Anindya Banerjee and Martin Wagner 645be re-centered and re-normalized for convergence to a well-behaved distribution.
Therefore,
tρ∗=tρ−NT ̃SˆN,TSTD(ρ)μˆ mT
σmT
⇒N(0, 1).HereT ̃denotes the effective average sample size, to take into account individual
specific numbers of lagged terms across the individual units, whileμmTandσmT
denote the mean and variance corrections for the three different specifications of
the deterministic terms. The latter are tabulated by LLC for various dimensions of
NandT.
These are interesting results, deriving from the application of sequential (with
firstT→∞followed byN→∞)central limit theorems, applicable under the
assumption ofindependentcross-sectional units whenever the individual building
blocks, which are identically distributed onceThas passed to infinity, have finite
second moments. As we show below, violations of this assumption can lead to
severe difficulties, which may be soluble for a number of cases using a range of
additional techniques.
Various modifications of the LLC procedure have been proposed,inter alia,by
Harris and Tzavalis (1999), who derive asymptotic results for fixedT, allowing only
theNdimension to tend to infinity, with closed form expressions for the correction
factors for serially uncorrelated errors. As for LLC above, appropriately scaled and
re-centeredt-statistics tend toN(0,1) densities asNtends to infinity, leading to tests
which have better properties when theTdimension is relatively small and little or
no serial correlation is permitted in theνi,tprocesses. Breitung (2000) develops, by
means of appropriate variable transformations, a modified LLC test which, while
coinciding with the LLC test when no deterministic terms are present, does not
require bias correction factors for the cases where a constant or trend is present.
Several features of the testing framework above lend themselves to exten-
sions and we shall try to deal with each of these in turn in the sections which
follow:
- relaxing homogeneity;
- relaxing cross-sectional independence;
- relaxing structural stability of the deterministic component.
13.2.1.2 Relaxing homogeneity – Im, Pesaran and Shin (2003)
Im, Pesaran and Shin (2003) (henceforth IPS), propose a class of group-mean panel
unit root tests to allow for a heterogeneous alternative specified as:
HA^1 :ρi<0 fori=1, 2,...,N 1 andρi=0 fori=N 1 +1,...,N, where lim
N→∞N 1
N
=k>0.IPS present two tests for the case of serially uncorrelated and correlated errors, and
for specifications of the deterministic terms allowing for a constant and a constant
and linear trend. The two tests are (i) at-test based on ADF regressions, denoted
IPSt, and (ii) a Lagrange multiplier test, denotedIPSLM. We concentrate here on (i),