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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Testing for Shifts in Trend With an Integrated or
Stationary Noise Component
Pierre Perron & Tomoyoshi Yabu
To cite this article: Pierre Perron & Tomoyoshi Yabu (2009) Testing for Shifts in Trend With an
Integrated or Stationary Noise Component, Journal of Business & Economic Statistics, 27:3,
369-396, DOI: 10.1198/jbes.2009.07268
To link to this article: http://dx.doi.org/10.1198/jbes.2009.07268

Published online: 01 Jan 2012.

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Testing for Shifts in Trend With an Integrated or
Stationary Noise Component
Pierre P ERRON
Department of Economics, Boston University, Boston, MA 02215 (perron@bu.edu)

Tomoyoshi YABU

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Faculty of Business and Commerce, Keio University, Tokyo 108-8345, Japan (tyabu@fbc.keio.ac.jp)
We consider testing for structural changes in the trend function of a time series without any prior knowledge of whether the noise component is stationary or integrated. Following Perron and Yabu (2009), we
consider a quasi-feasible generalized least squares procedure that uses a super-efficient estimate of the
sum of the autoregressive parameters α when α = 1. This allows tests of basically the same size with

stationary or integrated noise regardless of whether the break is known or unknown, provided that the
Exp functional of Andrews and Ploberger (1994) is used in the latter case. To improve the finite-sample
properties, we use the bias-corrected version of the estimate of α proposed by Roy and Fuller (2001). Our
procedure has a power function close to that attainable if we knew the true value of α in many cases. We
also discuss the extension to the case of multiple breaks.
KEY WORDS: Generalized least squares procedure; Median-unbiased estimate; Structural change;
Super-efficient estimate; Unit root.

1. INTRODUCTION
This article considers the problem of testing for structural
changes in the trend function of a univariate time series without
any prior knowledge as to whether the noise component is stationary or contains an autoregressive (AR) unit root. This problem is of practical importance for several reasons. For example, one often is interested in testing whether the rate of growth
of some macroeconomic variables, such as real GDP, exhibits
a structural change. With data in logarithmic form, the coefficient on the trend component represents this average growth
rate. In many cases, however, we have no prior knowledge about
whether the noise component can be perceived as being stationary, or I(0), or as having an AR unit root, or I(1). Performing
a structural change test using first-differenced data or growth
rates, as is commonly done, is tantamount to assuming an I(1)
noise component and leads to tests with very poor properties
when the series has an I(0) noise component. But performing a

structural change test on the slope of a linear trend function for
the level of the data entails different limit distributions in both
cases.
On the other hand, to get information about whether the noise
component is I(0) or I(1), it is useful to have information about
whether or not a structural change is present, at least in the unknown break date. If the break date is known, as was considered by Perron (1989), then we can have unit root tests with no
knowledge about the presence or absence of a break, because it
is possible to have tests that are invariant to the parameters of
the trend function with the inclusion of the appropriate dummy
variables. Things are rather different in the case where the break
date is unknown, where the usual tests, suggested by Zivot and
Andrews (1992) and others, are no longer invariant to the magnitude of change if present (see Vogelsang and Perron 1998 and
Perron 2006 for an extensive discussion). In the presence of a
change in trend, both the size and the power of commonly used
tests can be affected by a change in slope or intercept. Devising

unit root tests with good power requires information about the
presence or absence of a change (see Kim and Perron 2009).
The way to solve this circular problem is to have tests for
structural changes in level and/or slope that are valid whether

the noise component is I(0) or I(1), the object of our study. Various works are related to the problem tackled here. For example, Perron (1991) considered extensions of Gardner’s (1969)
test for a change in the coefficients of a polynomial trend function. The limit is different in the I(0) and I(1) cases, but the
normalization to obtain a nondegenerate limit distribution is the
same when accounting for correlation in the noise component
using a parametric AR approximation. One strategy is then to
use the set of largest critical values across the I(0) and I(1)
cases. But this test suffers from important problems of nonmonotonicity in power; that is, the power function decreases as
the magnitude of the change increases. Perron (1991) showed
that applying the test to first-differenced data can mitigate this
problem when testing for a change in slope. Vogelsang (1997)
considered a similar strategy but using a Wald-type procedure
and showed that it has superior power. But in both cases the
critical values are substantially smaller in the I(0) case than in
the I(1) case, indicating that the tests are quite conservative in
the I(0) case. The solution is then to devise a procedure that
has the same limit distribution in both the I(0) and I(1) cases.
Vogelsang (2001) was the first to provide such a solution in the
context of a change in the coefficients of a linear trend function, building on previous work related to hypothesis testing on
the coefficients of a polynomial time trend that he reported previously (Vogelsang 1998). He considered a setup in which the
correlation is accounted for by a parametric AR approximation,

so that the Wald test has a nondegenerate limit distribution in

369

© 2009 American Statistical Association
Journal of Business & Economic Statistics
July 2009, Vol. 27, No. 3
DOI: 10.1198/jbes.2009.07268

Downloaded by [Universitas Maritim Raja Ali Haji] at 17:28 12 January 2016

370

both the I(0) and I(1) cases. The novelty is that he weighed the
statistic by a unit root test scaled by some parameters. For any
given significance level, a value of this scaling parameter can be
chosen so that the asymptotic critical values will be the same.
But simulations showed that the test has little power in the I(1)
case, so he resorted to advocating the joint use of that test and
a normalized Wald test that has good properties in the I(1) case

but otherwise has very little power in the I(0) case.
We propose a new approach that builds on the work of Perron
and Yabu (2009), who analyzed the problem of hypothesis testing on the slope coefficient of a linear trend model when no
information about the nature of the noise component, I(0) or
I(1), is available. It is based on a quasi-feasible generalized
least squares (FGLS) approach that uses a super-efficient estimate of the sum of the AR parameters α when α = 1. The
estimate of α is the ordinary least squares (OLS) estimate obtained from an autoregression applied to detrended data and is
truncated to take a value of 1 when the estimate is in a T −δ
neighborhood of 1. This makes the estimate “super-efficient”
when α = 1 and implies that inference on the slope parameter
can be performed using the standard normal or chi-squared distribution whether α = 1 or |α| < 1. Theoretical arguments and
simulation evidence demonstrate that δ = 1/2 is the appropriate
choice.
We extend the analysis of Perron and Yabu (2009) to the case
of testing for changes in level or slope of the trend function of a
univariate time series. When the break date is known, things are
relatively straightforward, because their asymptotic results apply directly; in particular, the standard Wald test from the GLS
regression with the truncated estimate of α has a chi-squared
limit distribution irrespective of the nature of the noise components, that is, with I(0) or I(1) errors. Our analysis demonstrates that in this case the procedure has good finite-sample
properties and a power function close to that attainable with the

