On the Existence of the Moments of the Asymptotic Trace
Statistic *
Deniz Dilan Karaman Orsal1 and Bernd Droge2
Humboldt-Universitat zu Berlin
18th February 2009
Abstract
In this note we establish the existence of the first two moments of the asymptotic trace
statistic, which appears as weak limit of the likelihood ratio statistic for testing the cointe-
gration rank in a vector autoregressive model and whose moments may be used to develop
panel cointegration tests. Moreover, we justify the common practice to approximate these
moments by simulating a certain statistic, which converges weakly to the asymptotic trace
statistic. To accomplish this we show that the moments of the mentioned statistic converge
to those of the asymptotic trace statistic as the time dimension tends to infinity.
Keywords: Cointegration, Trace statistic, Asymptotic moments, Uniform integrability.
JEL classification: C32, C33, C12
1 Motivation and Framework
Cointegration tests play an important role in the empirical analysis of long-run relationships
among integrated variables, but they often suffer from low power properties due to the small
time span of the available time series. The performance of the tests could be improved by
enlarging the data basis, e.g. by considering additional cross-sectional units (individuals)
with similar data. Therefore the cointegration methodology has been extended to the panel
data framework. Similar to the case of testing for unit roots, panel cointegration tests may
be based on standardizing the average of individual cointegration test statistics. By some
central limit theorem, standard normal quantiles may then serve as critical values. However,
the justification of such a procedure requires the existence of the first two moments of some
distribution. For example, Larsson et al. (2001) used the likelihood framework to present a
test for the cointegrating rank in heterogeneous panels. Their test, which they refer to as
standardized LR-bar test, is based on the likelihood ratio (LR) test statistic developed by
Johansen (1995) for vector autoregressive (VAR) models. Under the null hypothesis and as
the time dimension approaches infinity, the LR statistic converges weakly to the asymptotic
trace statistic, whose moments are thus used for standardizing the average of the individual
LR test statistics.
* This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 “Economic
Risk”.
1Institute for Statistics and Econometrics, School of Business and Economics, Humboldt-Universitat zu
Berlin, Spandauer Str. 1, 10099 Berlin, Germany, E-mail: [email protected]
2Institute for Statistics and Econometrics and CASE - Center for Applied Statistics and Economics, School
of Business and Economics, Humboldt-Universitat zu Berlin, Spandauer Str. 1, 10099 Berlin, Germany,
E-mail: [email protected]