Bruce E. Hansen, Econometrics Chapter12. Instrumental Variables

Exercise 12.12 Consider the structural equation and reduced form Y=sX2+e X: yZ+u |E|Ze|=0 |E|Zul=0 with X2 treated as endogenous so that [E [X29] 0. For simplicity assume no intercepts. Y, Z. and X are scalar. Assume y 0. Consider the following estimator. First. estimate y by OLS of X on Z and constrttct the tted values if; = ?Z;. Second. estimate )3 by OLS of Y; on [53,-]2. [a] Write out this estimator S explicitly as a function ofthe sample. (b) Find its probability limit as :1 ~ 00. (c) In general. is 6 consistent for [5? Is there a reasonable condition under which is consistent