Question: The observed data is {Yi ,Xi ,Zi } 2 RRkR`, k 1 and ` 1, i 1, ...,n. An econometrician first estimates

The observed data is {Yi ,Xi ,Zi } 2 R£Rk£R`, k È 1 and ` È 1, i Æ 1, ...,n. An econometrician first estimates Yi Æ X0 i

b¯Å bei by least squares. The econometrician next regresses the residual bei on Zi , which can be written as bei Æ Z0 i e°Å eui .

(a) Define the population parameter ° being estimated in this second regression.

(b) Find the probability limit for e°.

(c) Suppose the econometrician constructs a Wald statistic W for H0 : ° Æ 0 from the second regression, ignoring the two-stage estimation process. Write down the formula forW.

(d) Assume E

Z X0¤

Æ 0. Find the asymptotic distribution forW under H0 : ° Æ 0.

(e) If E

Z X0¤

6Æ 0 will your answer to

(d) change?

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