infeasible GLS procedure that uses the true value of α. When
the break dates are unknown, the limit distributions of the Exp,
Mean, and Sup functionals of the Wald test across all permissible breaks dates (see Andrews and Ploberger 1994) are no
longer the same in the I(0) and I(1) cases. It turns out, however,
that the limit distribution is nearly the same when considering
the Exp functional; thus it is possible to have tests with nearly
the same size in both cases. To improve the finite-sample properties of the test, we also use a bias-corrected version of the
OLS estimate of α (obtained from an autoregression based on
the residuals from estimating the parameters of the trend function by OLS), as suggested by Roy and Fuller (2001). This allows for a testing procedure that has good size and power properties in finite samples.
Because the only other procedure that delivers a testing procedure valid with both I(0) and I(1) errors is that of Vogelsang
(2001), we make extensive comparisons of the finite-sample
power of both set of tests. We show that our procedure is more
powerful and also has a power function close to what would be
attainable if we knew the true value of α in many cases.
The article is organized as follows. Section 2 considers the
basic case where the noise component has an AR(1) structure
and there is a single break. This allows a presentation of the
main ingredients and properties of our suggested procedure.
Section 3 presents an evaluation of the finite-sample properties of our tests. Section 4 extends the analysis to the case of a

Journal of Business & Economic Statistics, July 2009

general correlation structure for the noise component, and Section 5 does so to the case of more than one break. Section 6
presents empirical applications related to real GDP series for a
variety of counties. Section 7 offers brief concluding remarks,
and the Appendix presents some technical derivations.
2.

THE AR(1) CASE

We start with the case of a single break where the noise component is an AR(1), to highlight the main issues involved. Extensions to the general case are presented in Sections 4 and 5.
The data-generating process for a scalar random variable yt is
assumed to be
yt = x′t  + ut ,

(1)

ut = αut−1 + et
for t = 1, . . . , T, where et ∼ iid(0, σ 2 ), xt is a (r × 1) vector of
deterministic components, and  is a (r × 1) vector of unknown

parameters, which are model-specific and defined later. For
simplicity, we let u0 be some finite constant. Here −1 < α ≤ 1,
so that both stationary and integrated errors are allowed.
We are interested in testing the null hypothesis R = γ ,
where R is a (q × r) full-rank matrix and γ is a (q × 1) vector, where q is the number of restrictions. The restrictions to be
considered pertain to testing whether or not a structural change
in a trend function occurs. Thus we consider three models involving a change of intercept and/or slope in a trend function.
Throughout, the break date is denoted by T1 = [λ1 T] for some
λ1 ∈ (0, 1), where [·] denotes the largest integer that is less than
or equal to the argument and 1(·) is the indicator function.
Model I (Structural change in intercept). Here xt = (1,
DU t , t)′ and  = (μ0 , μ1 , β0 )′ , where DU t = 1(t > T1 ). This
specification allows for one-time change in the intercept. The
hypothesis of interest is μ1 = 0.
Model II (Structural change in slope). Here xt = (1, t, DT t )′
and  = (μ0 , β0 , β1 )′ , where DT t = 1(t > T1 )(t − T1 ). This
specification allows for a one-time change in the slope of the
trend without a change in level, so that the trend function is
joined at the time of break. The hypothesis of interest is β1 = 0.
Model III (Structural change in both intercept and slope).

Here xt = (1, DU t , t, DT t )′ and  = (μ0 , μ1 , β0 , β1 )′ . This
specification allows for a simultaneous change in the intercept
and slope. The hypothesis of interest is μ1 = β1 = 0.
2.1 The Feasible Generalized Least Squares Estimate
We first consider the case where the break date T1 is known;
we return to the unknown break date case in Section 2.4. Suppose that the AR coefficient α is known. The GLS estimate of
the parameters is then obtained by applying OLS to the regression
(1 − αL)yt = (1 − αL)x′t  + (1 − αL)ut
for t = 2, . . . , T, together with y1 = x′1  + u1 . It is well known
that in such a case, the Wald statistic for testing the null hypothesis, R = γ , is asymptotically distributed as a chi-squared
random variable for any values of α. In practice, however, α

Perron and Yabu: Testing for Shifts in Trend With an Integrated or Stationary Noise Component

is unknown, and using an estimate can make things very different. Consider the FGLS estimate of  that uses the following
estimate of α:

T
T


α̂ =
ût ût−1
û2t−1 ,
(2)

Perron and Yabu (2009) proved that (a) T 1/2 (α̂S − α) →d
N(0, 1 − α 2 ) when |α| < 1 and (b) T(α̂S − 1) →p 0 when α =
1. Thus κ, the limit of T(α̂S − 1), is zero when the errors are
I(1). The following theorem (the proof of which is given in the
Appendix) shows that tests based on the FGLS procedure using
α̂S have a χ 2 (q) distribution in both the I(0) and I(1) cases.

where {ût } are the OLS residuals from the regression of yt on xt .
When |α| < 1, T 1/2 (α̂ − α) →d N(0, 1 − α 2 ), and the Wald statistic obtained from the FGLS procedure still has a χ 2 (q) limit
distribution. Things are different when α = 1. From standard
results,
 1
 1
T(α̂ − 1) ⇒
W ∗ (r) dW(r)
W ∗ (r)2 dr ≡ κ,

ˆ be the FGLS estimate of  using α̂S , that
Theorem 1. Let 
α̂
α̂
is, the OLS estimate from a regression of yt S on xt S , where
α̂S
α̂S
yt = (1 − α̂S L)yt and xt = (1 − α̂S L)xt for t = 2, . . . , T,
α̂
α̂
y1 S = y1 , and x1 S = x1 . Denote the residuals associated with
this regression by êt . The Wald statistic for testing the null hypothesis R = γ is

t=2

t=2

0

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371

0

where W ∗ (r), 0 ≤ r ≤ 1 is the continuous-time residual function from a projection of a Wiener process W(r) on the
continuous-time version of the deterministic components [e.g.,
{1, 1(r > λ1 ), r} for model I]. The limit of the Wald statistic
constructed from the FGLS regression no longer has a chisquared limit distribution. To illustrate the problem, consider
the simple case with xt = DT t and  = β1 . The limit distribution of the t-statistic for testing β1 = 0 is given by (see the
Appendix for details):

