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14 215 are exactly the o 1 n o directly replacing the

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Unformatted text preview: tly replacing the population objects µy , µx , var (x ) , cov (x , y ) in (4) with their sample counterparts. ˆ MM ˆ MM ... Recall, β , β o 1 regarded as functions of the RVs X and Y are estimators of ( βo , β1 ); Melissa Tartari (Yale) Econometrics 27 / 93 The Method of Moments Approach VI More generally, the LLN allows us to rewrite the system (2.12 & 2.13) in terms of the sample counterparts (or analogues) of E [u ] and E [xu ]. We can thus de…ne the MoM estimators as the solution to the system: ∑n=1 xi (yi βo β1 xi ) = 0 i . ∑n=1 yi βo β1 xi = 0 i (2.14 & 2.15) It goes without saying that the estimators obtained as solutions to the ˆ MM ˆ MM obtained by system (2.14 & 2.15) are exactly the βo , β1 n o directly replacing the population objects µy , µx , var (x ) , cov (x , y ) in (4) with their sample counterparts. ˆ MM ˆ MM ... Recall, β , β o 1 regarded as functions of the RVs X and Y are estimators of ( βo , β1 ); evaluated at a speci…c data sample are estimates of ( βo , β1 ). Melissa Tartari (Yale) Econometrics 27 / 93 The Method of Moments Approach VI You may want to to do the following for practice: Melissa Tartari (Yale) Econometrics 28 / 93 The Method of Moments Approach VI You may want to to do the following for practice: Assume ZCMA but allow E [U ] = λ 6= 0. Melissa Tartari (Yale) Econometrics 28 / 93 The Method of Moments Approach VI You may want to to do the following for practice: Assume ZCMA but allow E [U ] = λ 6= 0. Show that the moment condition E [(u E (u )) x ] = 0, implied by the ZCMA, is su¢ cient to identify β1 . Melissa Tartari (Yale) Econometrics 28 / 93 The Method of Moments Approach VI You may want to to do the following for practice: Assume ZCMA but allow E [U ] = λ 6= 0. Show that the moment condition E [(u E (u )) x ] = 0, implied by the ZCMA, is su¢ cient to identify β1 . Derive the implied sample analogue-based MoM estimator of β1 . Melissa Tartari (Yale) Econometrics 28 / 93 The Method of Moments Approach VI You may want to to do the following for practi...
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