A Principal Components Approach to Cross-Section Dependence in Panels

where σ²_x = ^p_i ∑_t(xit — x)²∕NT and σ_x,i_j = ^p_i ∑_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 σ_ij∙ = ɪ'i ^(Ni-^j/T for σU and σ_ij∙, respectively.

Second, although ui_t is orthogonal to xi_t 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)

NTσ²_x   NT(σ²_x)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
X_it = x_it — X_i and Xj_t = Xj_t — 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(U_it,U_j∙_t) = 0, and dependence between
residuals and regressors for different units, cov(U_it,X_j∙_t) = 0 arising from
an omitted global variable or shock z_t. 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 σχp_q σ²_u + N (N — 1)T φpq

^for p^,q = ¹,∙∙∙,K, ^{where σ}X_p = Pi=1 PT _l(⅜,√ ^{— X}p^{)2/NT and σ}xpq =
i^N=1 t^T=₁(X_p,_it — X_p)(X_q,_it — Xq)∕NT. A consistent estimator vαr(Θ) can

⁵For our simple DGP u_it = γ(z_t — δx_it) + ε_it with δ = σ2∕(σ2 + σ²_d). It follows that
cov(uj_t,Xj_t) = —γδσd + γ(1 — δ)σ2 = 0. However, cov(uj_t,x_it) = γ(1 — δ)σ2 = 0.

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