 1
tF ⇒ W(1) − W(λ1 ) − κ
W(r) dr
λ1

−κ



1

λ1

(r − λ1 ) dW(r) − κ



1

(r − λ1 )W(r) dr

λ1



[(1 − λ1 ) − κ(1 − λ1 )2 + κ 2 (1 − λ1 )3 /3]1/2 ,


(3)

which differs from a standard Normal distribution. Accordingly, the limit of the Wald statistic, WF = tF2 , is no longer chisquared distributed.
Suppose now that κ, the limit of T(α̂ − 1), is zero; then
tF ⇒ [W(1) − W(λ1 )]/(1 − λ1 )1/2 =d N(0, 1), and thus WF ⇒
[W(1) − W(λ1 )]2 /(1 − λ1 ) =d χ 2 (1). We would then recover
the same limiting distribution in the unit root case as in the stationary case, and no discontinuity would be present. Perron and
Yabu (2009) exploited this fact to derive a testing procedure
with the same limit distribution in both the I(0) and I(1) cases.
2.2 A Super-Efficient Estimate When α = 1
Perron and Yabu (2009) proposed using the following superefficient estimate of α:

α̂ if T δ |α̂ − 1| > d
α̂S =
(4)
1 if T δ |α̂ − 1| ≤ d
for δ ∈ (0, 1) and d > 0. Thus when α̂ is in a T −δ neighborhood
of 1, it is assigned a value of 1. Our use of a super-efficient estimate when α = 1 warrants some comments. As is well known,
such estimates have better properties only on a set of measure
zero (here α = 1) and have less precision in a neighborhood of
this value. This is not a problem here, because α is a nuisance
parameter in the context of our testing procedure, which pertains to the coefficients of the trend function. Thus having α̂S
less efficient than α̂ for values of α close to 1 does not imply
that the tests will be badly behaved, as we will show.

ˆ − )]′ [s2 R(X′ X)−1 R′ ]−1 [R(
ˆ − )], (5)
WFS (λ1 ) = [R(
α̂

where X = {xt S } and s2 = T −1
WFS (λ1 )
 
⇒ R

1

T

2
t=1 êt .

F(r, λ1 )F(r, λ1 )′ dr

Then if |α| < 1,

−1 

1

′
F(r, λ1 ) dW(r)

R

′

0

0

 
× R

1

 
× R

1

′

F(r, λ1 )F(r, λ1 ) dr

0
′

F(r, λ1 )F(r, λ1 ) dr

0
d

−1

−1 

−1

0

1


F(r, λ1 ) dW(r)

2

≡ G0 (λ1 ) = χ (q),
where F(r, λ1 ) = [1, 1(r > λ1 ), r]′ for model I, [1, r, 1(r >
λ1 )(r − λ1 ))]′ for model II, and [1, 1(r > λ1 ), r, 1(r > λ1 )(r −
λ1 )]′ for model III. If α = 1, then
⎧
e2[λ1 T]+1
⎪
⎪
⎪
for model I
lim
⎪
⎪
T→∞
σ2
⎪
⎪
⎪
⎪
⎨ [λ1 W(1) − W(λ1 )]2
for model II
WFS (λ1 ) ⇒
λ1 (1 − λ1 )
⎪
⎪
⎪
2
2
⎪
⎪
⎪ lim e[λ1 T]+1 + [λ1 W(1) − W(λ1 )]
⎪
⎪
⎪
λ1 (1 − λ1 )
⎩ T→∞ σ 2
for model III
≡ G1 (λ1 ).
The limit distribution is χ 2 (q) for model II and, if the errors are
Normally distributed, for models I and III as well.
Therefore, constructing the GLS regression with the truncated estimate, α̂S , effectively bridges the gap between the I(0)
and I(1) cases, and the chi-squared asymptotic distribution is
obtained in both cases.
Remark 1. The limit chi-squared distribution requires the
assumption of Normality of the errors for models I and III
to ensure that limT→∞ e2[λ1 T]+1 /σ 2 ∼ χ 2 (1). This is unavoidable, because when using a GLS-type procedure when α = 1, a
change in level becomes a single outlier. For tests of hypotheses that do not involve the change in intercept, the Normality
assumption is not required. See also the discussion in Remark 5
(Sec. 2.4).

372

Journal of Business & Economic Statistics, July 2009

2.3 The Case With α Local to Unity

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The result obtained in Theorem 1 is pointwise in α for
−1 < α ≤ 1 and does not hold uniformly, particularly in a local neighborhood of 1, although, using the results of Mikusheva
(2007), we could show that our test has an asymptotic size no
greater than the stated nominal size uniformly over |α| < 1.
Thus it is of interest to see what happens when the true value
of α is close to but not equal to 1. Adopting the standard local
to a unity approach, we have the following result, the proof of
which is given in the Appendix.
Theorem 2. Suppose that α = 1 + c/T; then
⎧
⎪
e2[λ1 T]+1
⎪
⎪
for model I
lim
⎪
⎪
⎪
T→∞
σ2
⎪
⎪
⎪
⎪
⎨ [λ1 Jc (1) − Jc (λ1 )]2
for model II
WFS (λ1 ) ⇒
λ1 (1 − λ1 )
⎪
⎪
⎪
⎪
e2[λ1 T]+1 [λ1 Jc (1) − Jc (λ1 )]2
⎪
⎪
⎪
lim
+
⎪
2
⎪
λ1 (1 − λ1 )
⎪
⎩ T→∞ σ
for model III
≡ Gc (λ1 ),
r
where Jc (r) = 0 exp(c(r − s)) dW(s) ∼ N(0, (exp(2cr) −
1)/2c).
The results are fairly intuitive. Because the true value of α
is in a T −1 neighborhood of 1 and α̂S truncates the values of α̂
in a T −δ neighborhood of 1 for some 0 < δ < 1 (i.e., a larger
neighborhood), in sufficiently large samples, α̂S = 1. Thus the
estimator of  is essentially the same as the first-difference estimator. Let us consider the case of model II. The first-difference
estimator of β1 is
T
−1
t=1 (DU t − T (T − T1 )) ut
β̂1FD = 
.
T
−1
2
t=1 (DU t − T (T − T1 ))
Then the Wald statistic has the following limit distribution:


