652 Panel Methods to Test for Unit Roots and Cointegration
The results, as reported in O’Connell (1998, p. 6, Table 1) for non-zero values of
ω, were quite dramatic. He showed that, for example forT= 20,N=10andω= 0.3,
the true size of the test at 5% (respectively 10%) nominal size, if LLC critical values
are used, was 9% (respectively 15%). These distortions increased asωincreased –
for example, whenω=0.9, for the same configuration ofTandN, the rejections
of the true null hypothesis at 5% (respectively 10%) increased to 37% (respectively
43%). The size distortions were not affected significantly by increasingTfor fixed
N. Thus the configurationsT =60,N=10,ω=0.3 andT=100,N=10,
ω=0.3 give rejections of 9% (respectively 15%) at 5% (respectively 10%) nominal
size, which are unchanged from theirT=20 values. The distortions, however, do
increase withNfor fixedTandω. ForT=20,N=50,ω=0.3, the rejection at 5%
(respectively 10%) increases to 21% (respectively 28%) and to 31% (respectively
37%) whenNis raised to 90, and these numbers are even higher whenωis also
increased.
Adjusting the critical values to control for the size of the test leads to a severe
reduction in power. For example, the power of thet-test to rejectH 0 :ρi= 0
against the alternativeH 0 :ρi=−0.04 (reported in O’Connell, 1998, p. 6), where
this non-zero value ofρigives a half-life of deviation from purchasing power parity
of between four and five years and is coherent with empirical estimates, is 8% when
T=20,N= 10, 14% whenT=20,N= 50, and 13% whenT=20,N=90.ωhere is
0.3 and the confidence level of the tests is 5%. ForT=60andN= 50, power drops
from 92% (whenω= 0) to 30% (whenω= 0.5) and to 9% (whenω= 0.9). Thus the
necessity to counteract distortions neutralizes any beneficial effects of increasing
N, the longitudinal dimension of the panel, and the loss of power involved – as
a result of the necessary adjustment – is clearly a serious one. Indeed, the trade-
off between theNandTdimensions, which is evident from both the asymptotic
theory and the empirical implementation, is a topic which is relatively ill-studied
in the literature.
In order to account for the non-zero off-diagonal terms in, O’Connell proposed
the following generalized least squares (GLS) estimator:
ρˆGLS=
tr(X′Y−^1 )
tr(X′X−^1 ),whereYis aT×Nmatrix of the first-differencedys andXis a matrix of lagged
ys. Whenω=0, we recover the usual ordinary least squares (OLS) estimator. Thus:
YT×N=⎛
⎜
⎜⎜
⎜
⎜⎜
⎜⎝y1,1 y2,1 ....yN,1
y1,2 y2,2 .. .yN,2
......
......
.....
y1,T y2,T. yN,T⎞
⎟
⎟⎟
⎟
⎟⎟
⎟⎠.Computation of the feasible GLS estimator, denoted byρˆFGLS, requires a consis-
tent estimator of. Allowing only for contemporaneous correlation, a consistent