Smith in the area of seasonal unit root testing (see, in particular, Smith andTaylor, 1998, anddel Barrio Castro, 2009), Tomás del Barrio Castro, Paulo Rodrigues, and Robert Taylor develop new semiparametric tests for the seasonal unit root hypothesis in their paper "Semi-Parametric Seasonal Unit Root Tests". They generalise the M class of unit root tests of Stock (1999), Perron and Ng (1996) and Ng and Perron (2001) to the seasonal context.…”