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A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks
WALTER ENDERS* and JUNSOO LEE**
* Department of Economics, Finance & Legal Studies, University of Alabama, Tuscaloosa, AL 35487-
0224, USA (email: wenders@cba.ua.edu)
** Department of Economics, Finance & Legal Studies, University of Alabama, Tuscaloosa, AL 35487-
0224, USA (email: 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.
JEL Classifications: C12, C22, E17
Keywords: Structural breaks, nonlinear models, Fourier approximation.
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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 paper 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 of the
break(s). We follow Becker, Enders and Lee (2006) and illustrate that the essential
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
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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 (exponential 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):
y
t
= d(t) +
y
t-1
+
·t +
t
(1)
where
t
is a stationary disturbance with variance
2
, and d(t) is a deterministic function of t.
We note that the initial value is assumed to be a fixed value, and
t
is weakly dependent. If the
functional form of d(t) is known, it is possible to estimate (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 mis-specified. Our test is based on the fact that it is often possible to
approximate d(t) using the Fourier expansion:
1
0
11
( ) sin(2 / ) cos(2 / ); / 2
nn
kk
kk
dt kt T kt T n T
(2)
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As 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.
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where n represents the number of cumulative frequencies contained in the approximation, k
represents a particular frequency, and T is the number of observations.
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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 n
is small so that equation (2) can be viewed as an application of Gallant’s (1981) flexible Fourier
form (FFF) to modeling 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, n should be small since it is important to allow the evolution of the nonlinear trend to
be gradual. There is little point in claiming that a series reverts to an arbitrarily evolving mean.
Finally, Becker, Enders and Hurn (2004) show that if the number of breaks is unknown, a test for
structural change that uses a single trigonometric component can have better power than the
well-cited Bai-Perron (1998) test.
One key issue for our test is whether a small number of frequency components can
replicate the types of breaks typically seen in economic data. To keep the problem tractable, we
begin by considering a Fourier approximation using a single frequency component, so that
d(t)
0
+
k
sin(2
kt/T) +
k
cos(2
kt/T), (3)
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When 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 n increases, the
tests lose power. As such, in finite samples, it is sufficient to treat n as a finite value (n << T), and the test
depends on n.
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where k represents the single frequency selected for the approximation, and
k
and
k
measure
the amplitude and displacement of the sinusoidal component of the deterministic term. Thus,
even with a single frequency (n = 1), we can allow for multiple smooth breaks.
Although (3) is especially suitable to mimicking smooth breaks, the solid lines in Panels
1 to 9 of Figure 1 show nine series for T = 500 containing various types of sharp and smooth
structural breaks. Panel 1 illustrates a temporary break, which often occurs empirically in macro
variables. Panels 2 and 3 allow for breaks in the intercept and slope of a trending series. As
indicated by Prodan (2008), the types of sharp breaks illustrated in Panels 1 through 3 are
difficult to detect using standard break-testing methodologies. In these cases, using smooth
breaks with our Fourier approximation can work better than using dummy variables. Series with
smooth breaks are shown in Panels 4 through 9. Panel 4 of the figure shows the following
LSTAR break with parameter values d
1
= 3,
= 0.05, T = 500, and
= 0.5:
y
t
= d
1
/[ 1 + exp(
(t
T)) ]. (4)
Panel 5 of the figure shows the same LSTAR break setting
= 0.75. Panel 6 shows the
following ESTAR break for d
1
= 3,
= 0.0003, T = 500, and
= 0.75:
y
t
= d
1
[ 1 exp(
(t
T)
2
) ]. (5)
Series with multiple smooth breaks are shown in Panels 7 through 9. Of course, the
essential features of all nine series are invariant to inverting their magnitudes or to reordering the
data from t = 500 to t = 1.
The dashed line (short dashes) in Panel 1, shows the time path of y
t
=
(t) + γt obtained
by setting k = 1,
0
= 1.26,
1
= 1.45,
1
= 1.24, and γ = –0.003. The values of
0
,
1
,
1
and γ
were selected by setting k = 1 and regressing y
t
on a constant, a linear time trend t, sin(2πkt/T)
and cos(2πkt/T). The sum of squared residuals (SSR = 302.62) was about one third of that
