Question: Consider the structural equation and reduced form Y X2 e X Z u E[Ze] 0 E[Zu] 0 with X2 treated as

Consider the structural equation and reduced form Y Æ ¯X2 Åe X Æ °Z Åu E[Ze] Æ 0 E[Zu] Æ 0 with X2 treated as endogenous so that E

X2e

6Æ 0. For simplicity assume no intercepts. Y , Z, and X are scalar. Assume ° 6Æ 0. Consider the following estimator. First, estimate ° by OLS of X on Z and construct the fitted values bXi Æ b°Zi . Second, estimate ¯ by OLS of Yi on

bXi

¢2.

(a) Write out this estimator b¯ explicitly as a function of the sample.

(b) Find its probability limit as n!1.

(c) In general, is b¯ consistent for ¯? Is there a reasonable condition under which b¯ is consistent?

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