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Consider the following linear regression model: ðŠ ð = ðŒ + ðœ ð¥ ð + ðŸ ð¥ ð 2 + ð ð , Assume i.i.d. data and ðŒ [ ð ð | ð¥ ð ] = 0 . To estimate ðŒ , ðœ and ðŸ by GMM, we use the three theoretical moment conditions ðŒ [ ðŠ ð â ðŒ â ðœ ð¥ ð â ðŸ ð¥ ð 2 ] = 0 ðŒ [ ( ðŠ ð â ðŒ â ðœ ð¥ ð â ðŸ ð¥ ð 2 ) ð¥ ð ] = 0 ðŒ [ ( ðŠ ð â ðŒ â ðœ ð¥ ð â ðŸ ð¥ ð 2 ) ð¥ ð 2 ] = 0 To compute the variance of the GMM estimator we need the matrices ð€ 0 and ð· 0 .
é项
A.The matrix
ð€
0
is:
ð€
0
=
ðŒ
[
1
ð¥
ð
ð¥
ð
2
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
3
ð¥
ð
2
ð¥
ð
3
ð¥
ð
4
]
.
B.The matrix
ð€
0
is:
ð€
0
=
ðŒ
[
â
1
â
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
3
]
.
C.The matrix
ð€
0
is:
ð€
0
=
ðŒ
[
â
1
â
ð¥
ð
â
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
2
â
ð¥
ð
3
]
.
D.The matrix
ð€
0
is:
ð€
0
=
ðŒ
[
â
1
â
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
â
ð¥
ð
2
â
ð¥
ð
3
â
ð¥
ð
2
â
ð¥
ð
3
â
ð¥
ð
4
]
.
E.There is not enough information to compute the matrix
ð€
0
.
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We begin by identifying the three moment conditions used in the GMM estimation:
- g1(Ξ) = y_i â α â β x_i â γ x_i^2
- g2(Ξ) = (y_i â α â β x_i â γ x_i^2) x_i
- g3(Ξ) = (y_i â α â β x_i â γ x_i^2) x_i^2
Here Ξ = (α, β, γ).
To compute the variance of the GMM estimator, we need Î0, which is the expected Jacobian matrix of the moment functions with respect to Ξ, i.e., Î0 = E[ âg(Ξ)/âΞ' ], where g(Ξ) stacks g1, g2, g3.
Now we differentiate each moment with respect to α, β, γ:
- For g1: âg1/âα = â1, âg1/âβ = âx_i, âg1/âγ = âx_i^2.
- Fo......Login to view full explanationç»åœå³å¯æ¥ç宿Žçæ¡
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Consider the following nonlinear regression model: yt=αx β t +εt Assume i.i.d. data and ðŒ[εt|xt]=0. To estimate α and β by GMM, we use the following moment conditions: ðŒ[ytâαx β t ]=0 ðŒ[(ytâαx β t )xt]=0 To compute the variance of the estimates, we need to estimate the matrices Î0 and Ί0.
Consider the following nonlinear regression model: yi=α+x β i +εi, Assume i.i.d. data and ðŒ[εi|xi]=0. To estimate α and β by GMM, we use the two theoretical moment conditions ðŒ[yiâαâx β i ]=0 ðŒ[(yiâαâx β i )xi]=0 To compute the variance of the GMM estimator we need the matrices Î0 and Ί0.
Consider the following nonlinear regression model: ðŠ ð = ðŒ + ðœ ð¥ ð + ð ð , Assume i.i.d. data and ðŒ [ ð ð | ð¥ ð ] = 0 . To estimate ðŒ and ðœ by GMM, we need at least two moment conditions, and we use ðŒ [ ðŠ ð â ðŒ â ðœ ð¥ ð ] = 0 ðŒ [ ( ðŠ ð â ðŒ â ðœ ð¥ ð ) ð¥ ð ðœ ð¥ ð â 1 ] = 0 Chose the correct answer below.
Consider the following nonlinear regression model: yt=αx β t +εt Assume i.i.d. data and ðŒ[εt|xt]=0. To estimate α and β by GMM, we chose among the following moment conditions: ðŒ[ytâαx β t ]=0 ðŒ[(ytâαx β t )xt]=0 ðŒ[(ytâαx β t ) 1 xt ]=0 Choose the most appropriate answer below:
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