Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 653

estimator ofis given by the (estimated) covariance matrix of the first differences
of the vectoryt, that is,

ˆ=T−^1

∑T

t= 1

yty′t

yt=(y 1 t,y2,t,...,yN,t)′

and, therefore,

ρˆFGLS=

tr(X′Yˆ−^1 )

tr(X′Xˆ−^1 )

.

Clearly, with this estimator not all correlations between the cross-section members
need to be identical, as assumed by O’Connell to illustrate the issues in the sim-
ulation part of his paper. The results reported in O’Connell (1998, p. 12, Table 3)
are encouraging because the rejection frequencies under both the null hypothesis
(size) and the alternative hypothesis (power) are now invariant to the value ofω
in the DGP, and depend onTandNalone. ProvidedTis much larger thanN,
the power of the FGLS test comes close to the OLS panel unit root test when the
disturbances are independently and identically distributed. Efficiency losses ensue
asNcomes closer and closer toT, which Hlouskova and Wagner (2006) refer to as
size divergence whenNis “too large” compared toT.
The discussion above serves to introduce and illustrate some of the main issues
involved, but is nevertheless too specialized, applying as it does to only one highly
restrictive specification of cross-sectional dependence. More general methods for
dealing with short-run cross-sectional dependence will be discussed below (includ-
ing dealing with nonparametric methods for estimating the variance covariance
matrix of the disturbances when there is dependence across the units).

13.2.2.2 Cross-sectional dependence via approximate factor models – Bai
and Ng (2004)

An important class of tests developed to allow for long-run cross-sectional depen-
dence is due to Bai and Ng (2004). Returning to the formulation given by
(13.5)–(13.7), the basic idea is to think of the series comprising the panel as consist-
ing of the sum of a set of deterministic terms, common factors and idiosyncratic
components. The detailed assumptions made by Bai and Ng are given by:

(i) For non-randomπi,||πi|| ≤A; for randomπi,

E||πi||^4 ≤A,N^1
∑N
i= 1 πiπ

′
i→,anr×rpositive definite matrix, where→denotes
convergence in probability.

(ii)ηt∼i.i.d.(0,η),E||ηt||^4 ≤A, var(Ft)=

∑∞

j= 0 CjηC

′

j>0,

∑∞

j= 0 j||Cj||<A,

C(1) has rankr 1 ,0≤r 1 ≤r.