[ Tt=1 DU t ut − T −1 (T − T1 ) Tt=1 ut ]2
WFD =

s2 Tt=1 (DU t − T −1 (T − T1 ))2
=

[T −1/2 (uT − uT1 ) − T −1 (T − T1 )T −1/2 (uT − u0 )]2

s2 T −1 Tt=1 (DU t − T −1 (T − T1 ))2

[λ1 Jc (1) − Jc (λ1 )]2
λ1 (1 − λ1 )

because T −1 Tt=1 (DU t − T −1 (T − T1 ))2 → λ1 (1 − λ1 ),
T −1/2 u[rT] ⇒ σ Jc (r), T −1/2 u0 = op (1), and s2 = σ 2 + op (1).
This limit distribution is the same as that in Theorem 2 for
model II.
Note that when c = 0, we recover the result of Theorem 1
for the I(1) case. But for models II and III, when c < 0, the
variance of [λ1 Jc (1) − Jc (λ1 )] is different from λ1 (1 − λ1 ), and,
accordingly, the upper quantiles of the limit distributions are
smaller than those of a χ 2 (q), so that, without modifications,
a conservative test may be expected for values of α close to 1,
relative to the sample size. Also, we note that the limit of the
variance of the component [λ1 Jc (1) − Jc (λ1 )] is 0 as c → −∞,
and we do not recover the same result that applies to the I(0)
case. As reported by Phillips and Lee (1996), the local to unity
⇒

asymptotic framework with c → −∞ involves a doubly infinite
triangular array such that the limit of the statistic depends on the
relative approach to infinity of c and T.
To address this issue and provide some guidelines on choosing the truncation parameter δ, Perron and Yabu (2009) analyzed the properties of the OLS (β̂ ols ), first-difference (β̂ FD ),
and FGLS (β̂ F ) estimates of the slope coefficient in a linear trend model using a local asymptotic framework in which
c is allowed to increase with the sample size. The specification is that c = bT 1−h for some 0 < h < 1 and b < 0; thus
α = αT = 1 + b/T h . When h approaches 1, we get back to the
local to unity case, and when h approaches 0, we get closer
to the fixed |α| < 1 case. Note that the FGLS estimate based
on the truncated estimate α̂S , β̂ FS , is such that β̂ FS = β̂ FD
when truncation applies and β̂ FS = β̂ F when it does not apply.
Consider first the issue of the relative efficiency of the estimates for different increasing sequences for c, that is, different values of h. When h < 1/2, β̂ F is as efficient as β̂ ols , and
both are more efficient than β̂ FD . This is consistent with the results of Phillips and Lee (1996), who considered the case with
α = 1 + c/T known and showed that to establish the equivalence between the GLS and OLS estimate as c increases, we
must consider values of c such that c2 /T is large. On the other
hand, β̂ F is more efficient than β̂ ols when h > 1/2. The FGLS
estimate β̂ F is more efficient than β̂ FD for all 0 < h < 1. Thus,
if the goal were to maximize power, then we would never truncate. So we must rely on size considerations, because a welldocumented fact is that the size distortions of tβF increase as α
approaches 1 if we are using the standard Normal critical values. To minimize these size distortions, we need to truncate.
Thus the issue is when to stop truncating. Perron and Yabu
(2009) recommended doing so when the region for which no
truncation applies is also the region for which we attain the asymptotic equivalence between β̂ F and β̂ ols . This occurs when
h < 1/2, which suggests that we should set δ = 1/2, so that
truncation applies only when h > 1/2. The rationale for not
truncating more is that when h < 1/2, β̂ F is as efficient as
β̂ ols , and we are then in a region that can be labeled stationary, and we can expect the standard Normal distribution theory
to be a good approximation. The obvious price to pay is that the
test will be conservative when 1/2 < h < 1 (in a neighborhood
close to 1), but as c increases further and we get away from the
nonstationary region, we effectively ensure a test with a limiting N(0, 1) distribution and estimates that are as efficient as the
OLS, thereby bridging the gap between the nonstationary and
stationary regions. Indeed, simulations reported by Perron and
Yabu (2009) demonstrate that this value leads to a procedure
that works best in small samples. The same considerations apply in the current context with breaks. We also verified by simulations that δ = 1/2 is the best choice for the models considered
here. Thus we continue to use this value and will calibrate the
appropriate value of d using simulations.
Remark 2. It is important to note that in model I, the limit
distribution does not depend on c, and in this case we can expect
the tests to be unaffected by size distortions for values of α close
to but not equal to 1. Simulation results will show that this is in
fact the case.

Perron and Yabu: Testing for Shifts in Trend With an Integrated or Stationary Noise Component

2.4 The Case With an Unknown Break Date
In general, the break date is unknown, and the analysis
must be extended accordingly. Following Andrews (1993) and
Andrews and Ploberger (1994), we consider the following three
functionals of the Wald test for different break dates:

Mean-WFS = T −1
WFS (λ′1 ),







1
′
−1
exp WFS (λ1 ) ,
Exp-WFS = log T
2

(6)




sup-WFS = sup WFS (λ′1 ),



Downloaded by [Universitas Maritim Raja Ali Haji] at 17:28 12 January 2016


= {λ′1 ; ε

≤ λ′1

≤ 1 − ε} for some ε > 0. Here T1′ (resp.
where
′
λ1 ) denotes a generic break date (resp. break fraction) used to
construct a particular value of the Wald test, whereas T1 (resp.
λ1 ) will continue to denote the true break date (resp. break fraction). Given that the regressors are trending and the errors may
have a unit root, none of these functionals need have any optimal properties in our case. Nevertheless, given that they have
some optimality properties in the stationary case, they are worth
considering. Their limit distributions are a simple consequence
of Theorem 1, stated in the following corollary.
Corollary 1. Let g(λ′1 ) denote G0 (λ′1 ) and G1 (λ′1 ) for the
I(0) and I(1) cases, as defined in Theorem 1. Then

