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Consider the following nonlinear regression model: š¦ š” = š¼ š„ š” š½ + š š” Assume i.i.d. data and š¼ [ š š” | š„ š” ] = 0 . To estimate š¼ and š½ by GMM, we use the following moment conditions: š¼ [ š¦ š” ā š¼ š„ š” š½ ] = 0 š¼ [ ( š¦ š” ā š¼ š„ š” š½ ) š„ š” ] = 0 We have an i.i.d. sample with š = 2000 observations, with ā š” = 1 š š„ š” = 3000 and ā š” = 1 š š„ š” 2 = 5000 . We obtain point estimates š¼ Ģ = ā 1 and š½ Ģ = 2 . To compute the variance of the estimates, we need to estimate the matrix š¤ 0 , š¤ Ģ 0 = [ š¤ Ģ 11 š¤ Ģ 12 š¤ Ģ 21 š¤ Ģ 22 ] Then, the value š¤ Ģ 11 is:
é锹
A.š¤
Ģ
11
=
ā
1.5
B.š¤
Ģ
11
=
1.5
C.There is not enough information to compute
š¤
Ģ
11
.
D.š¤
Ģ
11
=
ā
2.5
E.š¤
Ģ
11
=
5000
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We start by restating the question setup to ensure clarity: we have a nonlinear regression model given by y_t = α x_t β (with ε_t as the error term and E[ε_t|x_t] = 0), and the GMM moment conditions are:
E[y_t ā α x_t β] = 0 and E[(y_t ā α x_t β) x_t] = 0.
An i.i.d. sample with T = 2000 observations is provided, along with sample averages: (1/T)ā x_t = 3000 and (1/T)ā x_t^2 = 5000. The GMM estimates are Ī±Ģ = ā1 and Ī²Ģ = 2. We are asked to determine the estimated value of ĪĢ11, the (1,1) element of the matrix ĪĢ0, where Ī0 is the Jacobian (matrix of partial derivatives) of the moment conditions with respect to the parameters Īø = (α, β).
Step-by-step analysis of each option:
Option A: ĪĢ11 = ā1.5
- To obtain ĪĢ11,......Login to view full explanationē»å½å³åÆę„ēå®ę“ēę”
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Consider the following nonlinear regression model: š¦ š” = š¼ š„ š” š½ + š š” Assume i.i.d. data and š¼ [ š š” | š„ š” ] = 0 . To estimate š¼ and š½ by GMM, we use the following moment conditions: š¼ [ š¦ š” ā š¼ š„ š” š½ ] = 0 š¼ [ ( š¦ š” ā š¼ š„ š” š½ ) š„ š” ] = 0 To compute the variance of the estimates, we need to estimate the matrices š¤ 0 and š· 0 .
Consider the following nonlinear regression model: yi=α+βxi+εi, Assume i.i.d. data and š¼[εi|xi]=0. To estimate α and β by GMM, we need at least two moment conditions, and we use š¼[yiāαāβxi]=0 š¼[(yiāαāβxi)xiβxiā1]=0 Chose the correct answer below.
Consider the following nonlinear regression model: š¦ š” = š¼ š„ š” š½ + š š” Assume i.i.d. data and š¼ [ š š” | š„ š” ] = 0 . To estimate š¼ and š½ by GMM, we use the following moment conditions: š¼ [ š¦ š” ā š¼ š„ š” š½ ] = 0 š¼ [ ( š¦ š” ā š¼ š„ š” š½ ) š„ š” ] = 0 We have an i.i.d. sample with š = 1000 observations, with ā š” = 1 š š„ š” = 3000 and ā š” = 1 š š„ š” 2 = 5000 . We obtain point estimates š¼ Ģ = ā 3 and š½ Ģ = 2 . To compute the variance of the estimates, we need to estimate the matrix š¤ 0 , š¤ Ģ 0 = [ š¤ Ģ 11 š¤ Ģ 12 š¤ Ģ 21 š¤ Ģ 22 ] Then, the value š¤ Ģ 11 is:
Consider the following nonlinear regression model: š¦ š” = š¼ š„ š” š½ + š š” Assume i.i.d. data and š¼ [ š š” | š„ š” ] = 0 . To estimate š¼ and š½ by GMM, we use the following moment conditions: š¼ [ š¦ š” ā š¼ š„ š” š½ ] = 0 š¼ [ ( š¦ š” ā š¼ š„ š” š½ ) š„ š” ] = 0 We have an i.i.d. sample with š = 1000 observations, with ā š” = 1 š š„ š” = 100 , ā š” = 1 š š„ š” 2 = 200 and ā š” = 1 š š„ š” 3 = 800 . We obtain point estimates š¼ Ģ = ā 1 and š½ Ģ = 3 . To compute the variance of the estimates, we need to estimate the matrix š¤ 0 , š¤ Ģ 0 = [ š¤ Ģ 11 š¤ Ģ 12 š¤ Ģ 21 š¤ Ģ 22 ] Then, the value š¤ Ģ 11 is:
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