A Principal Components Approach to Cross-Section Dependence in Panels



3.1 Baseline regression (RI)

RI is misspecified due to an unobserved global effect zt 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

ʌ

plim 6* =

T→∞


N-1  i SNx

i yx

N-1 Pi SN

i xx


βσ2d + (β + γ)σz2


σ2d + σz2


= β + γδ


and letting N → ∞ subsequently, results in plimN,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 uit orthogonal to xjt. Assum-
ing spherical disturbances or
E(uu0) = a^INT = Σ ® IT with Σ = σ2u IN ,
we have var(θ) = σ2u (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

σ2u

σij     • • •

σij

Σ = E(utu0t) =

σij
.

σ2u

.

.

.

.

.

.

.

.

σij

σij

• ∙ ∙     σij

σ2u

since the errors are groupwise homoskedastic E(ui2t) = σ2u and have equal
covariances
E(uitujt) = σij . 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 )]

var(θ)


_________σu__ i j=i_________________________________
Pt P,(χi< - x)2     j      [Pt P,(χit - X)2]2

(11)


σU + σ (N - 1)σx,ij

NT σ2x + ij NT X)2



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