Mean-WFS ⇒
g(λ′1 ) dλ′1 ,



Exp-WFS ⇒ log






1
exp g(λ′1 ) dλ′1 ,
2



373

Accordingly, the limit distributions of the three tests differ in
the I(0) and I(1) cases. Table 1 presents the asymptotic critical
values with a value for the trimming parameter ε = 0.01 and
0.15. These were calculated by simulations using iid N(0, 1)
random variables to approximate the Wiener process. The integrals are approximated by normalized sums with 2000 steps,
and 10,000 replications are used. What transpires from these results is that although the limit distributions are different, the relevant quantiles are very similar for the I(0) and I(1) cases when
considering the Exp functional. For example, for model II, the
5% critical values with ε = 0.01 are 1.97 for the I(0) case and
2.02 for the I(1) case. Thus taking the larger critical value, corresponding to 2.02, would be expected to bring a powerful robust statistic for both stationary and integrated errors.
Table 2 presents an extended grid of critical values for the
Exp version of the test, which also includes a wider range of
values for the trimming parameter ε. From this table, we can
deduce the asymptotic size of the test in both cases when the
larger critical value is used. For example, for model I, when doing a 5% test, the critical values are largest for the I(0) case,
and in the I(1) case, the asymptotic size varies between 0.045
for ε = 0.25 and 0.035 for ε = 0.10. For model II, the critical
values are largest in the I(1) case, but in the I(0) case, the asymptotic size is between 0.045 and 0.05 for all values of ε. For
model III, the relevant critical values correspond to those in the
I(1) case, in the I(0) case, the size is between 0.035 and 0.045.
The discrepancies for other significance levels can be obtained
from the table. Thus the minimum value of the size for the I(0)
and I(1) cases is close to 5%, implying tests that are not very
conservative, which is a useful feature to get decent power.
Remark 3. We computed the limit distribution of the test
Exp-WFS under a sequence of local to unity alternatives (i.e.,
αT = 1 + c/T for c < 0) and verified that, as in the known break
date case, the test is conservative. Details are omitted.

sup-WFS ⇒ sup g(λ′1 ).



Here things are not as simple as in the known break date case.
Even though the distributions of G0 (λ′1 ) and G1 (λ′1 ) are both
chi-squared for any fixed λ′1 , the two functions are different.

Remark 4. We may be interested in using model III, and
test only whether a shift in slope is present (β1 with μ1 unrestricted), that is, allow the possibility of both a shift in intercept

Table 1. Asymptotic distributions for one break occurring at an unknown date
ε = 0.01
sup-WFS
%

ε = 0.15

Mean-WFS

Exp-WFS

sup-WFS

Mean-WFS

Exp-WFS

I(0)

I(1)

I(0)

I(1)

I(0)

I(1)

I(0)

I(1)

I(0)

I(1)

I(0)

I(1)

Model I
0.900
0.950
0.975
0.990

9.99
11.64
13.08
15.03

∞
∞
∞
∞

1.72
2.11
2.50
3.04

0.98
0.98
0.98
0.98

1.59
2.07
2.60
3.33

1.60
1.92
2.32
2.99

8.96
10.60
12.17
14.30

∞
∞
∞
∞

1.31
1.64
1.98
2.47

0.70
0.70
0.70
0.70

1.22
1.74
2.30
3.12

1.26
1.58
1.99
2.64

Model II
0.900
0.950
0.975
0.990

6.38
7.78
9.19
11.00

8.86
10.42
11.94
13.97

2.12
2.84
3.59
4.54

1.93
2.49
3.06
3.90

1.45
1.97
2.55
3.30

1.52
2.02
2.57
3.37

4.91
6.27
7.63
9.33

7.14
8.68
10.24
12.18

1.66
2.28
2.92
3.82

1.50
2.01
2.54
3.19

1.07
1.61
2.18
2.97

1.13
1.67
2.26
3.06

Model III
0.900
0.950
0.975
0.990

12.79
14.52
16.13
18.13

∞
∞
∞
∞

3.36
4.16
4.96
6.11

2.91
3.47
4.04
4.88

2.68
3.34
3.94
4.67

2.96
3.55
4.15
5.02

11.11
12.86
14.38
16.66

∞
∞
∞
∞

2.58
3.17
3.83
4.70

2.20
2.71
3.24
3.89

2.25
2.84
3.52
4.35

2.48
3.12
3.75
4.47

374

Journal of Business & Economic Statistics, July 2009

Table 2a. Asymptotic distribution of the Exp test: model I

Downloaded by [Universitas Maritim Raja Ali Haji] at 17:28 12 January 2016

ε = 0.01

ε = 0.05

ε = 0.10

ε = 0.15

Table 2c. Asymptotic distribution of the Exp test: model III

ε = 0.25

ε = 0.01

ε = 0.05

ε = 0.10

ε = 0.15

ε = 0.25

%

I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1)

%

I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1)

0.900
0.905
0.910
0.915
0.920
0.925
0.930
0.935
0.940
0.945
0.950
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995

1.59
1.60
1.62
1.66
1.71
1.76
1.81
1.85
1.92
1.98
2.07
2.14
2.22
2.33
2.44
2.60
2.74
3.02
3.33
3.91

0.900
0.905
0.910
0.915
0.920
0.925
0.930
0.935
0.940
0.945
0.950
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995

2.68
2.70
2.74
2.81
2.85
2.92
2.99
3.05
3.12
3.22
3.34
3.37
3.50
3.63
3.76
3.94
4.13
4.44
4.67
5.55

1.60
1.61
1.62
1.65
1.67
1.70
1.74
1.78
1.81
1.86
1.92
1.97
2.04
2.11
2.23
2.32
2.50
2.81
2.99
3.62

1.47
1.51
1.56
1.59
1.63
1.67
1.71
1.75
1.82
1.89
1.97
2.05
2.16
2.25
2.40
2.53
2.69
2.90
3.24
3.84

1.52
1.55
1.57
1.59
1.62
1.65
1.69
1.72
1.76
1.82
1.86
1.92
1.99
2.07
2.16
2.29
2.41
2.60
2.81
3.26

1.33
1.36
1.40
1.44
1.50
1.55
1.59
1.66
1.72
1.80
1.88
1.96
2.05
2.18
2.30
2.45
2.60
2.79
3.05
3.60

