Suppose that for the random variables (Y ,X) with X 0 an economic model implies E[Y j X] X 1/2 . (4.64)

Suppose that for the random variables (Y ,X) with X È 0 an economic model implies E[Y j X] Æ

¡

°ÅµX

¢1/2 . (4.64)

A friend suggests that you estimate ° and µ by the linear regression of Y 2 on X, that is, to estimate the equation Y 2 Æ ®Å¯X Åe. (4.65)

(a) Investigate your friend’s suggestion. Define u Æ Y ¡

¡

°ÅµX

¢1/2 . Show that E[u j X] Æ 0 is implied by (4.64).

(b) Use Y Æ

¡

°ÅµX

¢1/2

Åu to calculate E

£

Y 2 j X

¤

. What does this tell you about the implied equation

(4.65)?

(c) Can you recover either ° and/or µ from estimation of (4.65)? Are additional assumptions required?

(d) Is this a reasonable suggestion?