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 (σX)2