We show that the power of the KPSS-test against integration, as measured by divergence rates of the test statistic under the alternative, remains the same when residuals from an OLS-regression rather than true observations are used. This is in stark contrast to residual based tests of the null of integration in a cointegration setting, where power is drastically reduced when residuals are used. --cointegration,power,long memory,KPSS-Test

Krämer, Walter

Sibbertsen, Philipp

English

Research Papers in Economics

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Sibbertsen, Philipp; Krämer, Walter
Working Paper
The Power of the KPSS-Test for
Cointegration when Residuals are
Fractionally Integrated
Technical Report / Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten
Datenstrukturen, No. 2004,31
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Sibbertsen, Philipp; Krämer, Walter (2004) : The Power of the KPSS-Test
for Cointegration when Residuals are Fractionally Integrated, Technical Report / Universität
Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten Datenstrukturen, No. 2004,31, http://
hdl.handle.net/10419/22543The Power of the KPSS–Test for Cointegration when
Residuals are Fractionally Integrated1
by
Philipp Sibbertsen2 and Walter Kr¨ amer
Fachbereich Statistik, Universit¨ at Dortmund, D-44221 Dortmund, Germany
Version April 2004
Abstract
We show that the power of the KPSS-test against integration, as
measured by divergence rates of the test statistic under the alter-
native, remains the same when residuals from an OLS-regression
rather than true observations are used. This is in stark contrast
to residual based tests of the null of integration in a cointegration
setting, where power is drastically reduced when residuals are used.
JEL-numbers: C12, C32.
Keywords: cointegration, power, long memory, KPSS-Test.
1Research supported by Deutsche Forschungsgemeinschaft under SFB 475
2Corresponding author:
Phone: +49- 231- 755-3886, Fax: +49- 231- 755-5284
e-mail: sibberts@statistik.uni-dortmund.de
11 Introduction and Summary
This paper is concerned with the Kwiatkowski–Phillips–Schmidt–Shin (1992,
KPSS) procedure for testing the null hypothesis that there is at least one
cointegrating relationship among m I(1) variables zt1;:::;ztm. If all potential
cointegrating relationships are known, this simply amounts to applying the
standard KPSS–test to these relationships. If a cointegrating relationship has
to be estimated, one applies the KPSS–test to the residuals from this regres-
sion.
Lee and Schmidt (1996) show that the KPSS–test statistic diverges when the
residuals from the cointegrating relationship are I(d)–processes with ¡0:5 <
d < 0:5; Marmol (1997) shows that the KPSS–test is also consistent against
I(d)–processes with d ¸ 0:5. However, both Lee and Schmidt (1996) and Mar-
mol (1997) assume that the true cointegrating relationship is known.
The case where the cointegrating relationship has to be estimated has been
considered in great detail by Phillips and Ouliaris (1990), but only for tests
of the complementary null hypothesis that there is no cointegration. Below
we consider the null hypothesis that cointegration does exist. It emerges that
the divergence rates under the alternatives from Lee and Schmidt (1996) and
Marmol (1997) remain the same. We also prove that the divergence rate does
not depend on the order of the cointegrating regressors. This is a rather sur-
prising result, and in stark contrast to the case of residual based tests of the
null hyposthesis of no cointegration, where power is drastically reduced when
regression residuals rather than true cointegration errors are used (Kr¨ amer and
Marmol 2004).
22 Power under the alternative
We consider the regression model
yt = ¯
0zt + ut; (1)
where the m-vector zt is deﬁned by zt =
Pt
s=1 xs and xs » I(dX). This implies
that zt » I(1 + dX) and that all elements of the m-vector of regressors are
integrated to the same order. We then postulate that yt » I(1 + dX), but
that ut » I(du) where ¡1=2 < du < 1=2 < 1 + dX, implying what is called
fractional cointegration. We consider the null hypothesis that there is exactly
one cointegrating relationship.
The KPSS–procedure for testing this hypothesis is based on the statistic
ˆ ´ =
1
T 2
T X
t=1
S
2
t=s
2(`); (2)
where
St =
t X
i=1
ui ¡ ¯ u; (3)
s
2(`) =
1
T
T X
t=1
(ut ¡ ¯ u)
2 +
2
T
` X
s=1
ws`
T X
t=s+1
(ut ¡ ¯ u)(ut¡s ¡ ¯ u) (4)
and where ws` = 1 ¡ s
`+1 s2(`) is the Newey–West estimator of the long–run
variance of the residuals. Its consistency in our long–memory set up follows
from Hosking (1996) for ` = 0. Therefore, we restrict our attention to the case
` = 0 and disregard the estimation of the variance when computing the rates
