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The Aitken estimator of δis given by

(5.61)

where

Φ= Σ⊗IG= Cov(r) (5.62)

Clearly, since Σwill typically be unknown, the estimator in (5.61) is not
feasible. If a consistent estimator of the (m×m) matrix Σexists, say , then
the feasible Aitken estimator is given by

(5.63)

and represents the covariance matrix of the 2SLS residuals of the system.
The estimator in (5.63) was termed the three-stage least squares (3SLS)
estimator, or perhaps the Aitken structural estimator, whereas the 2SLS
should be termed the OLS structural estimator. The term 3SLS has the fol-
lowing intuitive interpretation: In stage one, we purge from the explana-
tory current endogenous variables their stochastic components; in the
second stage we obtain a consistent estimator for Σ; in the third stage we
obtain the desired estimator of the structural parameters. Although a
rather cumbersome view of the process, it is, for better or worse, the his-
torically initial and established view.
Actually, and despite the terminology, the 3SLS estimator in (5.63) can
be computed in one operation, as we do in Chapter 6 using SAS. The rea-
son is that the typical element of the matrix can be expressed solely in
terms of the moment matrices of the data. For example,

(5.64)

where Aijis the T×Tmatrix

Aij= [I– Z(Q′iQi)–1Q′iR–1X′]′[I– Zj(Q′jQj)–1Q′jR–1X′] (5.65)

Having defined what we wish to mean by the 3SLS estimator, let us
take a more detailed view of it. Expanding (5.63), one sees that

σˆij ij j ,,,,

T

= yAy⋅⋅′ ij= m

1

1 12 K

Σˆ

δ=ˆ (QQQw′′ΦΦΦΣˆ−− −^11 )^1 ˆˆ=⊗IG

Σˆ

δ=()QQQw′′ΦΦ−− −^111

Three-Stage Least Squares (3SLS) 91