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Consider the following nonlinear regression model: 𝑦 𝑡 = 𝛼 𝑥 𝑡 𝛽 + 𝜀 𝑡 Assume i.i.d. data and 𝔼 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we use the following moment conditions: 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 𝑥 𝑡 ] = 0 We have an i.i.d. sample with 𝑇 = 1000 observations, with ∑ 𝑡 = 1 𝑇 𝑥 𝑡 = 1000 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 4000 . We obtain point estimates 𝛼 ̂ = 1 and 𝛽 ̂ = 2 . To compute the variance of the estimates, we need to estimate the matrix 𝛤 0 , 𝛤 ̂ 0 = [ 𝛤 ̂ 11 𝛤 ̂ 12 𝛤 ̂ 21 𝛤 ̂ 22 ] Then, the value 𝛤 ̂ 11 is:
Options
A.𝛤
̂
11
=
−
4
B.𝛤
̂
11
=
4000
C.There is not enough information to compute
𝛤
̂
11
.
D.𝛤
̂
11
=
−
1
E.𝛤
̂
11
=
1000
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Step-by-Step Analysis
We start by restating the problem setup and the moment conditions used for GMM estimation. The model is y_t = α x_t β + ε_t with i.i.d. data and E[ε_t | x_t] = 0. The moment conditions are:
1) E[y_t − α x_t β] = 0
2) E[(y_t − α x_t β) x_t] = 0
We are given T = 1000, ∑ x_t = 1000, and ∑ x_t^2 = 4000, with the point estimates α̂ = 1 and β̂ = 2.
To form the Γ0 matrix, which is the expected Jacobian of the moment conditions with respect to the parameters θ = (α, β) evaluated at the true values, we compute the derivatives of the moments:
- For g1_t = y_t − α x_t β, the part......Login to view full explanationLog in for full answers
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Similar Questions
Consider the following nonlinear regression model: 𝑦 𝑡 = 𝛼 𝑥 𝑡 𝛽 + 𝜀 𝑡 Assume i.i.d. data and 𝔼 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we need two moment conditions. Choose the best answer below.
Consider the following nonlinear regression model: 𝑦 𝑡 = 𝛼 𝑥 𝑡 𝛽 + 𝜀 𝑡 Assume i.i.d. data and 𝔼 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we chose among the following moment conditions: 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 𝑥 𝑡 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 1 𝑥 𝑡 ] = 0 Choose the most appropriate answer below:
Consider the following nonlinear regression model: 𝑦 𝑖 = 𝛼 + 𝛽 𝑥 𝑖 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we use the two theoretical moment conditions 𝔼 [ 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 𝑥 𝑖 ] = 0 To compute the variance of the GMM estimator we need the matrices 𝛤 0 and 𝛷 0 .
Consider the following linear regression model: 𝑦 𝑖 = 𝛼 + 𝛽 𝑥 𝑖 + 𝛾 𝑥 𝑖 2 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 , 𝛽 and 𝛾 by GMM, we use the three theoretical moment conditions 𝔼 [ 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ) 𝑥 𝑖 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ) 𝑥 𝑖 2 ] = 0 To compute the variance of the GMM estimator we need the matrices 𝛤 0 and 𝛷 0 .
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