Trend and Initial Condition in Stationarity Tests: The Asymptotic Analysis
| Author | Anton Skrobotov |
| Date | 01 April 2015 |
| DOI | http://doi.org/10.1111/obes.12057 |
| Published date | 01 April 2015 |
254
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 0305–9049
doi: 10.1111/obes.12057
Trend and Initial Condition in Stationarity Tests:The
Asymptotic Analysis*
Anton Skrobotov
Russian Presidential Academy of National Economy and Public Administration, Institute of
Applied Economic Research, 82, Vernadsky pr., 117571, Moscow, Russia
(e-mail: antonskrobotov@gmail.com)
Abstract
In this article, we investigate the behaviour of stationarity tests proposed by M¨uller [Jour-
nal of Econometrics (2005) Vol. 128, pp. 195–213] and Harris et al. [Econometric Theory
(2007) Vol. 23, pp. 355–363] with uncertainty over the trend and/or initial condition. As
different tests are efficient for different magnitudes of local trend and initial condition,
following Harvey et al. [Journal of Econometrics (2012) Vol. 169, pp. 188–195], we pro-
pose decision rule based on the rejection of null hypothesis for multiple tests. Additionally,
we propose a modification of this decision rule, relying on additional information about
the magnitudes of the local trend and/or the initial condition that is obtained through
pre-testing. The resulting modification has satisfactory size properties under both uncer-
tainty types.
I. Introduction
The influence of linear trend and/or the initial condition can be very important in unit root
testing. In recent works, Harvey, Leybourne andTaylor (2009) and Harvey, Leybourne and
Taylor (2012) [hereafter HLT, see also Harvey, Leybourne and Taylor (2008)] considered
the issue of deterministic trend inclusion in the unit root test and investigated the behaviour
of tests with different initial conditions. Harvey et al. (2009) showedthat under uncertainty
over the linear trend, the optimal unit root test is a simple union of rejections of the two
tests (i.e., the null hypothesis of the unit root is rejected, if it is rejected by at least one of the
tests): the first with inclusion of the linear trend, the second only with the constant. Both
of these tests need to be effective for their respective types of deterministic components in
the absence of large initial condition. Simultaneously, knowing the type of deterministic
components (i.e. whether the trend is present in the data or not) and uncertainty over the
initial condition, the union of rejection testing strategy of two tests (one effective at a
small initial condition and the second effective at a large initial condition for a given type
of deterministic component) will be the best. HLT extended the procedure by assuming
*We thank two anonymous referees, Vladimir Nosko and Marina Turuntseva for helpful comments and recom-
mendations for the improvement of an earlier version of this article.
JEL Classification numbers: C12, C22.
Trend and initial condition in stationarity tests 255
uncertainty over both the trend and initial condition and suggested a union of rejection
testing strategy for all four tests. Additionally, HLT also suggested a modification of this
test with pre-testing of the linear trend coefficient and the initial condition. Then, if there
was evidence of a large local trend and/or evidence of a largeinitial condition, it was more
likely that a trend and/or a large initial condition were actually present in the data. Thus,
this information can be used to construct the union of rejection testing strategy.
There is a need of a similar procedure for stationarity tests, as hypothesis testing
opposed to unit root is important for confirmatory analysis [see, e.g., Maddala and Kim
(1998, Ch. 4.6)]. Harris et al. (2007) (hereafter HLM) proposed a modification of the stan-
dard Kwiatkowski et al. (1992) test (hereafter KPSS) in the near integration using (quasi)
GLS-detrending.1Asymptotic properties of the test in the constant case were compared
with point-optimal test proposed by M¨uller (2005) for different initial conditions. The re-
sults revealed that in the case of small initial conditions, the test proposed in M¨uller (2005)
is effective. Given a large, or even moderate initial conditions, this test has serious liberal
size distortions, tending to unity for strongly autocorrelated stationary data generating
processes. Additionally, it is dominated by the HLM test with large initial conditions.
In this article, we consider the asymptotic properties of stationarity tests proposed by
HLM and M¨uller (2005) following the approach of HLT. In section II, we introduce the
HLM test and point-optimal test proposed in M¨uller (2005) and obtain corresponding
limiting distributions assuming local behaviour of the trend. Additionally, we use para-
metrization of the initial condition following the M¨uller and Elliott (2003) approach. In the
subsection ‘Asymptotic behaviour under a local trend’ in section III, weanalyze these tests
in the case of an asymptotically negligible initial condition, assuming the local behaviour
of the trend. As in HLM, in this case the asymptotic size curves showthat the point-optimal
tests of M¨uller (2005) is superior to the HLM test. Additionally, the test with only constant
has serious liberal size distortions, driven by increase of the magnitude of the local trend
parameter. We propose to use the intersection of rejections testing strategy2of two tests
with and without a trend in deterministic component, that is, we reject the null hypothesis,
if both tests simultaneously reject it. We also propose a modification of this decision rule,
using pre-testing of a trend parameter and using this information to perform only the test
with trend, if there is evidence of a large local trend. As simulations show, this procedure
has advantages over a simple intersection rejection of two test (with- and without-trend).
In the subsection ‘Asymptotic behaviour under various initial conditions’in section III, we
analyze a similar procedure, assuming knowledge of deterministic term, but no knowledge
of initial condition magnitude. In this case, as in the subsection ‘Asymptotic behaviour
under a local trend’ in section III, simple intersection of rejections of corresponding tests
is the best solution, as well as the modification with pre-testing of the initial condition.
In the subsection ‘Asymptotic behaviour under uncertainty over both the trend and initial
1M¨uller (2005) revealed that the use of conventional KPSS test with the bandwidth parameter in the long-run
variance estimator increasing at a slowerrate than the length of the sample, leads to an asymptotic size equal to unity
under the null hypothesis about near integration. Therefore, in our analysis (local to unit root), we do not consider
conventional OLS-detrended KPSS test.
2HLT used the ‘union of rejections’ term and their test rejected the null hypothesis, if at least one of the tests
rejected it. As weconsider stationarity tests in our procedure, we reject the null hypothesis if all of tests reject it, and
we call this strategy the ‘intersection of rejections’.
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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