A Principal Components Approach to Cross-Section Dependence in Panels



where σ2x = pit(xit x)2∕NT and σx,ij = pit(xit Xi)(xjt Xj)∕NT.
Since θ is consistent for θ, var(θ) can be consistently estimated by substitut-
ing
σU = U0u∕(NT k) where k is the number of freely estimated parameters
in RII and
σij= ɪ'i ^(Ni-^j/T for σU and σij, respectively.

Second, although uit is orthogonal to xit by construction, it is correlated
with
xjt for j 6= i.5 Hence (11) is still incorrect in assuming that elements
of the form
E(xituitxjtujt) in E (X 0uu0X) are equal to E (xitxjt)E (uitujt) or,
more generally, that
E (X0uu0X) = X0E(uu0)X in (9). The appropriate
asymptotic covariance matrix is

var(θ) = Λ⅛ + N2               (12)

NTσ2x   NT(σ2x)2

where φ = E (xituitxjtujt). An estimator for var(θ) is straightforward to
compute using
σU and φ = PN=1 PN=i(T-1 PT=1 XitUitXjtUjt)/N(N1) where
Xit = xit Xi and Xjt = Xjt XjThe first term in (12) is the usual POLS
variance formula and the second term may be viewed as a correction for cross-
section dependence of residuals,
cov(Uit,Ujt) = 0, and dependence between
residuals and regressors for di
fferent units, cov(Uit,Xjt) = 0 arising from
an omitted global variable or shock
zt. Consequently, POLS not only gives
biased and inconsistent (for both
N, T →∞) estimators for RI but also
biased standard errors. Formula (12) suggests that the bias of the latter is of
order
T-1 and hence it will be non-negligible for small T even if N is large.

In a more general K-regressor setup, Y = XΘ+ u, the appropriate asymp-
totic covariance matrix of the POLS estimator
Θ is

var(Θ) = (X 0X )-1E (X 0uu0X )(X 0X )-1 = (X 0X )-1Π(X 0X )-1     (13)

where Π is a K × K matrix with elements

pp = NT σXp σU + N (N 1)T φpp, pq = NT σχpq σ2u + N (N 1)T φpq

for p,q = 1,∙∙∙,K, where σXp = Pi=1 PT l(⅜,√ Xp)2/NT and σxpq =
iN=1 tT=1(Xp,it Xp)(Xq,it Xq)∕NT. A consistent estimator vαr(Θ) can

5For our simple DGP uit = γ(zt δxit) + εit with δ = σ2(σ2 + σ2d). It follows that
cov(ujt,Xjt) =γδσd + γ(1 δ)σ2 = 0. However, cov(ujt,xit) = γ(1 δ)σ2 = 0.



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