1.41
1.43
1.45
1.46
1.49
1.52
1.55
1.58
1.62
1.66
1.70
1.76
1.82
1.88
1.98
2.07
2.19
2.39
2.67
3.29

1.22
1.26
1.30
1.35
1.40
1.46
1.50
1.56
1.61
1.69
1.74
1.85
1.93
2.02
2.14
2.30
2.50
2.74
3.12
3.55

1.26
1.28
1.30
1.34
1.36
1.40
1.43
1.47
1.51
1.56
1.58
1.67
1.73
1.81
1.88
1.99
2.14
2.36
2.64
3.28

0.83
0.86
0.90
0.93
0.98
1.03
1.09
1.15
1.20
1.28
1.33
1.41
1.51
1.61
1.73
1.92
2.11
2.35
2.83
3.33

0.91
0.93
0.96
0.98
1.01
1.04
1.08
1.11
1.16
1.21
1.26
1.33
1.40
1.49
1.58
1.68
1.81
2.01
2.32
2.91

and slope but only test if the latter is present. In that case, the
limit distribution in the I(1) case is the same as in model II.
In the I(0) case, it is different but very close to that in the I(1)
case, indeed even closer than for model II. Thus we can still use
the critical values corresponding to model II.
Remark 5. The limit distribution of the Exp-WFS is still sensitive to the Normality assumption on the errors when α = 1
in models I and III that have a change in level. Size distortions
occur when the distribution of the errors have fat tails so that
Table 2b. Asymptotic distribution of the Exp test: model II
ε = 0.01

ε = 0.05

ε = 0.10

ε = 0.15

ε = 0.25

%

I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1)

0.900
0.905
0.910
0.915
0.920
0.925
0.930
0.935
0.940
0.945
0.950
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995

1.45
1.47
1.51
1.55
1.60
1.64
1.70
1.76
1.84
1.89
1.97
2.06
2.19
2.30
2.41
2.55
2.74
3.00
3.30
3.88

1.52
1.58
1.62
1.65
1.70
1.74
1.80
1.86
1.92
1.98
2.02
2.10
2.19
2.29
2.46
2.57
2.82
3.09
3.37
4.01

1.34
1.38
1.42
1.47
1.53
1.58
1.64
1.70
1.76
1.84
1.90
1.98
2.07
2.19
2.29
2.42
2.61
2.83
3.07
3.61

1.40
1.45
1.49
1.53
1.57
1.62
1.68
1.73
1.78
1.86
1.93
2.01
2.10
2.23
2.36
2.51
2.71
2.98
3.27
3.95

1.20
1.24
1.29
1.32
1.37
1.43
1.47
1.54
1.61
1.68
1.75
1.85
1.94
2.06
2.19
2.34
2.56
2.79
3.20
3.80

1.28
1.33
1.36
1.41
1.46
1.51
1.57
1.62
1.69
1.78
1.86
1.93
2.05
2.14
2.24
2.39
2.55
2.81
3.18
3.68

1.07
1.11
1.15
1.19
1.24
1.30
1.34
1.39
1.47
1.54
1.61
1.71
1.80
1.91
2.06
2.18
2.42
2.64
2.97
3.62

1.13
1.19
1.23
1.27
1.32
1.37
1.44
1.49
1.53
1.59
1.67
1.73
1.80
1.91
2.10
2.26
2.39
2.61
3.06
3.46

0.71
0.74
0.79
0.83
0.87
0.92
0.97
1.03
1.11
1.19
1.25
1.32
1.40
1.52
1.65
1.81
1.97
2.20
2.60
3.28

0.74
0.79
0.84
0.89
0.93
0.98
1.02
1.08
1.14
1.22
1.28
1.37
1.46
1.59
1.73
1.84
2.02
2.24
2.61
3.27

2.96
2.98
3.00
3.02
3.07
3.12
3.17
3.23
3.32
3.40
3.55
3.60
3.66
3.77
3.93
4.15
4.28
4.54
5.02
5.72

2.51
2.56
2.60
2.65
2.71
2.75
2.82
2.89
2.97
3.04
3.12
3.23
3.36
3.46
3.63
3.83
4.06
4.39
4.78
5.42

2.82
2.86
2.90
2.94
3.00
3.05
3.11
3.17
3.24
3.30
3.36
3.46
3.57
3.68
3.81
3.99
4.15
4.42
4.76
5.42

2.35
2.39
2.45
2.50
2.56
2.62
2.68
2.74
2.79
2.88
2.98
3.10
3.22
3.33
3.48
3.67
3.86
4.11
4.57
5.15

2.65
2.70
2.74
2.79
2.83
2.87
2.93
2.98
3.03
3.10
3.16
3.25
3.34
3.44
3.59
3.77
4.00
4.29
4.59
5.16

2.25
2.27
2.31
2.36
2.40
2.46
2.53
2.59
2.66
2.74
2.84
2.92
3.03
3.14
3.30
3.52
3.62
3.92
4.35
5.02

2.48
2.53
2.57
2.63
2.67
2.76
2.83
2.90
2.97
3.05
3.12
3.20
3.32
3.47
3.60
3.75
3.96
4.23
4.47
5.25

1.86
1.90
1.96
2.01
2.06
2.11
2.18
2.26
2.35
2.43
2.50
2.62
2.76
2.89
3.06
3.24
3.46
3.73
4.04
4.78

2.15
2.20
2.25
2.29
2.35
2.41
2.47
2.54
2.62
2.69
2.79
2.88
3.01
3.15
3.31
3.50
3.69
4.06
4.57
5.58

outliers can be present. We do not consider this a defect of our
suggested procedure. In the unit root case, levels shifts induced
by exogenous events or by outliers are basically observationally equivalent. Thus if a test procedure is to have power for detecting exogenous level shifts, then it also must have increased
probability of rejection when outliers occur. As we document
later, the testing procedure of Vogelsang (2001) is such that the
limit and finite-sample distributions are not sensitive to errors
with a fat-tailed distribution; consequently, however, it also has
no power for detecting exogenous level shifts. Thus either we
reject for neither case or we reject for both cases. There seems
to be no way around this problem. We prefer a procedure to indicate a rejection even if it is due to an outlier, because indeed
such outliers acts as important level shifts.
The results also demonstrate that the Mean and Sup versions
of the test are of little use in our context. Given the quite large
discrepancies between the two sets of critical values, these tests
would imply substantial size distortions in either the I(0) or I(1)
case, depending on which case has the largest critical value.
Some of the results in Table 1 can be explained using theoretical arguments. First, for model I, the limit of the Mean test in the
I(1) case is (1 − 2ε), or 0.98 for ε = 0.01 and 0.7 for ε = 0.15.
The reason for this is that in large samples, WFS (λ′1 ) is equiva
lent to e2T ′ +1 /σ 2 and thus the Mean test is T −1
WFS (λ′1 ) =
1

