and Yi and Xi are, respectively, the matrices of observations on the
explanatory current endogenous and predetermined variables appearing in
the ith equation.
Let us define
(5.49)(5.49)And thus write the entire system in equation (5.47) as
w= Qδ+ r (5.50)The 2SLS estimator of all the parameters of the system is therefore
(5.51)The covariance matrix of r⋅ 1 with r⋅j, i≠jwill not vanish, and one sus-
pects that δ
~
is an asymptotically inefficient estimator of δ. We shall return
to this problem in the next section when we consider three-stage least
squares estimators. Let us now examine the precise nature of the estimator
exhibited in equation (5.51).
We have
(5.52)′ = (5.52)′
′′
′ =′
′′ ⋅
⋅⋅QQQQ
QQQQQwQw
Qwmm Qwnm11
2211
2200O
Mδδ= ̃ ()QQ Qw′′−−^11 =+()QQ Qr′′QQQQrr
rrmm==
⋅
⋅⋅diag(, ,, ) 121K^2
Mww
wwmm=
=
⋅
⋅⋅⋅
⋅⋅1
21
2
MMδδ
δδ