Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 703

very similar to the analysis relating to single-equation estimators. In panel VAR co-
integration analysis, however, the testing for cointegration and the estimation
steps are much more intertwined than in the single-equation tests discussed above,
for most of which the null hypothesis is that of no cointegration. The two tests to
be discussed differ in this respect, since the test of Larsson, Lyhagen and Löthgren
(2001) does not impose cross-sectional homogeneity in the construction of the test
statistic (but needs this assumption for the derivation of the asymptotic distribution
of the test statistic) whereas the test of Breitung (2005) incorporates this restriction.

13.3.3 Larsson, Lyhagen and Löthgren (2001)

Larsson, Lyhagen and Löthgren (2001) (henceforth LLL) consider testing for co-

integration in the above framework under the assumption thati=αiβ

′

i=αβ

′
=
for alli=1,...,N. For some further technical assumptions we refer the reader to
their paper. Note here that their test is based simply on the cross-sectional aver-
age of the Johansen (1995) trace statistic, where the cross-sectional homogeneity
assumption just mentioned is not put in place anywhere in the construction of
their test. However, the asymptotic distribution (under the null hypothesis) of
their test statistic is established only under this assumption (see Assumption 3 and
Theorem 1 of LLL). The null hypothesis of their test isH 0 :rk()=kand the test
is consistent against the alternative hypothesisH 1 :rk(i)>kfor a non-vanishing
fraction of the cross-section members.
The construction of the test statistic is similar to that of IPS and it is given
by a suitably centered and scaled cross-sectional average of the individual trace
statistics. Thus, denote byLRsi(k,m)the trace statistic for the null hypothesis of a
k-dimensional cointegrating space for cross-section uniti, where the superscript
s=1,..., 5 indicates the specification of the deterministic components. Further,

denote byμsLR(k,m)andσLR2,s(k,m)the expected value and variance ofLRsi(k,m).In
this respect LLL is a most notable but unfortunately underrated exception in the
panel unit root and cointegration literature, insofar as the authors derive, admit-
tedly only for the model without deterministic components and with normally
distributed innovations, the existence of the necessary moments as well as uniform
integrability and Lindeberg-type conditions. These are needed to derive formally
the asymptotic normality of the test statistic.^31 Using a sequential limit with first
T→∞followed byN→∞, it holds that:

LLLs(k,m)=N−^1 /^2

∑N

i= 1

LRsi(k,m)−μsLR(k,m)

(σLR2,s(k,m))^1 /^2

⇒N(0, 1). (13.31)

Finite-sample correction factors for the LLL test statistic are given in Hlouskova
and Wagner (2008) for all five mentioned specifications of the deterministic com-
ponents, where we note again that the theoretical result is only derived for the case
without deterministic components and with normally distributed innovations.
LLL do not explicitly consider estimation of the cointegrating space. Given that
their test is based on the assumption of a cross-sectionally identical cointegrating
space, one possibility to obtain an estimate of the cointegrating space is given by the
cross-sectional average of identically normalized individual specific cointegrating