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CRITICALVALUES FOR COINTEGRATION TESTS IN HETEROGENEOUS PANELS WITH MULTIPLE REGRESSORS CRITICALVALUES FOR COINTEGRATION TESTS IN
HETEROGENEOUS PANELS WITH MULTIPLE
REGRESSORS
Peter PedroniÃ
I. INTRODUCTION
In this paper we describe a method for testing the null of no cointegration in
dynamic panels with multiple regressors and compute approximate critical
values for these tests. Methods for non-stationary panels, including panel
unit root and panel cointegration tests, have been gaining increased accep-
tance in recent empirical research. To date, however, tests for the null of no
cointegration in heterogeneous panels based on Pedroni (1995, 1997a) have
been limited to simple bivariate examples, in large part due to the lack of
critical values available for more complex multivariate regressions. The
purpose of this paper is to ®ll this gap by describing a method to implement
tests for the null of no cointegration for the case with multiple regressors
and to provide appropriate critical values for these cases. The tests allow for
considerable heterogeneity among individual members of the panel, includ-
ing heterogeneity in both the long-run cointegrating vectors as well as
heterogeneity in the dynamics associated with short-run deviations from
these cointegrating vectors.
1.1. Literature Review
Initial theoretical work on the non-stationary panel data focused on testing
for unit roots in univariate panels. Early examples include Quah (1994),
who studied the standard unit root null in panels with homogeneous
dynamics, and Levin and Lin (1993) who studied unit root tests in panels
with heterogeneous dynamics, ®xed effects, and individual-speci®c determi-
nistic trends. These tests take the autoregressive root to be common under
both the unit root null and the stationary alternative hypothesis. More
recently, Im, Pesaran and Shin (1997) and Maddala and Wu (1999) suggest
several panel unit root tests which also permit heterogeneity of the
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, SPECIAL ISSUE (1999)
0305-9049
653
# Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford
OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
ÃAcknowledgments: I thank Anindya Banerjee, David Canning and Peter Dunne for helpful
comments and I thank Terri Ziacik for valuable research assistance. A computer program which
implements these tests is available upon request from the author at ppedroni@indiana.edu autoregressive root under the alternative hypothesis. Applications of panel
unit root methods have included Bernard and Jones (1996), Coakley and
Fuertes (1997), Evans and Karras (1996), Frankel and Rose (1996), Lee,
Pesaran and Smith (1997), MacDonald (1996), O'Connell (1998), Oh
(1996), Papell (1997), Wei and Parsley (1995), and Wu (1996).
However, many empirical questions involve multivariate relationships
and a researcher is interested to know whether or not a particular set of
variables is cointegrated. Consequently, Pedroni (1995, 1997a) studied the
properties of spurious regression, and tests for the null of no cointegration
in both homogeneous and heterogeneous panels. For the case with hetero-
geneous panels, Pedroni (1995, 1997a) provides asymptotic distributions for
test statistics that are appropriate for various cases with heterogeneous
dynamics, endogenous regressors, ®xed effects, and individual-speci®c
deterministic trends. Pedroni (1997a) includes tests that are appropriate both
for the case with common autoregressive roots under the alternative hypoth-
esis as well as tests that permit heterogeneity of the autoregressive root
under the alternative hypothesis in the spirit of Im et al. (1997).
Applications of the panel cointegration tests developed in Pedroni (1995,
1997a) for the case with heterogeneous cointegrating vectors have included,
among others, Butler and Dueker (1999), Canzoneri, Cumby and Diba
(1996), Chinn (1997), Chinn and Johnston (1996), Neusser and Kugler
(1998), Obstfeld and Taylor (1996), Ong and Maxim (1997), Pedroni
(1996b), and Taylor (1996). However, to date, these applications have been
limited to cases in which the cointegrating regressions involved a single
regressor. Many topics, on the other hand, involve applications in which
more than a single regressor is likely to be required. Therefore, the purpose
of this paper is to provide critical values that are appropriate for these
situations based on the heterogeneous panel cointegration statistics devel-
oped in Pedroni (1995, 1997a).
