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Applied Economics Letters
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Further results on the detection of changes in
persistence in linear time series
Steven Cook a
a
Department of Economics, University of Wales Swansea, Swansea, SA2 8PP
First Published: February 2007
To cite this Article: Cook, Steven (2007) 'Further results on the detection of changes
in persistence in linear time series', Applied Economics Letters, 14:2, 145 - 150
To link to this article: DOI: 10.1080/13504850500426053
URL: http://dx.doi.org/10.1080/13504850500426053
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Applied Economics Letters, 2007, 14, 145–150
Further results on the detection
of changes in persistence in linear
time series
Steven Cook
Department of Economics, University of Wales Swansea, Singleton Park,
Swansea, SA2 8PP
E-mail: [email protected]
Recent research concerning the properties of ‘change in persistence’ tests in
the presence of structural change is extended. It is found that the recently
proposed tests of Leybourne, Kim and Taylor (2004) are subject to severe
size distortion in the presence of both breaks in level and drift under the
unit root null hypothesis. The present analysis suggests that any results
obtained from these tests should be treated with caution to avoid drawing
a spurious inference of a change in persistence.
I. Introduction
Following the seminal study of Dickey and Fuller
(1979), a large literature has emerged examining the
issue of testing the unit root hypothesis in economics.
The diversity of this literature is illustrated by recent
research which has appeared in the companion
journal Applied Economics. For example, at a
theoretical level Patterson and Heravi (2003) have
employed response surface analysis to derive critical
values for the weighted symmetric unit root test of
Park and Fuller (1995), while Cook and Manning
(2003) have examined the properties of asymmetric
unit root tests under threshold and consistentthreshold estimation and Cook (2005) has examined
the robustness of the Dickey–Fuller test under
rank-based estimation, in the presence of structural
change. At an empirical level, the US macroeconomy
(Gil-Alana, 2004; Sen, 2004), purchasing power
parity (Narayan, 2005) and the dynamic behaviour
of inflation (Charemza et al., 2005) have all been the
subject of attention using a range of unit root testing
procedures. In this article, the ‘unit root literature’ in
the companion journal is complemented and
extended by considering the behaviour of tests for a
change in persistence in linear time series.
In recent research, Cook (2004) has examined the
properties of tests of a change of persistence in the
presence of structural change. In particular, the size
properties of the GLS-based tests of Leybourne et al.
(2003) were noted when applied to unit root processes
subject to breaks in level or drift. Interestingly,
the results obtained showed the tests of Leybourne
et al. (2003) to exhibit severe oversizing in the
presence of level breaks and severe undersizing in
the presence of breaks in drift. Therefore, use of the
above tests can lead to the spurious inference of a
change in persistence between I(0) and I(1) status
when structural change occurs in the level of an
I(1) series. More recently, Leybourne et al. (2004),
hereafter referred to as LKT, have extended the
analysis of changes in persistence by proposing a
testing framework which is not only consistent in the
presence of constant I(1) status, but also when
applied to series which remain I(0) throughout the
sample considered. In the present article, the finitesample properties of these more recent tests are
examined when applied to unit root processes
experiencing breaks in level or drift. The results
obtained are of particular interest for two reasons.
First, it is shown that both forms of break (level
and drift) result in severe oversizing of the LKT tests.
Applied Economics Letters ISSN 1350–4851 print/ISSN 1466–4291 online ß 2007 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/13504850500426053
145
S. Cook
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146
This contrasts with previously presented results for
the GLS-based tests of Leybourne et al. (2003)
where level breaks alone generate oversizing.
Second, the degree of oversizing is substantially
larger for the LKT tests than for the earlier GLSbased tests. In combination, these points show
that increased likelihood of conflating a structural
change with a change in persistence when using the
LKT tests.
This article will proceed as follows. Section II
outlines the LKT tests. Section III presents the
properties of the LKT tests in the presence of level
breaks, while Section IV presents analogous results
for breaks in drift. Section V concludes.
a change in persistence is then derived using the
following regression:
y~ dt ¼ ðÞy~dt1 þ "t ,
ydt ¼ ðÞydt1 þ "t ,
t ¼ 1, 2, . . . , ½T
ð1Þ
where the variable of interest (yt) is demeaned
or detrended as appropriate via preliminary regression upon an intercept or intercept and trend to
generate the revised series ydt . Given a sample of
T observations, a series of subsamples are
considered using the break fraction , where
2 ¼ ½0:2, 1. For each subsample, the t-ratio
^
associated with ðÞ
is calculated, these statistics
being denoted as DF f.1 As the date of the break is
unknown, the minimum of the calculated values
of DF f is taken as the test of persistence against H01
as below:
DF f
inf
inf DF f ðÞ
2
ð2Þ
To provide a test against a change in persistence
against in the opposite direction from I(1) to I(0)
status (H10), LKT follow the above approach using
reversed realizations of the series of interest yt. This
series is denoted as y~ t ð¼ yTtþ1 Þ. The resulting test of
ð3Þ
denotes the residuals obtained from the
where
regression of y~ t upon either an intercept or intercept
^
and trend. Again a series of t-ratios is derived for ðÞ
for the subsamples considered, these being denoted
as DF r(). The resulting test statistic is then given as
DF r inf where:
DF r
inf
inf DF r ðÞ
LKT note that the
diverge under the null
therefore propose a
which does not diverge
given as:
II. LKT Tests of a Change in Persistence
To test for the presence of a change in persistence,
LKT draw upon the approach of Banerjee et al.
