Hence the ith subvector of δ
~
is given by(5.53)(5.53)which is exactly the 2SLS estimator and, incidentally, is also the computa-
tionally efficient method for obtaining it.
Three-Stage Least Squares (3SLS)
We reported that the problem of estimating the parameters of a structural
system of equations by 2SLS can be reduced to the problem of estimating
by ordinary least squares, the parameters of a single equation in the nota-
tion of (5.50).
However, it was also shown earlier that in this context whether such a
procedure is efficient or not depends on the covariance structure of the er-
ror terms in the various equations of the system. In particular, it was
shown that if the error terms between any two equations were correlated,
then a gain in efficiency would result by applying Aitken’s procedure pro-
vided that not all equations contain the same variables. A procedure that
would take into account this postulated covariance structure will be effi-
cient relative to the 2SLS procedure, which does not take it into account. In
general, different equations will contain different explanatory variables. If
their respective error terms are correlated, then by focusing our attention
on one equation at a time we are neglecting the information conveyed by
the rest of the system. If we could use such information, then clearly we
would improve on 2SLS.
The Three-Stage Least Squares Estimator
Consider again the system
y⋅i= Yiβ⋅i+ Xiγ⋅i+ u⋅i i= 1, 2,...,m (5.54)̃ ()() ()δ⋅ − ⋅− − −
⋅
⋅= ′′= ′′ ′ ′
′′
′′ ′
′
iiiiiiiii
ii i iiii
iiQQ QwYXXX XY YX
XY X XYXXX Xy
Xy11 1 1