Anindya Banerjee and Martin Wagner 719only established with certainrate restrictions, for example, of the form
√
N
T →^0
orNT →0. Phillips and Moon (1999) contains an insightful discussion in this
respect.
To return to our example, as long asTis finite, the individual specifict-statistics
will in general not be identically distributed. One way of overcoming this problem
is to assume cross-sectionally identically distributed seriesyi,t, which is, however,
far too strong an assumption to be of any practical relevance. Therefore, since
identical distributions for finiteTsamples are usually out of the question, other
conditions that allow the use of joint central limit theorems for independent but
not identically distributed random variables have to be established. A prime can-
didate in this respect is to establish a Lindeberg-type condition, or some special
cases formulated in terms of uniform integrability conditions, in the joint limit.
On a detailed discussion and an example of this approach see Phillips and Moon
(1999).
The major problem for the panel unit root and cointegration literature is to
establish the required conditions for finite values ofT, which are partly not even
derived for the time series limits of the building blocks of the panel statistics. One
underrated but good exception in this respect is the work of Larsson, Lyhagen and
Löthgren (2001), discussed in section 13.3.2.1, who work out the proofs in detail
for a panel version of the Johansen (1995) trace test for VAR models without deter-
ministic components and with normally distributed errors. However, substantial
parts of the current literature need theoretical strengthening. It is clearly a major
task for the literature to work out the correctness of necessary intermediate results
for the numerous test statistics and estimators. Until this task has been accom-
plished, the panel unit root and cointegration literature urgently needs to make
further significant progress in terms of establishing mathematical rigor.
Clearly, relaxing the assumption of cross-sectional independence complicates
matters even further, since now limit theory for dependent doubly indexed random
sequences has to be invoked to establish results. It is clear that without modeling
the extent and form of cross-sectional dependence carefully it will not, in general,
be possible to establish any well-defined limiting behavior. Up to now no general
modeling strategies for cross-sectionally dependent unit root non-stationary panels
have been devised and only certain special modeling approaches are in use to date.
Two approaches appear to be prominent. One is actually to treat theNdimension
as finite and fixed, which means that essentially high-dimensional time series prob-
lems are considered (see, for example, Pedroniet al., 2008). The other approach is
given by modeling cross-sectional dependence by resorting to factor models, as we
have discussed in the main body of the chapter.
Note as a final remark that the usage of panel techniques with both the time
series and the cross-sectional dimension tending to infinity allows us to solve some
problems that cannot be addressed in a pure time series setting. These include, for
example, the consistent estimation of the non-centrality parameter in a local-to-
unity framework or of distant initial conditions (see Moon and Phillips, 2000).