A Principal Components Approach to Cross-Section Dependence in Panels

3.1 Baseline regression (RI)

RI is misspecified due to an unobserved global effect z_t which is corre-
lated with each country regressor. It follows that the POLS estimator θ =
(X⁰ X )^-1X ⁰Y = θ + (X⁰ X )^-1X ⁰u is biased for the true parameter since
E(θ) = θ = β + γδ. The plim of ^{^} as T → ∞ for any fixed N is

and letting N → ∞ subsequently, results in plim_N,T→∞ θ = θ = β + γδ.
Hence, θ is inconsistent for β for both large T and N. Its variance is

var(θ) ≡ E (θ - θ)(0 - θ)0 = E ^£(X⁰X)^-1X⁰uu⁰X(X⁰X)^-1]     (9)
or var(θ) = (X⁰X) ¹X⁰E(uu⁰)X(X⁰X) ¹ for ui_t orthogonal to x_jt. Assum-
ing spherical disturbances or E(uu⁰) = a^INT = Σ ® IT with Σ = σ²_u IN ,
we have var(θ) = σ²_u (X⁰X)^-1 . However, for (6) the disturbances of RI con-
tain a random omitted variable which makes the latter inappropriate on two
accounts.

First, the contemporaneous covariance matrix has the following structure

since the errors are groupwise homoskedastic E(u_i²_t) = σ²_u and have equal
covariances E(u_ituj_t) = σ_ij . It follows that

N     NN      N

vαr(θ) = (£ XiXi)^-1(∑ ∑>ij XiXj )(£ XJXi)^-1      (10)

i=1            i=1 j=1             i=1

which for our DGP particularizes to

2             ∑∑[∑ t(xit - Xi)(Xjt - Xj )]

_________^σu__∣ i ^j=i_________________________________
Pt P,(χi< - x)^{2     j}      [Pt P,(χit - X)²]²

^σU _{+ σ} ^{(N - 1})^σx,ij

NT σ²_x ⁺ i^j NT (σX)²

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