slides_Ch2_W[1]

Slides_Ch2_W[1]

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Unformatted text preview: β1 = 1 Ex 2 2 =4 2 Ey 2 µy =4 =4 Melissa Tartari (Yale) (Ex ) Ex 2 Ey Ex 2 Exy Ex 2 2 E xExy (Ex )2 E xEy (Ex )2 Ex 2 Ex 3 5 E xy E xEy Ex Ex 2 (Ex )2 Exy E xEy Ex 2 (Ex )2 Ex 1 3 5 E [(y µy )(x µx )] µx E [(x µx )2 ] E [(y µy )(x µx )] E [(x µx )2 ] Econometrics Ey Eyx 3 5 (4) 24 / 93 The Method of Moments Approach IV Observe that (4) express the uknown parameters ( βo , β1 ) as functions n o of the unknown population objects µy , µx , var (x ) , cov (x , y ) . Have we then made any progress? Melissa Tartari (Yale) Econometrics 25 / 93 The Method of Moments Approach IV Observe that (4) express the uknown parameters ( βo , β1 ) as functions n o of the unknown population objects µy , µx , var (x ) , cov (x , y ) . Have we then made any progress? It is true that the expressions in (4) do not provide us with estimators of ( βo , β1 ); however, they suggest estimators. How? Melissa Tartari (Yale) Econometrics 25 / 93 The Method of Moments Approach IV Observe that (4) express the uknown parameters ( βo , β1 ) as functions n o of the unknown population objects µy , µx , var (x ) , cov (x , y ) . Have we then made any progress? It is true that the expressions in (4) do not provide us with estimators of ( βo , β1 ); however, they suggest estimators. How? The idea of the MoM approach is to replace the unknown population n o objects µy , µx , var (x ) , cov (x , y ) in (4) with their sample counterparts (or analogues) by exploting the LLN which tells us that: 1n 1n 1n p p xi ! µx , ∑ yi ! µy , ∑ (xi n i∑ n i =1 n i =1 =1 and Melissa Tartari (Yale) 1n (yi n i∑ =1 y ) (xi Econometrics p x )2 ! Var (x ) ¯ p x ) ! cov (x , y ) . ¯ 25 / 93 The Method of Moments Approach V Thus, using (4) we de…ne MoM estimators: β1 = βo cov (x , y ) Var (x ) = µy Melissa Tartari (Yale) β1 µx MoM Approach ) MoM Approach ) ¯ ∑n (yi y ) (xi x ) ˆ MM (2.19) β 1 = i =1 n 2 ¯ ∑i =1 (xi x ) ˆ MM βo = y ¯ Econometrics ˆ MM ¯ β1 x (2.17) 26 / 93 The Method of Moments Approach VI More generally, the LLN allows us to rewrite the system (2.1...
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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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