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that 2SLS is optimal in some sense within the class of consistent estima-
tors of the parameters β·1, γ·1, which use only information conveyed by
the equation containing β·1and γ·1and thus disregard information con-
veyed by the rest of the system. By assumption, rank (X) = G, and thus
X′Xis a positive definite matrix; hence there exists a nonsingular matrix
Rsuch that

X′X= RR′ (5.44)

Now transform equation (5.40) by R–1to obtain

R–1X′y·1= R–1X′Y 1 β·1+ R–1X′X 1 γ·1+ R–1X′u·1 (5.45)

Let

(5.46)

(5.46)

The 2SLS is simply the OLS estimator applied to equation (5.46).
But this particular formulation of the problem opens the way to a rou-
tine derivation of the 2SLS estimator of all the parameters of the entire sys-
tem of m-structural equations. Every equation of the system may be put in
the form exhibited in (5.46). Thus we can write

w·i= Qiδ·i+ r·i i= 1, 2,...,m (5.47)

where

(5.48)

wRXy

QRXYRXX

rRXu

ii

iii

i

i

i

ii

⋅

−

⋅

−−

⋅

⋅

⋅

⋅

−

⋅

= ′

= ′′

=













= ′

1

11

1

(, )

δ

β

γ

wRXy

QRXYRXX

rRXu

⋅

−

⋅

−−

⋅

⋅

⋅

⋅

−

⋅

= ′

= ′′

=













= ′

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(, )

δ

β

γ

Two-Stage Least Squares (2SLS) 87