of divergence below.
If the ut’s are known and ut » I(0), we have
ˆ ´
d ¡!
Z 1
0
V (r)
2dr; (5)
where V (r) is a standard Brownian Bridge (see Kwiatkowski et al. 1992, or
Lee and Schmidt 1996 for details). We consider the limiting distribution of ˆ ´
3when the residuals ut in (5) are I(d) with 0 < du < 1. For the case where the
ut’s are known, and 0 < du < 0:5, Lee and Schmidt (1996, Theorem 3) show
that
ˆ ´t = Op (T)
2du : (6)
For the case where du ¸ 0:5, Marmol (1997) shows that
ˆ ´t = Op(T); (7)
so that the KPSS–test is consistent against both stationary and nonstationary
long memory alternatives.
So far, however, it has remained an open problem whether the above diver-
gence rates carry over to the case where the ut’s are replaced by OLS–residuals.
We consider this case in what follows by computing the divergence rate of the
KPSS-test under fractional alternatives and allowing the regressors to be like-
wise fractionally integrated. This extends the usual situation of I(0) regressors.
But it can be shown that the structure of the regressors has no inﬂuence on
the divergence rate of the test statistic.
THEOREM: Assume that xt » I(dx); ¡1
2 < dx < 1
2 and ut » I(du); 0 <
du < 1 + dx and that the regressors are strictly exogenous. Then the rate of
divergence of the KPSS–test is of order Op(T 2du) irrespective of dx.
REMARK: In the case of endogenous regressors we have to assume also
dx +du > 0 to obtain the result. In the case of dX +du · 0 the rate of conver-
gence of ˆ ¯ is of order Op(T 1+2dX) and therefore the considerations below are
no longer valid. For details see Davidson (2003).
PROOF: We have:
ˆ u = u ¡ Z
0(Z
0Z)
¡1Z
0u: (8)
4From Davidson (2003) we obtain for the least squares estimator of the cointe-
gration parameter
T
1+dX¡du(ˆ ¯ ¡ ¯)
d ¡! M; (9)
with a matrix M which is speciﬁed in Davidson (2003). In addition we obtain
from Marinucci and Robinson (2000) that
T
¡3=2¡dX
[¸T] X
t=1
zt
d ¡! Bd(¸); (10)
where Bd(¸) denotes fractional Brownian motion with parameter d = dX + 1.
Using the relation
ˆ ut = ut ¡ (ˆ ¯ ¡ ¯)zt (11)
we obtain that T ¡1=2¡du P[¸T]
t=1 ut converges in distribution to some random
variable, say »(¸). The rate of convergence follows from (9), (10) and the
application of Slutsky’s Theorem to (11). The rate of convergence for the test
statistic follows from the continuous mapping theorem, applying the same
arguments as in Lee and Schmidt (1996). ²
REMARK: The limiting distribution in this case is also clearly not Gaussian.
Therefore, under the alternative of fractionally integrated cointegration resid-
uals we obtain divergence rates independent of the structure of the regressors
themselves. Furthermore, the limiting distribution is non-Gaussian. The KPSS-
test is thus powerful against fractional alternatives even when the cointegrating
relationship has to be estimated and whether the regressors themselves have
long memory or not.
5References
Davidson, J. (2003) : ”Convergence to stochastic integrals with fractionally
integrated integrator processes: Theory and applications to cointegrating
regression.” Working Paper, Cardiﬀ Business School, Cardiﬀ.
Hosking, J. (1996) : ”Asymptotic distribution of the sample mean, autoco-
variances and autocorrelations of long-memory time series.” Journal of
Econometrics 73, 261 – 284.
Kr¨ amer, W. and Marmol, F. (2004): ”The power of residual-based tests
for cointegration when residuals are fractionally integrated.” Economics
Letters 82, 63 – 69.
Kwiatkowski, D.; Phillips, P.C.B.; Schmidt, P. and Shin, Y. (1992):
”Testing the null hypothesis of stationarity against the alternative of a
unit root: How sure are we that economic time series have a unit root?”
Journal of Econometrics 54, 159 – 178.
Lee, D. and Schmidt, P. (1996): ”On the power of the KPSS test of sta-
tionarity against fractionally integrated alternatives.” Journal of Econo-
metrics 73, 285 – 302.
Marinucci, D. and Robinson, P. M. (2000): ”Weak convergence of mul-
tivariate fractional processes.” Stochastic Processes and their applications
86, 103 – 120.
Marmol, F. (1997): ”Searching for fractional evidence using unit root
tests.” Mimeo, Universidad Carlos III, Madrid.
Phillips, P.C.B. and Ouliaris, S. (1990): ”Asymptotic
properties of residual based tests for cointegration.” Econometrica 58,
165 – 193.
6Phillips, P.C.B. and Perron, P. (1988) : ”Testing for a unit root in time
series regression.” Biometrika 75, 335 – 346.
7

The Power of the KPSS-Test for Cointegration when Residuals are Fractionally Integrated

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The Power of the KPSS-Test for Cointegration when Residuals are Fractionally Integrated

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