712 Panel Methods to Test for Unit Roots and Cointegration
Series for 1996:1–2001:3 for Austria and Finland.
Our “new” dataset 1995–2005– 1995:1–2005:3 for ten out of eleven countries of the
CM dataset (Belgium+Luxembourg excluded, Austria and Finland start 1996:1,
Portugal and Austria stop 2004:12)
Full panel – reduced version of 1995–2005 dataset: trimmed in order to obtain a
balanced panel. Covers 1996:1–2004:12 for all ten countries.
13.6 Appendix B: Cross-sectional dependence
Allowing for cross-sectional dependence complicates matters substantially com-
pared to the case of cross-sectionally independent panels. In the early literature on
non-stationary panels, cross-sectional dependence had been modeled by allow-
ing for common fixed (respectively random) effects. It was, however, quickly
realized that in a non-stationary time series panel context the modeling of cross-
sectional dependence needed to allow for richer types of dependencies that allow
us, in particular, to model both transitory (short-run) and permanent (long-run)
cross-sectional dynamic dependencies.
During the course of the chapter, in modeling cross-sectional dependence,
we have typically followed the route prescribed by Bai and Ng (2004), inter
alia, in using approximate factor models. Dependence can also be introduced,
as in Banerjeeet al. (2004), by considering cointegrating relationships across
the variables in the cross-sectional units. In this brief appendix, we attempt
to provide an exploration of these issues, including a discussion of the restric-
tions on the cointegrating space (of the full panel) implied by factor models
and of the links between these alternative formulations of cross-sectional depen-
dence.
In order to clarify the concepts, consider the case of a panel ofunivariatetime
series, neglecting for simplicity deterministic components, and denote the stacked
joint vector (for givenN)asyt=(y1,t,...,yN,t)′.
The first assumption that is usually, unfortunately typically only implicitly, made
is that the joint vector process is a vectorI(1) process, or, in the case of all series
being stationary, a multivariate stationary process. Clearly, this is an assumption
that has to be put in place over and above the assumption that all theindividual
series areI(1) or stationary. This stems from the fact that stacking stationary pro-
cesses does not in general lead to a jointly stationary vector process. Stationarity
of the stacked process occurs if and only if all the processes arestationary correlated,
that is, if all cross-correlation functions between the individual processes are sta-
tionary. For cross-sectionally independent processes this latter condition is fulfilled
by construction. Since stacked stationary processes are not necessarily stationary
it is clear also that stackedI(1) processes are not necessarilyI(1) processes.
Given that we consider panels of non-stationary time series, dependence con-
cepts well-established in the time series literature are also of prime importance
in this context. In particular we will distinguish between short-run and long-run
dependence. To be precise, in these definitions we will focus on the dependence