## Simultaneous Equations Model

11.1 The Inconsistency of OLS. The OLS estimator from Eq. (11.14) yields 8ols =

E Ptqt/ E Pt2 where pt = Pt — l3 and qt = Qt — Q. Substituting qt = 8pt C

t=i t=i

TT

(u2t — U2) from (11.14) we get 8ois = 8 C E Pt(u2t — N/Y, P2. Using (11.18),

t=i t=i

T

we get plim £ Pt(U2t — U2)/T = (012 — 022)/(8 — ") where Oij = cov. Uit, Ujt)

t=1

for i, j = 1,2 and t = 1,2,.., T. Using (11.20) we get

Plim 8ois = 8 C [(CT12 — 022)/(8 — ")]/[(011 C CT22 — 2ст12)/(8 — ")2] = 8 C (o12 — 022)(8 — ")/(o11 C 022 — 2°12).

11.2 When Is the IVEstimator Consistent?

a. ForEq.(11.30)y1 = a^y2 C ‘1зУз C "11X1 C "12X2 C U1.Whenweregress

T

У2 on X1, X2 and X3 to get У2 = у2 C V2, the residuals satisfy E y2tV2t = 0

t=1

TTT

and 22 V2tXu = 22 V2tX2t = 22 V2tX3t = 0.

t=1 t=1 t=1

Simi...

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