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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.
 Fall '10
 DonaldBrown
 Econometrics

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