Can Tests for Stochastic Unit Roots Provide Useful Portmanteau Tests for Persistence?*
| Author | A. M. Robert Taylor,Dick Dijk |
| Date | 01 September 2002 |
| DOI | http://doi.org/10.1111/1468-0084.00029 |
| Published date | 01 September 2002 |
Can Tests for Stochastic Unit Roots Provide
Useful Portmanteau Tests for Persistence?*
A. M. Robert Taylor
and Dick van Dijk
à
Department of Economics, University of Birmingham, Edgbaston, Birmingham B15
2TT, United Kingdom, Email: R.Taylor@bham.ac.uk
à
Econometric Institute, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands,
E-mail: djvandijk@few.eur.nl
I. Introduction
In a recent paper McCabe and Smith (1998) investigate the power of the
Dickey-Fuller [DF] tests against stochastic unit root processes. Their results
lead them to argue that ‘‘... the DF tests and variants thereof may not offer
particularly useful statistics for detecting whether a time series exhibits forms
of nonstationarity other than difference stationarity...’’ (op. cit., p.751). In this
paper we turn the question around and investigate whether the recently
developed tests of the unit root null hypothesis against the alternative of a
stochastic unit root [SUR] of McCabe and Tremayne (1995), Leybourne,
McCabe and Treymane (1996) [LMT hereafter], and Leybourne, McCabe and
Mills (1996) [LMM], can form a useful statistical device in discriminating
between series whose degree of persistence is no more than that of a unit root
[Ið1Þ] process and those which display a greater degree of persistence than
Ið1Þseries. The SUR process is an example of the latter but so, for example, is
the fractionally integrated process of order d[IðdÞ], when dis greater than
unity. Conversely, a process which, for example, displays a shift in persistence
from Ið1Þto short memory [Ið0Þ] behaviour, or vice versa, at some point in its
history is less persistent than a fixed unit root process, as is an IðdÞprocess
with d<1.
*Helpful comments and suggestions from the Associate Editor and an anonymous referee are
gratefully acknowledged. Any remaining errors are ours. The second author thanks the Netherlands
Organization for Scientific Research (N.W.O.) for its financial support.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 64, 4 (2002) 0305-9049
381
ÓBlackwell Publishers Ltd, 2002. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350
Main Street, Malden, MA 02148, USA.
The analysis in this paper is based on an extensive set of Monte Carlo
experiments, where we apply the stochastic unit root tests to both stationary
and non-stationary random coefficient autoregressive [RCAR] processes,
stationary and non-stationary fractionally integrated [FI] processes, threshold
unit root [TUR] processes, changing persistence [CP] processes and Markov
switching autoregressive [MSAR] processes. The first of these is closely
related to the SUR process, while the remaining processes have been put
forward as alternatives to the standard unit root model, as has the SUR
process. We consider FI, RCAR and MSAR processes which display both
greater and lesser degrees of persistence than the Ið1Þprocess, while TUR
and CP processes are always of lower persistence than the Ið1Þprocess. If
the SUR tests are to be efficacious tools in detecting processes whose
degree of persistence exceeds that of the Ið1Þprocess, then the above set of
data generating processes [DGPs] should form a useful benchmark for
comparison.
We briefly discuss the SUR process and the tests against SUR in section II.
For more elaborate introductions to this topic we refer to LMT, LMM and
Granger and Swanson (1997). In section III we describe the set-up of the
Monte Carlo experiments, together with some discussion and interpretation of
the results. Overall, with one exception, our results suggest that SUR tests
reject the null less (more) frequently for processes with smaller (greater)
persistence than a unit root process. The SUR tests consequently appear, at
least on the basis of our numerical evidence, to be useful diagnostic tests
against processes whose degree of persistence exceeds that of a conventional
unit root process. Section IV concludes.
II. Stochastic Unit Roots: Model and Test Statistics
Consider the following first order random coefficient autoregressive
[RCAR(1)] process,
yt¼atyt1þet;t¼1;...;T;ð1Þ
where at¼aþdt,d0¼0 and
dt¼qdt1þgt;t¼1;...;T;ð2Þ
with jqj1. In (1), and throughout this paper unless otherwise stated, etis
i:i:d:ð0;r2Þ, while gtin (2) is i:i:d:ð0;x2Þ, independent of et.Ifa¼1 and
x2>0, ytis an RCAR(1) process with mean unit root, which therefore is
called a stochastic unit root process by LMT and Granger and Swanson
(1997). If q<1 in (2), atis a stationary AR(1) process with a mean of unity,
while if q¼1, atis a random walk. Also note that if x2¼0, at¼1 for all t
and, consequently, ytfollows a random walk.
382 Bulletin
ÓBlackwell Publishers 2002
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