(1992). To test the unit root hypothesis, denoted H1,
against an alternative of a change in persistence from
I(0) to I(1) status, denoted H01, LKT use the
following recursive regression:
t ¼ 1, 2, . . . , ½T
y~dt
2
ð4Þ
DF f inf and DF r inf tests
of stationarity (H0). LKT
further ratio-based test
under H0. This statistic is
DF f inf R ¼ r inf DF
ð5Þ
III. Level Breaks
To analyse the behaviour of the LKT tests in
the presence of breaks under the null, the data
generation processes (DGPs) of Cook (2004) are
employed. To analyse the properties of the tests
under breaks in level, the following DGP is
employed:2
yt ¼ st þ t ;
t ¼ 1, . . . , T
ð6Þ
t ¼ t1 þ t
pffiffiffiffi
¼k T
ð7Þ
t i:i:d: Nð0, 1Þ
0 for t TB
st ¼
1 for t > TB
ð9Þ
ð8Þ
ð10Þ
The error series {t} is generated using the RNDNS
procedure in the Gauss programming language with
all experiments performed over 10 000 replications.
Following the analysis of LKT, a sample size of
120 observations is considered (T ¼ 120).3 Breaks in
level are considered at all points in the sample, with
the time of the break given by TB ¼ {1, 2, . . . , 119}.
1
The superscript ‘f ’ is used to denote the use of the forward realizations of the series of interest. Later, reversed realizations of
the series will be employed, with the superscript ‘r’ used.
2
The following DGP was also employed by Leybourne et al. (1998) and Leybourne and Newbold (2000) to analyse the
properties of the Dickey–Fuller test and weighted symmetric Dickey–Fuller test of Park and Fuller (1995) in the presence of
breaks under the null.
3
Results for this single sample size are reported in the interests of brevity. Further similar results for alternative sample sizes
are available from the author upon request.
DF(f)
147
DF(r)
0.4
0.35
Rejection frequency
0.3
0.25
0.2
0.15
0.1
0.05
0
1
11
Fig. 1.
21
31
The DF f
DF(f)
inf
41
and DF r
inf
51
DF(r)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
11
21
31
41
51 61 71
Breakpoint
81
91
61
71
Breakpoint
81
91
101
111
tests in the presence of level breaks (k ^ 0.5)
0.9
Rejection frequency
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Further results on the detection of changes in persistence
101 111
Fig. 2. The DF f inf and DF r inf tests in the presence of level
breaks (k ^ 1.0)
With regard to the magnitude of the break, the
analysis follows Leybourne and Newbold (2000) with
two break sizes considered. In each case the size of
the break is dependent upon the sample size and k,
where k ¼ {0.5, 1.0}. Rejections under the above
DF f inf, DF r inf and R tests are noted across all
replications at the 5% nominal level of significance
with all tests performed in ‘intercept only’ form.
While rejections under the DF f inf and DFr inf tests
are calculated at the lower tail of the distribution,
rejections under the R test are calculated at both the
lower and upper tails of the distribution.
Given the large number of breakpoints considered,
all of the simulation results obtained are presented
graphically. Figures 1 and 2 present the observed
rejection frequencies for the DF f inf and DF r inf tests
in the presence of level breaks. From inspection of
these graphs it is clear that severe oversizing of the
DF f inf test is observed when a break occurs following
the initial observation in the sample period (TB ¼ 1),
with oversizing more apparent for the larger break in
Fig. 2 (k ¼ 1.0). However, as the timing of the break is
delayed, undersizing is observed, with an empirical
size of 2% observed when (k, TB) ¼ (1.0, 13). As the
break is further delayed, reversion to the nominal size
of 5% is noted. Similar reversed findings are observed
for the DF r inf test. Therefore, a very early break in
level results in a mistaken inference of a change in
persistence from I(0) to I(1) status using the DF r inf
test, while a very late break results in an opposite
change in persistence using the DF r inf test. To
complete the analysis in the presence of breaks in
level, the finite-sample rejection frequencies of the
R test are presented in Figs 3 and 4 for the smaller
and larger break respectively. It is apparent from
these graphs that the severe oversizing noted for
the individual DF f inf and DF r inf tests is reduced
when considering the ratio-based R test. However,
the test does exhibit oversizing in the upper (lower)
tail for very early (late) breaks. A secondary,
more moderate period of oversizing is noted in the
upper (lower) tail when the break occurs later
(early) in the sample due to previously noted undersizing for the individual tests. In summary, while
the size of the R test is not as severely inflated by
level breaks as the individual tests, it is still capable
of generating a spurious inference a change in
persistence.
