A Principal Components Approach to Cross-Section Dependence in Panels

be obtained substituting

for φ_pp and φ_pq , respectively.

3.2 Augmented regression (RII)

We start by asking how well w_t proxies the unobserved global variable z_t .
The plim of their squared sample correlation as T →∞ for any fixed N is

σ_z²

σ_z² +N^-1δ²τ²σ²_d+N^-1γ^-2τ²σ_ε²

where τ = (1 — δ)-1. Letting N → ∞ also, we have plim_τN→∞ p2_w = 1 and it
follows that for large N and T the first factor for RI residuals is a consistent
estimator of z_t . The plim of b as T →∞ for any fixed N is

(N-1 ∑i SNw )(N-1 ∑i SNy ) — (N-1 ∑i SNw )(N-1 ∑i SNw ), 14
(N-1 Pi SXX)(N-1 Pi SNw) — (N-1 Pi SNw)²     ( )

βγ²(1 — δ)²σZσd + N-1f1 + N-²g1
γ²(1 — δ)²σZσd + Nf + N-²g2

where

f1 = γ²δβσ_d⁴ + (2γ²βδ — γ²βδ² + γ³δ)σ²_dσ_z² + (β + γδ)σ_ε²σ_z² + βσ_ε²σ_d²,
f2 = γ²δ²σ⁴d +(2γ²δ — γ²δ²)σ²dσz² + σε²σz² + σε²σd²),

g2 = —γ2^δ2^σd and g1 = γ^δσ2^σd — γ2^δ2^βσd

ʌ

Making N →∞ also it follows that b is a consistent estimator for β since

plim (βγ²(1 — δ)²σ_z²σ²_d + N^-1f1 + N ^-2g1)

N→∞                           _ n

plim (γ²(1 — δ)²σZσd + N-1f2 + N-²g2) = ^β

N→∞

Some remarks are in order. First, the inconsistency of b for fixed N re-
flects the endogeneity bias which arises because the principal components

10

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