650 Panel Methods to Test for Unit Roots and Cointegration
research is the main motivation for this section, with a view to describing some of
the general principles involved.
Our discussion is in several parts. We start by discussing O’Connell’s contribution
to the debate on testing for purchasing power parity in panels. By setting out the
data-generation processes in some detail, we describe how O’Connell introduces
short-run dependence into the panel. Second, we present some of O’Connell’s
results to describe the size distortions arising from this dependence. We discuss
some solutions proposed to deal with that dependence. Based on this discussion, we
then proceed in the following sub-sections to a discussion of the second generation
of unit root tests aimed at modeling cross-sectional dependence, focusing on the
work of Bai and Ng (2004) and other authors who have made use of factor models.
Consider, for illustration, the DGP given by:
yi,t=εi,t, i=1, 2,...,N; t=2, 3,...,T,where, for the purposes of the illustration, we may takeεi,t∼i.i.d.N(0, 1)without
much loss of generality and where we also abstract from deterministic components.
yi,tis therefore a simple random walk process. The model to be estimated is given
by a simplified version of equation (13.10):
yi,t=μi+ρiyi,t− 1 +εi,t, i=1, 2,...,N; t=2, 3,...,T.The model is used to test:
H 0 :μi=ρi=0,against the alternative:
HA:μi =0; ρi<0,although the results described below (taken from O’Connell, 1998) are based on
looking only at the t-statistic for the estimate of the autoregressive parameter. It
should be noted that, as usual within the LLC framework,ρiis restricted to be
the same across allNunits under both the null and alternative hypotheses. The
testing framework can be augmented by polynomials of time, in particular by a
linear trendt.
Now, under the normality and independent and identically distributed (i.i.d.)
assumptions, cross-sectional dependence between the series is fully characterized
by thedynamiccovariance structure of the joint processεt=
(
ε1,t,...,εN,t)′
.To
make the discussion as simple as possible, consider a situation, like O’Connell
(1998) does implicitly, where the joint random vector is also normally distributed
and where the only correlations occur contemporaneously, in which case the
dependence structure is fully characterized by=E(εtεt′). The joint vector of
all disturbancesε=
(
ε 1 ,...,εT)′
has its block-diagonal covariance, in our set-up