A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks*
| Author | Walter Enders,Junsoo Lee |
| Date | 01 August 2012 |
| DOI | http://doi.org/10.1111/j.1468-0084.2011.00662.x |
| Published date | 01 August 2012 |
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©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011. Published by Blackwell Publishing Ltd,
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 74, 4 (2012) 0305-9049
doi: 10.1111/j.1468-0084.2011.00662.x
A Unit Root Test Using a Fourier Series to
Approximate Smooth BreaksÅ
Walter Enders and Junsoo Lee
Department of Economics, Finance & Legal Studies, University of Alabama, Tuscaloosa, AL
35487-0224, USA (e-mails: wenders@cba.ua.edu; jlee@cba.ua.edu)
Abstract
We develop a unit-root test based on a simple variant of Gallant’s (1981) flexible Fourier
form. The test relies on the fact that a series with several smooth structural breaks can often
be approximated using the low frequency components of a Fourier expansion. Hence, it is
possible to test for a unit root without having to model the precise form of the break. Our
unit-root test employing Fourier approximation has good size and power for the types of
breaks often used in economic analysis. The appropriate use of the test is illustrated using
several interest rate spreads.
I. Introduction
As shown in Perron (1989), traditional unit-root tests lose power if structural breaks present
in the data-generating process are ignored. If the break date is known, these unit-root tests
can be modified by including dummy variables to capture changes in the level and trend.
Typically, structural breaks in a series are assumed to occur instantaneously and manifest
themselves contemporaneously. However, a number of authors have recognized that the
effects of structural change on the level or slope of a series can be gradual. For example,
Leybourne, Newbold and Vougas (1998) and Kapetanios, Shin and Snell (2003) develop
unit-root tests such that the deterministic component of the series is a smooth transition
process. To properly use this type of unit-root test, it must be assumed that there is a single
gradual break with a known break date and functional form. However, the break dates and
the number of breaks are likely to be unknown. The existing literature assumes, a priori,
the presence of only one or two structural breaks in the series in question. Although it is
possible to allow for more breaks, such tests are not powerful, as many parameters need
to be estimated. As such, it does not seem fruitful to develop a new test for the purpose of
capturing multiple structural breaks with unknown break dates.
The aim of this article is to develop a unit-root test that can be used in the presence of a
small number of smooth breaks in the deterministic components of a series. Specifically,
we use a variant of Gallant’s (1981) flexible Fourier form to control for the unknown nature
ÅThe authors wish to thank Ralf Becker, Mark Wohar, Kyung-so Im, Michael McCracken and Mark Strazicich
for their helpful comments.
JEL Classification numbers: C12, C22, E17.
A unit root test using a Fourier series 575
of the break(s). We follow Becker, Enders and Lee (2006) and illustrate that the essen-
tial characteristics of a series containing a small number of structural breaks can often be
captured using the low frequency components of a Fourier approximation. A key feature
of the approximation is that we do not need to assume that the break dates, the precise
number of breaks, and/or the exact form of the breaks are known a priori. Moreover, the
Fourier approximation can reduce the need to estimate a large number of parameters and,
hence, results in a test with good size and power properties. The test is designed to work
when breaks are gradual and we show that it has good size and power properties in the
presence of either LSTAR (logistic smooth transition autoregressive) or ESTAR (expo-
nential smooth transition autoregressive) breaks. Nevertheless, we show that our test can
perform reasonably well in the presence of sharp breaks. The appropriate use of our test is
illustrated using several interest rate spreads.
II. Approximating a nonlinear trend with a Fourier series
A simple modification of the Dickey–Fuller (DF) type test is to allow the deterministic
term to be a time-dependent function denoted by d(t):
yt=d(t)+yt−1+·t+t(1)
where tis a stationary disturbance with variance 2
, and d(t) is a deterministic function
of t. Wenote that the initial value is assumed to be a fixed value, and tis weakly dependent.
If the functional form of d(t) is known, it is possible to estimate equation (1) directly and
to test the null hypothesis of a unit root (i.e. =1). When the form of d(t) is unknown,
any test for =1 is problematic if d(t) is misspecified. Our test is based on the fact that it
is often possible to approximate d(t) using the Fourier expansion:1
d(t)=0+
n
k=1
ksin(2kt/T )+
n
k=1
kcos(2kt/T ); n≤T/ 2(2)
where nrepresents the number of cumulative frequencies contained in the approximation,
krepresents a particular frequency, and Tis the number of observations.2
In the absence of a nonlinear trend, all values of k=k=0 so that the standard
Dickey–Fuller specification emerges as a special case. There are several reasons why
it is inadvisable to use a large value for n. As we demonstrate below,the presence of many
frequency components uses degrees of freedom and can lead to an over-fitting problem.
As such, we keep the value of nis small so that equation (2) can be viewed as an appli-
cation of Gallant’s (1981) flexible Fourier form (FFF) to modelling d(t). As evidenced by
Gallant (1981), Davies (1987), Gallant and Souza (1991) and Bierens (1997), a Fourier
approximation using a small number of frequency components can oftentimes capture the
essential characteristics of an unknown functional form. Moreover, nshould be small since
1As indicated in Becker, Enders and Hurn (2004), structural change can be captured by the relatively low frequency
components of a series since breaks shift the spectral density function towards zero. Becker et al. also show that the
higher frequency components of a series are most likely to be associated with stochastic parameter variation.
2When the sample size gets very large, it will be natural to expect that the number of frequencies (n) will also
increase accordingly.In the limit, we may let n=n(T)→∞as T→∞. However, as nincreases, the tests lose power.
As such, in finite samples, it is sufficient to treat nas a finite value (nT), and the test depends on n.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011
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