Consider regression equations
у = Za + u
Z = ХП + V,
where Ем = О, ЕУ = 0, Em’ = T*, and u and V are possibly correlated. We assume that is a known nonsingular matrix. Although the limited information model can be written in this way by assuming that certain elements of V are identically equal to 0, the reader may regard these two equations in a more abstract sense and not necessarily as arising from the limited information model considered in Section 7.3.1.
Premultiplying (7.3.21) and (7.3.22) by *P-I/2, we obtain
l/2y = 4′-i/2za + ‘F"i/2u (7.3.23)
^-i/2Z = ip-i/2Xn + <ir-i/2V. (7.3.24)
We define the G2SLS estimator of a as the 2SLS estimator of a applied to
(7.3.23) and (7.3.24); that is,
Ob2S = [Z’4′-1X(X,4′-1X)-,X,4′-1Z]-1Z/4′-‘X(X’4′-1X)-1X’4′-,y.
Given appropriate assumptions on ‘F-1/2X, *P-1/2u, and Ч’_1/2У, we can show
VTXdcas ~ a) ->N[0, (lim 7’-1ІГХ,’Р-1ХПГ1]. (7.3.26)
As in Section 7.3.6, we can show that G2SLS is asymptotically the best instrumental variables estimator in the model defined by (7.3.21) and (7.3.22). The limit distribution is unchanged if a regular consistent estimator of*P is substituted.
The idea of G2SLS is attributable to Theil (1961), who defined it in another asymptotically equivalent way: (Z/P*P~1Z)-1Z/P*P_1y. It has the same asymptotic distribution as Oo2s-