slides_Ch2_W[1]

S in the two unknowns o 1 in matrix notation 1 ex

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Unformatted text preview: 2 & 2.13) Observe that (2.12 & 2.13) imply 2 restrictions on fY ,X . Melissa Tartari (Yale) Econometrics 21 / 93 The MoM: the Basics Observe that because of E [u jx ] = 0 we also have that E [ux ] = E [E [ux jx ]] = E [xE [u jx ]] = E [x 0] = 0. Thus, we have two moment conditions : E [u ] = 0 ) E [xu ] = 0 E [y βo β1 x ] = 0 E [x (y βo β1 x )] = 0 (2.12 & 2.13) Observe that (2.12 & 2.13) imply 2 restrictions on fY ,X . (2.12 & 2.13) is a linear system of two eq.s in the two unknowns ( βo , β1 ). In matrix notation: 1 Ex Melissa Tartari (Yale) Ex Ex 2 βo β1 = Econometrics Ey Eyx or A β = h 21 / 93 The MoM: the Solution of the System I We can easily solve the system (2.12 & 2.13) for ( βo , β1 ) as β=A Melissa Tartari (Yale) Econometrics 1 h. 22 / 93 The MoM: the Solution of the System I We can easily solve the system (2.12 & 2.13) for ( βo , β1 ) as β=A 1 h. The solution exists and is unique as long as the matrix A is not singular, hence invertible. Melissa Tartari (Yale) Econometrics 22 / 93 The MoM: the Solution of the System I We can easily solve the system (2.12 & 2.13) for ( βo , β1 ) as β=A 1 h. The solution exists and is unique as long as the matrix A is not singular, hence invertible. The system being determinate is synonymous with ( βo , β1 ) being identi…ed. Melissa Tartari (Yale) Econometrics 22 / 93 The MoM: the Solution of the System II In the next slide we solve the system. De…ne the 2 by 2 matrix A= Melissa Tartari (Yale) ab cd Econometrics . 23 / 93 The MoM: the Solution of the System II In the next slide we solve the system. De…ne the 2 by 2 matrix A= ab cd . Recall that A Melissa Tartari (Yale) 1 = 1 1 Aadj = det A ad bc Econometrics d c b a . 23 / 93 The MoM: the Solution of the System II In the next slide we solve the system. De…ne the 2 by 2 matrix A= ab cd . Recall that A 1 = 1 1 Aadj = det A ad bc d c b a . For convenience we let µx denote Ex and µy denote Ey . Melissa Tartari (Yale) Econometrics 23 / 93 The MoM: the Solution of the System II In the next slide we solve the system. De…ne the 2 by 2 matrix A= ab cd . Recall that A 1 = 1 1 Aadj = det A ad bc d c b a . For convenience we let µx denote Ex and µy denote Ey . Feel free to verify the derivations in the next page. Melissa Tartari (Yale) Econometrics 23 / 93 The MoM: the Solution of the System III βo...
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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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