slides_Ch2_W[1]

X 00 7 93 identication the claim proof let x be any

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Unformatted text preview: Let x 0 and x 00 be two di¤erent values of X , then E Y jX = x 0 E Y jX = x 00 = βo + β1 x 0 . = βo + β1 x 00 . Subtract the 2nd eq from the …rst and solve for β1 : β1 = Melissa Tartari (Yale) E [Y jX = x 0 ] (x 0 Econometrics E [Y jX = x 00 ] . x 00 ) 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 = βo + β1 x 0 . = βo + β1 x 00 . Subtract the 2nd eq from the …rst and solve for β1 : β1 = E [Y jX = x 0 ] (x 0 E [Y jX = x 00 ] . x 00 ) Take the …rst eq (or the 2nd) and solve for βo : βo = E Y jX = x 0 Melissa Tartari (Yale) E [Y jX = x 0 ] (x 0 Econometrics E [Y jX = x 00 ] 0 x. x 00 ) 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 = βo + β1 x 0 . = βo + β1 x 00 . Subtract the 2nd eq from the …rst and solve for β1 : β1 = E [Y jX = x 0 ] (x 0 E [Y jX = x 00 ] . x 00 ) Take the …rst eq (or the 2nd) and solve for βo : βo = E Y jX = x 0 E [Y jX = x 0 ] (x 0 E [Y jX = x 00 ] 0 x. x 00 ) Finally, observe that E [Y jX = x 0 ],E [Y jX = x 00 ] , and E [Y jX = x 0 ] are known if we have access to the population (recall LLN). Melissa Tartari (Yale) Econometrics 7 / 93 Identi…cation: the road-map We just showed that in the SLRM the ZCMA is su¢ cient for identication of ( βo , β1 ). Melissa Tartari (Yale) Econometrics 8 / 93 Identi…cation: the road-map We just showed that in the SLRM the ZCMA is su¢ cient for identi…cation of ( βo , β1 ). We next delve into the meaning of the ZCMA. Melissa Tartari (Yale) Econometrics 8 / 93 Identi…cation: the road-map We just showed that in the SLRM the ZCMA is su¢ cient for identi…cation of ( βo , β1 ). We next delve into the meaning of the ZCMA. We …nally introduce three estimation methods, that is, three procedures to obtain estimates of...
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