Consider the structural equation Y 0 1X 2X2 e (12.93) with X 2 R treated as endogenous so that E[Xe] 6 0. We have

Consider the structural equation Y Æ ¯0 ů1X ů2X2 Åe (12.93)

with X 2 R treated as endogenous so that E[Xe] 6Æ 0. We have an instrument Z 2 R which satisfies E[e j Z] Æ 0 so in particular E[e] Æ 0 , E[Ze] Æ 0 and E

£

Z2e

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Æ 0.

(a) Should X2 be treated as endogenous or exogenous?

(b) Suppose we have a scalar instrument Z which satisfies X Æ °0 Å°1Z Åu (12.94)

with u independent of Z and mean zero.

Consider using (1,Z,Z2) as instruments. Is this a sufficient number of instruments? Is (12.93)

just-identified, over-identified, or under-identified?

(c) Write out the reduced form equation for X2. Under what condition on the reduced form parameters

(12.94) are the parameters in (12.93) identified?