slides_Ch2_W[1]

2b after some algebra we have e u jx 1 fa pb pa a

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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...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.

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