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U : for any given value x of X , the mean of U is the same hence it is
equal to the mean of U in the entire population. Formally,
E [U jX ] = E [U ] Melissa Tartari (Yale) Econometrics by 2.5 =0 (2.6) 6 / 93 Identi…cation: the claim
The zero conditional mean assumption (ZCMA)
Consider the following restriction on the relationship between X and
U : for any given value x of X , the mean of U is the same hence it is
equal to the mean of U in the entire population. Formally,
E [U jX ] = E [U ] by 2.5 =0 (2.6) Claim
Consider the SLRM Y = βo + β1 X + U , E [U ] = 0 and suppose that
the RV X takes at least two values. If the ZCMA holds then ( βo , β1 )
are identi…ed. Melissa Tartari (Yale) Econometrics 6 / 93 Identi…cation: the claim
The zero conditional mean assumption (ZCMA)
Consider the following restriction on the relationship between X and
U : for any given value x of X , the mean of U is the same hence it is
equal to the mean of U in the entire population. Formally,
E [U jX ] = E [U ] by 2.5 =0 (2.6) Claim
Consider the SLRM Y = βo + β1 X + U , E [U ] = 0 and suppose that
the RV X takes at least two values. If the ZCMA holds then ( βo , β1 )
are identi…ed.
Working De…nition of Identi…cation: a parameter is identi…ed we can
learn it from data on the entire population (or an in…nitely large
sample from the population).
Melissa Tartari (Yale) Econometrics 6 / 93 Identi…cation: the claim (proof)
Let x be any value of X . Use ZCMA and the normalization E [U ] = 0
to obtain
E [Y jX = x ] = βo + β1 x + 0. Melissa Tartari (Yale) Econometrics 7 / 93 Identi…cation: the claim (proof)
Let x be any value of X . Use ZCMA and the normalization E [U ] = 0
to obtain
E [Y jX = x ] = βo + β1 x + 0.
Let x 0 and x 00 be two di¤erent values of X , then
E Y jX = x 0 E Y jX = x 00 Melissa Tartari (Yale) = βo + β1 x 0 .
= βo + β1 x 00 . Econometrics 7 / 93 Identi…cation: the claim (proof)
Let x be any value of X . Use ZCMA and the normalization E [U ] = 0
to obtain
E [Y jX = x ] = βo + β1 x + 0....
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 Fall '10
 DonaldBrown
 Econometrics

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