Consider regression equations

у = Za + u

(7.3.21)

Z = ХП + V,

(7.3.22)

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)

and

^-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.

(7.3.25)

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-

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