be obtained substituting
ʌ
φpp
∑∑(τ 1 ∑Xp,itUitXp,jtUjt)
i j=i________t____________________ ^
N2 , φpq
ΣΣ(T 1 Σ Xp,it'UitXq,jt'Ujt)
i j 6=i t
N2
for φpp and φpq , respectively.
3.2 Augmented regression (RII)
We start by asking how well wt proxies the unobserved global variable zt .
The plim of their squared sample correlation as T →∞ for any fixed N is
plim pZw
T→∞
(SN )2
( ɔzw )
S N S N
zz ww
σz2
σz2 +N-1δ2τ2σ2d+N-1γ-2τ2σε2
where τ = (1 — δ)-1. Letting N → ∞ also, we have plimτN→∞ p2w = 1 and it
follows that for large N and T the first factor for RI residuals is a consistent
estimator of zt . The plim of b as T →∞ for any fixed N is
plim b
T→∞
(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 = γ2δβσd4 + (2γ2βδ — γ2βδ2 + γ3δ)σ2dσz2 + (β + γδ)σε2σz2 + βσε2σd2,
f2 = γ2δ2σ4d +(2γ2δ — γ2δ2)σ2dσz2 + σε2σz2 + σε2σd2),
g2 = —γ2δ2σd and g1 = γδσ2σd — γ2δ2βσd
ʌ
Making N →∞ also it follows that b is a consistent estimator for β since
plim (βγ2(1 — δ)2σz2σ2d + N-1f1 + N -2g1)
plim b
N,T →∞
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