(1 − 2ε)[(σ 2 (1 − 2ε)T)−1
e2T ′ +1 ] + op (1) →p (1 − 2ε),
1
using the fact 
that, from a standard law of large numbers,
[(1 − 2ε)T]−1
e2T ′ +1 →p σ 2 . To explain why for the I(1)
1
case the Sup version diverges in model I (the explanation is
similar for model III), we again use the fact that in large samples, WFS (λ′1 ) is equivalent to e2T ′ +1 /σ 2 . Thus the sup sta1

tistic is sup
e2T ′ +1 /σ 2 = max
{e2T ′ +1 /σ 2 }. The probability
1

1

of sup-WFS < η is Pr(
∩ {e2T ′ +1 /σ 2 < η}) = Pr(e2t /σ 2 <
1

Perron and Yabu: Testing for Shifts in Trend With an Integrated or Stationary Noise Component

η)(1−2ε)T using the assumption of independence; therefore,
Pr(sup-WFS < η) → 0 as T → ∞ if Pr(e2t /σ 2 < η) < 1, which
is the case for any η < ∞. Thus sup-WFS diverges to infinite as
T → ∞.

Downloaded by [Universitas Maritim Raja Ali Haji] at 17:28 12 January 2016

2.5 Some Modifications to Improve
Finite-Sample Properties
It is well known that the OLS estimate of α is biased downward, especially when α is near 1. Thus in many cases, no
truncation may apply when some truncation would be desirable. Perron and Yabu (2009) recommended using a medianunbiased estimate. In the current context, however, obtaining an
exactly median unbiased estimate is computationally very demanding, especially in the unknown break date case and more
so when a more general AR(p) structure for the noise component is entertained (as we do in Sec. 4). An alternative approach with similar finite-sample properties is the bias correction proposed by Roy and Fuller (2001), which we adopt here.
We consider that, based on the OLS estimate, Roy, Falk, and
Fuller (2004) and Perron and Yabu (2009) used a similar biascorrected estimate but based on a weighted symmetric least
squares estimate of α instead of the OLS estimate used here;
both lead to tests with similar properties. It is a function of a
unit root test, namely the t-ratio τ̂ = (α̂ − 1)/σ̂α , where α̂ is
the OLS estimate and σ̂α is its standard deviation. The biascorrected estimate is given by
α̂M = α̂ + C(τ̂ )σ̂α ,
⎧
−τ̂ if τ̂ > τpct
⎪
⎪
⎪
⎪
⎪
Ip T −1 τ̂ − (1 + r)[τ̂ + c2 (τ̂ + a)]−1
⎪
⎪
⎪
⎨
if − a < τ̂ ≤ τpct
C(τ̂ ) = I T −1 τ̂ − (1 + r)τ̂ −1
p
⎪
⎪
⎪
1/2
⎪
⎪
if − c1 < τ̂ ≤ −a
⎪
⎪
⎪
⎩
1/2
0
if τ̂ ≤ −c1 ,

(7)

where c1 = (1 + r)T, with r the number of parameters estimated
in the trend function [i.e., the number of elements in the vec2 (I + T)][τ (a +
tor  from model (1)], c2 = [(1 + r)T − τpct
p
pct
−1
τpct )(Ip + T)] , a is some constant, and τpct is a percentile
of the limit distribution of τ̂ when α = 1. The relevant limit
distribution is that pertaining to the known break date case, because the correction is applied to each given break fractions
T1′ /T, and, accordingly, the critical values are those given by
Perron (1989). We simulated the percentage points for each values of T1′ /T, but we could use the values in the tables of Perron
(1989) with interpolation. In addition, Ip = [(p + 1)/2], where
p is the order of the AR process considered for the noise component (here 1, but different in the generalizations considered
in Sec. 4). The parameters for which specific values need to
be selected are τpct and a. Based on extensive simulation experiments, we selected a = 10, because it leads to tests with
better properties, and for τpct , we use τ0.95 for the known break
date case and τ0.99 for the unknown break date case (see Yabu
2005 for details). Also, again based on extensive simulations,
we found that the value d = 1 for the truncation leads to the
best results in finite samples. Thus our suggested procedure involves the following steps:

375

1. For any given break date, detrend the data by OLS to obtain residuals, say ût .
2. Estimate an AR(1) for ût yielding the estimate α̂ and the
t-ratio τ̂ .
3. Use α̂ and τ̂ to get the Roy and Fuller (2001) biased corrected estimates α̂M .
4. Apply the truncation

α̂M if |α̂M − 1| > T −1/2
α̂MS =
1
if |α̂M − 1| ≤ T −1/2 .
5. Apply a GLS procedure with α̂MS to obtain the estimates
of the coefficients of the trend and the estimate of the
variance of the residuals and construct the standard Waldstatistic, which we denote by WFMS .
6. When dealing with the case of an unknown break date,
repeat steps 1–5 for all permissible break dates and construct the Exp-Wald statistic, denoted by Exp-WFMS .
Remark 6. Using the biased-corrected versions, α̂M , instead
of the OLS estimates does not change the stated large-sample
results (Theorems 1 and Corollary 1). All that is needed for
these asymptotic results to hold is that T(α̂M −1) = Op (1) when
α = 1, and T 1/2 (α̂M − α) →d N(0, 1 − α 2 ) when |α| < 1. These
conditions are satisfied.
3. SIMULATION EVIDENCE
In this section we assess the finite-sample properties of the
procedure. We first consider the size of the tests. For this, we
generate the data through a simple AR(1) process of the form
yt = αyt−1 + et , with et ∼ iid N(0, 1) and y0 = 0. (Setting the
constant and trend parameters to zero is without loss of generality, because the tests are invariant to them.) The nominal
size of the tests is 5% throughout. We consider the following
sample sizes: T = 100, 250, 500 for a known break date and
T = 100, 250 for an unknown break date. The value of the trimming parameter for the Exp-WFMS test in the unknown break
date case is set to ε = 0.01. The numbers of replications is 5,000
for the known break date case and 2000 for the unknown break
date case. Using these specifications, we evaluate the size of
the tests for values of α in the range of [0, 1] with increments
of 0.05.
Figure 1 presents the results. In the case of a known break
date, the size of the test is very close to 5% for all values of the
AR parameters in all cases considered even when T = 100. As
expected, the test is slightly conservative for models II and III
when α is close to but not equal to 1. In the unknown break date
case, the test shows some liberal size distortions when T = 100
and a change in intercept is involved (models I and III). But
these distortions are virtually eliminated when T = 250, except
when α is close to but not equal to 1, where the test is somewhat
conservative, as expected.
We now consider the power of the tests. The specifications
are the same, except that the data are generated by the following
processes:
• For model I: yt = ηDU t + ut
• For model II: yt = ηDT t + ut
• For model III: yt = η(10DU t + DT t ) + ut .

