Anindya Banerjee and Martin Wagner 651equal to the correlation, matrix given by:
=⎛
⎜⎜
⎜
⎜⎜
⎝ 0 ... 00 ..
...
.
..
...
.
... 0
0 ... 0
⎞
⎟⎟
⎟
⎟⎟
⎠.In order to analyze the effect of cross-sectional dependence, O’Connell (1998) stud-
ied the effects of relaxing the assumption of diagonality of the matrixon the
performance of the LLC tests, using the original LLC correction factors and critical
values derived under the assumption of cross-sectional independence. In particu-
lar, O’Connell considers the following simple shape of the correlation matrix, also
used by Hlouskova and Wagner (2006) as one of their designs which they label, for
obvious reasons, constant correlation:
=⎛
⎜
⎜⎜
⎜⎜
⎝1 ω ... ω
ω 1..
...
.
..
...
.
... ω
ω ... ω 1
⎞
⎟
⎟⎟
⎟⎟
⎠.This formulation imples that the “distance” between the cross-sectional distur-
bances is not considered relevant economically. Various justifications for such a
covariance structure can be offered from examples dealing with spatial structures
or for purchasing power parity based applications.^13 However, here it serves only
for the purposes of illustration within the context of a Monte Carlo study. The
simple O’Connell design also ensures that there is no long-run dependence for
− 1 <ω<1.
The simulations were conducted with the following choices of sample sizes and
parameters in the DGP:^14
N∈{10, 50, 90}
T∈{20, 60, 100}
ω∈{0, 0.3, 0.5, 0.7, 0.9},with the disturbances distributed as described above. It is worth pointing out that
the{N,T}combinations considered do have relevance for assessing the results of
the simulations, since a number of the asymptotic results on the properties of unit
root tests in panels impose joint conditions onNandTtending to infinity in order
to prove the theorems. This is also a crucial issue that arises later when discussing
factor estimation.
For given values ofN,Tandω, thet-statisticρ/ˆ+σρˆ(denotedtOLS)was computed
and compared (for each of the 5,000 replications/panels) with critical values (at
1%, 5% and 10%) tabulated by LLC, designed for the case without correlation,
that is, forω=0.