Maximum Eigenvalue Test for Seasonal Cointegrating Ranks*
| DOI | http://doi.org/10.1111/j.1468-0084.2006.00174.x |
| Author | Sinsup Cho,Sung K. Ahn,Byeongchan Seong |
| Published date | 01 August 2006 |
| Date | 01 August 2006 |
Maximum Eigenvalue Test for Seasonal
Cointegrating Ranks*
Byeongchan Seong,Sinsup Choand Sung K. Ahn
Department of Management and Operations, Washington State University, Pullman,
WA 99164-4736, USA (e-mail: bcseong@wsu.edu; ahn@wsu.edu)
Department of Statistics, Seoul National University, Seoul 151-747, Korea
(e-mail: sinsup@snu.ac.kr)
Abstract
The maximum eigenvalue (ME) test for seasonal cointegrating ranks is
presented using the approach of Cubadda [Oxford Bulletin of Economics and
Statistics (2001), Vol. 63, pp. 497–511], which is computationally more
efficient than that of Johansen and Schaumburg [Journal of Econometrics
(1999), Vol. 88, pp. 301–339]. The asymptotic distributions of the ME test
statistics are obtained for several cases that depend on the nature of
deterministic terms. Monte Carlo experiments are conducted to evaluate the
relative performances of the proposed ME test and the trace test, and we
illustrate these tests using a monthly time series.
I. Introduction
In empirical studies of systems of non-stationary economic time series, the
identification of the cointegrating (CI) rank is often of major interest because
it affects the model set-up and inference procedures at other stages of the
analysis (Lu¨tkepohl, Saikkonen and Trenkler, 2001). Therefore, the CI rank of
a system is usually investigated at an earlier stage of the analysis. If a vector
autoregressive (VAR) model is an adequate description of the data generating
*The authors thank two anonymous referees for their helpful comments that lead to a significant
improvement of this paper. Byeongchan Seong’s research was supported by the Post-doctoral
Fellowship Program of the Korea Science & Engineering Foundation (KOSEF). The research of
Sinsup Cho and Sung K. Ahn was supported by the Korea Research Foundation Grant (KRF-2005-
070-C00022) funded by the Korean Government (MOEHRD).
JEL Classification numbers: C12, C22, C32.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 68, 4 (2006) 0305-9049
497
Blackwell Publishing Ltd, 2006. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK
and 350 Main Street, Malden, MA 02148, USA.
process (DGP), the likelihood ratio type tests proposed by Johansen (1988,
1996) are commonly used to identify the CI rank. Two variants of these tests
are available – the maximum eigenvalue (ME) test and the trace (TR) test –
both of which are frequently applied in empirical studies, particularly in non-
seasonal cointegration analysis.
For seasonal cointegration, the TR tests for seasonal CI ranks have been
considered by Lee (1992), Johansen and Schaumburg (henceforth, JS) (1999),
Cubadda (2001) and Cubadda and Omtzigt (2005) among others. Franses and
Kunst (1999) calculated the finite sample critical values of the TR and ME
tests in the quarterly model used by Lee (1992) that does not incorporate
polynomial (non-synchronous) cointegration at p/2. However, the ME test for
seasonal CI ranks in the general model as in JS and Cubadda (2001) has not
been considered in the literature, although it appears natural to consider the
ME test in the seasonal case as in the non-seasonal case.
In this study, we develop the ME test in the general unrestricted model and
compare its performance with the TR test in the seasonal case. With regard to
the non-seasonal case, through Monte Carlo experiments, Toda (1994) showed
that neither of the two tests is uniformly superior while the TR test performs
better in some situations where the power is low. Furthermore, Lu¨tkepohl
et al. (2001) showed that there may be slight differences in small samples, but
the TR test tends to have more distorted sizes whereas its power is, in certain
situations, superior to that of the ME test.
We consider a VAR model for an m-dimensional process {y
t
} generated by
UðLÞyt¼ImX
p
j¼1
UjLj
!
yt¼PDtþet;ð1Þ
where e
t
are i.i.d. N
m
(0, X), and D
t
is a deterministic term that may contain a
constant, a linear term, or seasonal dummies. We assume that the initial values
y
0
,...,y
p+1
are fixed and that the roots of the determinant |U(z)| ¼0 are on
or outside the unit circle. We also assume that U(x
k
) is of rank r
k
, where
x
k
¼exp (ih
k
) is a root of |U(z)| ¼0 and i ¼ffiffiffiffiffiffiffi
1
p. Through full rank
factorization, U(x
k
) can be factored as UðxkÞ¼akb
k, where a
k
and b
k
are
complex-valued m·r
k
matrices of rank r
k
, and b
kdenotes the conjugate
transpose of b
k
. As the data and the coefficients in U(z) are real-valued, the
(non-stationary) complex roots of |U(z)| ¼0 come in complex conjugate pairs.
In general, the coefficients a
k
and b
k
are complex-valued for these complex
roots. Hence, the reduced rank structure of U(x
k
) at one complex root implies
that of Uð
xkÞat its conjugate because Uð
xkÞ¼
ak
b
k, where for any complex
matrix C,
Cdenotes its complex conjugate.
The seasonal CI rank at frequency h
k
/2pwith h
k
2(0, p) is the number
of linearly independent CI relations among the components of y
t
and is the
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