Journal of Business & Economic Statistics, July 2009

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376

Figure 1. Finite sample size of WFMS test.

Here ut = αut−1 + et with et ∼ iid N(0, 1) and u0 = 0. The
break date is set to T1 = [0.5T].
We compare the power of our test with that of three other
tests: the Wald test based on the infeasible GLS estimates,
which uses the true values of α and T1 , as well as the T −1 WT
and PST tests of Vogelsang (2001) and the Exp version in the

unknown break case. We also experimented with the tests of
Sayginsoy and Vogelsang (2004). Our simulations revealed that
it is somewhat better than the test of Vogelsang (2001) for
model I. For model II, it is not better than the T −1 WT when
α is close to 1, and neither has better power than PST when α
is not close to 1. In addition, their test is not applicable to the

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known break date case or to model III with an unknown break
date. Thus here we report only comparisons with the original
tests of Vogelsang (2001). For his tests, we use a 5% nominal
size, and the comparisons should be made assuming that they
are applied independently. The power curves are plotted for
α = 1.0, 0.95, 0.90, or 0.80 and a range of values of η > 0. We
report raw non–size-adjusted power. With size-adjusted power,
the qualitative results do not change appreciably, and all of the
conclusions continue to hold. Note that our test is constructed
assuming that the order of the autoregression is known to be 1.
Thus the power may be overstated compared with what can be
achieved in practice. However, as we show in Section 4.3, our
test continues to have good properties when no such knowledge
is assumed and a nonparametric correction is applied.
First, consider Figure 2, which presents the results for
model I in the case of a known break date for T = 100. For
any values of α, our test is as powerful as that based on the
infeasible GLS regression that uses the true value of α. The
test also is substantially more powerful than those of Vogelsang
(2001). These results are quite remarkable, because our test
(like any other) is inconsistent when α is local to 1 (which is
the case for the values of α considered), because the intercept
shift is then basically only an outlier. But an inconsistent test
still can have decent power in finite samples. As expected, however, power does not increase with the sample size but increases
only with the magnitude of the change; accordingly, the results
for T = 250 and 500 are basically identical and are not reported
here. The results for the case of an unknown break date, presented in Figure 3 for T = 100, yield similar conclusions, with

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the exception that the power of our test is somewhat lower than
what can be achieved with the infeasible GLS procedure. Still,
power rises rapidly with the magnitude of the change.
Now consider model II. The power functions for the known
break date case are presented in Figure 4, for T = 100, 250,
and 500. When α = 1, our test is as powerful as that based on
the infeasible GLS estimates for any T. When α = 0.8, 0.9, or
0.95 and T = 100, the power of our test is lower, due largely to
the fact that in this case the size is conservative. Nonetheless,
it offers important power improvements over Vogelsang’s tests.
As T increases, the power function of our test becomes closer to
that of the infeasible GLS test. For instance, the power functions
are equivalent when T = 500 and α = 0.8 or 0.9. The power
functions for the unknown break date case are presented in Figure 5, for T = 100 and 250. The results are qualitatively similar,
with the power functions being lower, as expected. Moreover,
our test no longer globally dominates both the T −1 W and PST
tests of Vogelsang (2001), though individually each of these
tests has very poor power for some values of α. Figure 6 for
the known break date case and Figure 7 for the unknown break
date case show that similar features hold for model III.
Remark 7. Our test is superior to Vogelsang’s PST test
in the unit root case is because it is defined by PST =
T −1/2 Wz exp(−b∗ JT ), where Wz is the appropriate Wald test
constructed from a version of Equation (1) in which the variables enter in the partial sum form. The component JT is some
unit root test that converges to zero when the errors are I(0) and
to some nondegenerate limit distribution when the errors are

Figure 2. Finite sample power, model I, known break date, T = 100.

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Journal of Business & Economic Statistics, July 2009

Figure 3. Finite sample power of the Exp tests, model I, unknown break date, T = 100.

Figure 4a. Finite sample power, model II, known break date, T = 100.

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Perron and Yabu: Testing for Shifts in Trend With an Integrated or Stationary Noise Component

Figure 4b. Finite sample power, model II, known break date, T = 250.

Figure 4c. Finite sample power, model II, known break date, T = 500.

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Journal of Business & Economic Statistics, July 2009

Figure 5a. Finite sample power of the Exp tests, model II, unknown break date, T = 100.

Figure 5b. Finite sample power of the Exp tests, model II, unknown break date, T = 250.

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Perron and Yabu: Testing for Shifts in Trend With an Integrated or Stationary Noise Component

Figure 6a. Finite sample power, model III, known break date, T = 100.

Figure 6b. Finite sample power, model III, known break date, T = 250.

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Journal of Business & Economic Statistics, July 2009

Figure 6c. Finite sample power, model III, known break date, T = 500.

Figure 7a. Finite sample power of the Exp tests, model III, unknown break date, T = 100.

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Figure 7b. Finite sample power of the Exp tests, model III, unknown break date, T = 250.

I(1). The scaling b∗ is chosen such that the limit distribution
of the PST test is the same in the I(1) and I(0) cases. Now the
original T −1/2 Wz has good properties in both cases. Because
PST is equivalent in large sample to T −1/2 Wz in the I(0) case,
the PST test accordingly has good properties when the errors
are I(0). But things are different in the I(1) case, because the
PST test is t