A Note with Quantiles of the Asymptotic Distribution of the Maximum Likelihood Cointegration Rank Test Statistics1
| DOI | http://doi.org/10.1111/j.1468-0084.1992.tb00013.x |
| Published date | 01 August 1992 |
| Author | Michael Osterwald‐Lenum |
| Date | 01 August 1992 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS,
54,3
(I
992)
0305-9049 $3.00
PRACTITIONERS' CORNER
A
Note with Quantiles
of
the Asymptotic Distribution
of
the Maximum Likelihood Cointegration
Rank
Test
Statistics
Michael
Osterwald-Lenurn?
I.
INTRODUCTION
The recent literature on maximum likelihood cointegration theory studies
Gaussian
VAR
models allowing for some deterministic components in the
form
of
polynomials in time. Here we are concerned with such models for
variables integrated at most
of
order one, when tests for cointegration involve
statistics with non-standard asymptotic distributions. Cf. Johansen
(
1988),
(1991a), (1991 b), Johansen and Juselius (1990). The asymptotic distributions
of
these test statistics are known
to
be
functions
of
the distribution
of
certain
matrices
of
stochastic. variables involving integrals
of
Brownian motions. In
fact, conditional on which restrictions on the coefficients
of
the polynomial in
time are valid, different asymptotic distributions are obtained. The cases
dealt with here .exhaust the hypotheses relevant to the cointegration rank
analysis
of
I(
1)
variables in models involving up to linear trends and possibly
seasonal dummies. This paper solves the numerical problem in making most
of the interesting quantiles
of
these asymptotic distributions available to the
applied econometrician. It thus includes recalculated and extended versions
of
the four tables presented in Johansen
(1988)
and Johansen and Juselius
(1990) as well as two new tables.
'This note
was
written during a visit
1
July
1989-1
June
1990
to the Department of
Economics, University of California,
San
Diego. Its previous title was 'Recalculated and
extended tables
of
the asymptotic distribution of some important maximum likelihood cointe-
gration test statistics' dated December
1989
and revised January
1990.
It
has since been
expanded by cases
2
and
2:.
ZThe author is indebted to Ssren Johansen for having supplied some of the programs (in
ISP)
used for the calculations in Johansen and Juselius
(1990).
and comments on a previous
version
of
this paper. Thanks
are
also due to Katarina Juselius and Niels Buus Christensen
for
useful comments on the structure of this note. Funding for the visit to UCSD from the Danish
Research Academy
is
gratefully acknowledged.
46
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