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Unformatted text preview: of the following
conditions holds (it is necessary that at least one holds ):
C.1 µA = 0 or 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 of the following
conditions holds (it is necessary that at least one holds ):
C.1 µA = 0 or
C.2 fA = 0 or
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 of the following
conditions holds (it is necessary that at least one holds ):
C.1 µA = 0 or
C.2 fA = 0 or
C.3 [(1 pB ) Pr (X = 1) (1 pA )] = 0.
Melissa Tartari (Yale) Econometrics 74 / 93 Failure of SLR.2: An Example (Q.2.b)
Let us comment on our …ndings:
C.1: µA = 0 implies µA = µB = 0  given the normalization  thus,
C.1 means that ZCMA holds whenever the distribution of U is the
same across the two subpop.s: mean ability/drive is the same among
2yr college students and 4yr college students. As long as the two
groups are not di¤erent in terms of their unobservables they can be
di¤erent in terms of their observables (pA 6= pB is allowed) and there
can be over/under sampling without either situation causing a bias in
the OLS estimator of β1 . Melissa Tartari (Yale) Econometrics 75 / 93 Failure of SLR.2: An Example (Q.2.b)
Let us comment on our …ndings:
C.1: µA = 0 implies µA = µB = 0  given the normalization  thus,
C.1 means that ZCMA holds whenever the distribution of U is the
same across the two subpop.s: mean ability/drive is t...
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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.
 Fall '10
 DonaldBrown
 Econometrics

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