Testing Stationarity in Small‐ and Medium‐Sized Samples when Disturbances are Serially Correlated*

Published date01 October 2011
DOIhttp://doi.org/10.1111/j.1468-0084.2010.00620.x
Date01 October 2011
669
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2011. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 73, 5 (2011) 0305-9049
doi: 10.1111/j.1468-0084.2010.00620.x
Testing Stationarity in Small- and Medium-Sized
Samples when Disturbances are Serially CorrelatedÅ
Kristian J ¨
onsson
Sveriges Riksbank, SE-103 37 Stockholm, Sweden (e-mail: kristian.jonsson@riksbank.se)
Abstract
In this article, we study the size distortions of the KPSS test for stationarity when serial
correlation is present and samples are small- and medium-sized. It is argued that two
distinct sources of the size distortions can be identied. The rst source is the nite-
sample distribution of the long-run variance estimator used in the KPSS test, while the
second source of the size distortions is the serial correlation not captured by the long-run
variance estimator because of a too narrow choice of truncation lag parameter. When the
relative importance of the two sources is studied, it is found that the size of the KPSS test
can be reasonably well controlled if the nite-sample distribution of the KPSS test statistic,
conditional on the time-series dimension and the truncation lag parameter, is used. Hence,
nite-sample critical values, which can be applied to reduce the size distortions of the
KPSS test, are supplied. When the power of the test is studied, it is found that the price
paid for the increased size control is a lower raw power against a non-stationary alternative
hypothesis.
I. Introduction
During the last three decades, one of the most investigated branches of econometrics in
general, and time-series econometrics in particular, is that of unit root, stationarity and
cointegration testing. The seminal contributions of Dickey and Fuller (1979), Engle and
Granger (1987) and Kwiatkowski et al. (1992) have all made their mark on applied eco-
nomics and are used as central tools for investigating various economic questions. When
using the test of Kwiatkowski et al. (1992), the so-called KPSS test, to test if a series is
stationary, I(0), against the alternative that the series contains a unit root, is I(1), several
ÅThe author would like to thank Anindya Banerjee, three anonymous referees, TommyAndersson, David Edger-
ton, Thomas Elger, Klas Fregert, Rolf Larsson, Johan Lyhagen, Joakim Westerlund and seminar participants at the
Department of Economics, Lund University, for useful suggestions and discussions on the topics covered in, and
relating to, this paper. The simulations in the current paper were carried out in Gauss on the LUNARC cluster.
The author acknowledges valuable technical support from Lars M Johansson. Financial support, from The Crafo-
ord Foundation, The Royal Swedish Academy of Sciences and The Jan Wallander and Tom Hedelius Foundation,
research grant number P2005-0117:1, is gratefully acknowledged. The views expressed in this paper are solely the
responsibility of the author and should not to be interpreted as reecting the views of the Executive Board of Sveriges
Riksbank.
JEL Classication numbers: C12, C14, C15, C22, E21.
670 Bulletin
implementations of the test, all with attractive asymptotic behaviour, are available.1The
implementations differ (most commonly) in their estimation of the so-called long-run vari-
ance. The alternatives available involve the use of various kernels to estimate the long-run
variance under the null hypothesis (see, e.g. Hobijn, Franses and Ooms, 2004), the use of
automatic procedures for the selection of the truncation lag or bandwidth parameter (see,
e.g. Hobijn et al., 2004; Carrion-i-Silvestre and Sans´o, 2006) and the application of a pre-
whitening lter in the long-run variance estimation (see, e.g. Sul, Phillips and Choi, 2005;
Carrion-i-Silvestre and Sans´o, 2006). Given that the choices made full certain regularity
conditions, the asymptotic distribution of the KPSS test statistic is the same regardless of
what choices that are made. The appropriateness of the various implementations depends
on how well the asymptotic approximation works for the specic sample size at hand.
In empirical applications, where it can be of interest to employ the KPSS test,
sample sizes are always limited and in addition often small. This applies especially
when postwar macroeconomic time series are investigated. Hence, when applied to inves-
tigate economic questions, the performance of the KPSS test relies to a large extent on
how well the nite-sample distribution of the test statistic corresponds to the asymptotic
distribution. Unfortunately, when serial dependence is present under the null hypothesis of
stationarity, the asymptotic approximation can be poor, which causes problems relating to
the size and power of the KPSS test (see, e.g. Lee, 1996; Caner and Kilian, 2001; Hobijn
et al., 2004; M¨uller, 2005). Methods to mitigate the size distortions within the framework
of Kwiatkowski et al. (1992) have been suggested by Hobijn et al. (2004), Sul et al. (2005)
and Carrion-i-Silvestre and Sans´o (2006). However, a common feature among these sug-
gestions is that their performance is investigated for rather large samples sizes, and that the
suggested remedies may be inappropriate in small-sample situations (see, e.g., J¨onsson,
2006). Hence, the performance of the KPSS test in small samples, when serial dependency
is allowed for, is still largely unknown and possibly open for improvement.
In this article, it is shown that the KPSS test for stationarity can be grossly oversized
in small samples when serial correlation is allowed for. When the sources of these size
distortions are studied, it is found that the long-run variance estimator is the main reason
for the size distortion, while the actual serial correlation of the data series exerts little
inuence on the test once the truncation lag has been accounted for.A natural suggestion is
then to proceed along the lines of Cheung, Chinn and Tran (1995) and Hornok and Larsson
(2000) and supply nite-sample critical values for the KPSS test.
Within the unit root testing framework of Dickey and Fuller (1979) and Said and
Dickey (1984), the use of small-sample critical values is well established. To control the
size of the so-called augmented Dickey–Fuller (ADF) test, Cheung and Lai (1995) suggest
that critical values should be obtained from simulating time series that are generated as
pure random walks, that is, unit root processes with serially independent disturbances,
and applying the ADF test to these time series. The authors argue that obtaining critical
values for the ADF test under the assumption that errors are serially independent intro-
duces nuisance parameters into the distribution of the test statistic when this assumption
is violated and disturbances display serial dependence. However, it is also argued that the
1By having the hypothesis that a time series is I(0) as the null hypothesis, the KPSS test differs from, for example,
the Dickey and Fuller (1979) test which has as null hypothesis that the series under consideration is I(1). Hence, the
KPSS test and the Dickey and Fuller (1979) tests can be considered as being complements to each other.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011

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