where σ2x = pi ∑t(xit — x)2∕NT and σx,ij = pi ∑t(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(N—1) where
Xit = xit — Xi and Xjt = Xjt — Xj∙ The 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,Uj∙t) = 0, and dependence between
residuals and regressors for different units, cov(Uit,Xj∙t) = 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.