Finally, the panel cointegration tests in this paper should be distinguished
from those which are based on a maintained assumption of homogeneity of
the cointegrating vectors among individual members of the panel. In addition
to the heterogeneous case, Pedroni (1995, 1997a) also studied properties for
the special case of homogeneous cointegrating vectors. Speci®cally, Pedroni
(1995, 1997a) showed that for panels with homogeneous cointegrating
vectors, an interesting special result holds such that residual-based tests for
the null of no cointegration have distributions that are asymptotically equiva-
lent to raw panel unit root tests if and only if the regressors are exogenous.
Kao (1999) further studied the special case in which cointegrating vectors
are assumed to be homogeneous, but the asymptotic equivalency result is
violated because of the endogeneity of regressors. An example of the
application of these techniques for a test of the null of no cointegration in
panels that are assumed to be homogeneous is the paper by Kao, Chiang and
Chen (1999). By contrast, McCoskey and Kao (1999) examine the reversed
null hypothesis of cointegration in their study of urbanization.
654
BULLETIN
# Blackwell Publishers 1999 The remainder of the paper is structured as follows. Next, in Section 2.1,
we begin with a description of how the statistics can be used and how to
construct them step by step. Then, in Section 2.2, we explain how the
critical values can be computed for these statistics, and provide a table of
adjustment terms that can be used to obtain appropriate critical values for
various cases of interest with multiple regressors. Finally, Section 3 offers a
few concluding remarks.
II. TESTS FOR THE NULL OF NO COINTEGRATION IN HETEROGENEOUS
PANELS WITH MULTIPLE REGRESSORS
In the conventional time series case, cointegration refers to the idea that for
a set of variables that are individually integrated of order one, some linear
combination of these variables can be described as stationary. The vector of
slope coef®cients that renders this combination stationary is referred to as
the cointegrating vector. It is well known that this vector is generally not
unique, and the question of how many cointegrating relationships exist
among a certain set of variables is also an important question in many cases.
In this study, we do not address issues of normalization or questions
regarding the particular number of cointegrating relationships, but instead
focus on reporting critical values for the case where we are interested in the
simple null hypothesis of no cointegration versus cointegration.
Consequently, it is important to keep in mind that we are implicitly
assuming, for example, that the researcher has in mind a particular normal-
ization among the variables which is deemed sensible and is simply
interested in knowing whether or not the variables are cointegrated. In this
case, it is also well known that conventional tests often tend to suffer from
unacceptably low power when applied to series of only moderate length,
and the idea of pooling the data across individual members of a panel is
intended to address this issue by making available considerably more
information regarding the cointegration hypothesis. Thus, in effect, panel
cointegration techniques are intended to allow researchers to selectively
pool information regarding common long-run relationships from across the
panel while allowing the associated short-run dynamics and ®xed effects to
be heterogeneous across different members of the panel.
In this case, one can think of such a panel cointegration test as being one
in which the null hypothesis is taken to be that for each member of the panel
the variables of interest are not cointegrated and the alternative hypothesis
is taken to be that for each member of the panel there exists a single
cointegrating vector, although this cointegrating vector need not be the
same for each member. Indeed, an important feature of these tests is that
they allow not only the dynamics and ®xed effects to differ across members
of the panel, but also that they allow the cointegrating vector to differ across
members under the alternative hypothesis. We consider this to be a valuable
feature of the test, since in practice the cointegrating vectors are often likely
CRITICALVALUES FOR COINTEGRATION TESTS
655
# Blackwell Publishers 1999 not to be strictly homogeneous in such panels. In such cases, incorrectly
imposing homogeneity of the cointegrating vectors in the regression would
imply that the null of no cointegration may not be rejected despite the fact
the variables are actually cointegrated.
1
2.1. How to Compute the Test Statistics
We now turn to a discussion of