This preview shows page 1. Sign up to view the full content.
Unformatted text preview: n E [U ] = 0.
Recall
E [U ] = E [U jA] Pr (A) + E [U jB ] Pr (B ) = µ A fA + µ B ( 1 fA ) Conclude E [U ] = 0 i¤
µB = Melissa Tartari (Yale) µA Econometrics fA
1 fA (11) 73 / 93 Failure of SLR.2: An Example (Q.2.b)
The question is: what does E [U jX = 1] = E [U jX = 0] imply for
(µA , µB , pA , pB )?
Before answering that question we need to take care of a little issue:
we need to impose the normalization E [U ] = 0.
Recall
E [U ] = E [U jA] Pr (A) + E [U jB ] Pr (B ) = µ A fA + µ B ( 1 fA ) Conclude E [U ] = 0 i¤
µB = µA fA
1 fA (11) We use (11) to replace µB in expressions (10) and (9): in so doing we
impose the normalization.
Melissa Tartari (Yale) Econometrics 73 / 93 Failure of SLR.2: An Example (Q.2.b)
After some algebra, we have
E [U jX = 1] = Melissa Tartari (Yale) µ fA (pB pA )
µA fA (1 pB )
; E [U jX = 0] = A
Pr (X = 1)
1 Pr (X = 1) Econometrics 74 / 93 Failure of SLR.2: An Example (Q.2.b)
After some algebra, we have
E [U jX = 1] = µ fA (pB pA )
µA fA (1 pB )
; E [U jX = 0] = A
Pr (X = 1)
1 Pr (X = 1) We subtract E [U jX = 1]
numerator
0 = (1 Pr (X = 1)) µA fA (1 = ...
= µA fA [(1 Melissa Tartari (Yale) E [U jX = 0] and equate to zero the pB ) pB ) Pr (X = 1) (1 Econometrics Pr (X = 1) µA fA (pB pA ) pA )] 74 / 93 Failure of SLR.2: An Example (Q.2.b)
After some algebra, we have
E [U jX = 1] = µ fA (pB pA )
µA fA (1 pB )
; E [U jX = 0] = A
Pr (X = 1)
1 Pr (X = 1) We subtract E [U jX = 1]
numerator
0 = (1 E [U jX = 0] and equate to zero the Pr (X = 1)) µA fA (1 = ...
= µA fA [(1 pB ) pB ) Pr (X = 1) (1 Pr (X = 1) µA fA (pB pA ) pA )] Thus, for E [U jX ] = 0 it is su¢ cient that one of the following
conditions holds (it is necessary that at least one holds ): Melissa Tartari (Yale) Econometrics 74 / 93 Failure of SLR.2: An Example (Q.2.b)
After some algebra, we have
E [U jX = 1] = µ fA (pB pA )
µA fA (1 pB )
; E [U jX = 0] = A
Pr (X = 1)
1 Pr (X = 1) We subtract E [U jX = 1]
numerator
0 = (1 E [U jX = 0] and equate to zero the Pr (X = 1)) µA fA (1 = ...
= µA fA [(1 pB ) pB ) Pr (X = 1) (1 Pr (X = 1) µA fA (pB pA ) pA )] Thus, for E [U jX ] = 0 it is su¢ cient that one...
View
Full
Document
This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.
 Fall '10
 DonaldBrown
 Econometrics

Click to edit the document details