S. Cook
R(upper)
R(upper)
R (lower)
0.6
Rejection frequency
0.25
0.2
0.15
0.1
0.5
0.4
0.3
0.2
0.1
0.05
0
0
1
Fig. 3.
R(lower)
0.7
0.3
Rejection frequency
11
21
31
41
51 61 71
Breakpoint
81
91
1
101 111
11
21
31
41
51
61
71
81
91
101 111
Breakpoint
The R test in the presence of level breaks (k ^ 1.0)
Fig. 4.
The R test in the presence of level breaks (k ^ 0.5)
DF(f)
DF(r)
0.12
0.1
Rejection frequency
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148
0.08
0.06
0.04
0.02
0
1
11
21
31
41
51
61
71
81
91
101
111
Breakpoint
Fig. 5.
The DF f
inf
and DF r
inf
tests in the presence of drift breaks (k ^ 10)
IV. Breaks in Drift
To analyse the properties of the above tests in the
presence of breaks in drift, the following DGP is
utilized:4
yt ¼ st þ yt1 þ t ,
ð12Þ
k
¼ pffiffiffiffi
T
ð13Þ
0
1
ð14Þ
4
ð11Þ
t i:i:d: Nð0, 1Þ
st ¼
The
now
t ¼ 1, . . . , T
for t TB
for t > TB
magnitude of the
determined by the
imposed break is
values k ¼ {10, 20}.
Empirical rejection frequencies at the 5% nominal
level of significance are now calculated for each of the
above tests following detrending of the series of
interest via preliminary regression upon an intercept
and trend.
In Figs 5 and 6 empirical rejection frequencies are
reported for the DF f inf and DF r inf tests for the
smaller and larger breaks (k ¼ 10, 20) respectively. It
can be seen that breaks relatively early in the sample
period (although not at the beginning) generate
oversizing of the DF f inf test, with oversizing positively related to the magnitude of the break.
To illustrate this, consider the maximum empirical
size of 59% observed for a break at TB ¼ 13. Given
the symmetry of the results for the DF f inf and DF r inf
tests, similar arguments can be presented for the
The treatment of initial conditions, method of random number generation, sample size, and number of replications and
discards for the break in drift experiments are the same as for the earlier level break experiments.
DF(f )
R(upper)
0.4
Rejection frequency
Rejection frequency
R(lower)
0.45
0.6
0.5
0.4
0.3
0.2
0.1
0
149
DF(r)
0.7
0.35
0.3
0.25
0.2
0.15
0.1
0.05
1
11
21
31
41
51
61
71
81
91 101 111
0
1
11
Breakpoint
Fig. 6. The DF f inf and DF r inf tests in the presence of drift
breaks (k ^ 20)
R(upper)
Fig. 8.
21
31
41
51 61 71
Breakpoint
81
91 101 111
The R test in the presence of drift breaks (k ^ 20)
while level breaks result in oversizing for the tests of
both Leybourne et al. (2003) and Leybourne et al.
(2004), the presently considered tests of the latter
have been found to exhibit more extensive oversizing.
The results of the current study therefore suggest that
any results obtained form the application of change
of persistence tests should be treated with caution as
spurious inferences may be drawn.
R(lower)
0.12
0.1
Rejection frequency
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Further results on the detection of changes in persistence
0.08
0.06
0.04
0.02
0
1
Fig. 7.
11
21
31
41
51 61 71
Breakpoint
81
91 101 111
The R test in the presence of drift breaks (k ^ 10)
DF r inf test with oversizing now observed for relatively late breaks. Figures 7 and 8 present the
properties of the R test. As with the results for level
breaks, it can be seen that again the ratio-based R test
exhibits less distortion than the individual tests.
However, the test is still over-rejects in the upper
(lower) tail oversized for breaks relatively early (late)
in the sample period, leading to a spurious inference
of a change in persistence.
V. Conclusion
In this article the finite-sample size properties of the
change in persistence test of Leybourne et al. (2004)
have been examined in the presence of structural
change under the null. The results obtained show that
both breaks in level and drift result in severe
oversizing of the tests, leading to the spurious
inference of a change in persistence. These findings
contrast with those for the alternative GLS-based
tests of Leybourne et al. (2003) where breaks in level
alone were found to lead to oversizing. In addition,
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