题目
题目
单项选择题

Consider the following linear regression model: 𝑊 𝑖 = 𝛌 + 𝛜 𝑥 𝑖 + 𝛟 𝑥 𝑖 2 + 𝜀 𝑖 , Assume i.i.d. data and 𝔌 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛌 , 𝛜 and 𝛟 by GMM, we use the three theoretical moment conditions 𝔌 [ 𝑊 𝑖 − 𝛌 − 𝛜 𝑥 𝑖 − 𝛟 𝑥 𝑖 2 ] = 0 𝔌 [ ( 𝑊 𝑖 − 𝛌 − 𝛜 𝑥 𝑖 − 𝛟 𝑥 𝑖 2 ) 𝑥 𝑖 ] = 0 𝔌 [ ( 𝑊 𝑖 − 𝛌 − 𝛜 𝑥 𝑖 − 𝛟 𝑥 𝑖 2 ) 𝑥 𝑖 2 ] = 0 To compute the variance of the GMM estimator we need the matrices 𝛀 0 and 𝛷 0 .

选项
A.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ 1 𝑥 𝑖 𝑥 𝑖 2 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 3 𝑥 𝑖 2 𝑥 𝑖 3 𝑥 𝑖 4 ] .
B.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .
C.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .
D.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 3 − 𝑥 𝑖 2 − 𝑥 𝑖 3 − 𝑥 𝑖 4 ] .
E.There is not enough information to compute the matrix 𝛀 0 .
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标准答案
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思路分析
We begin by identifying the three moment conditions used in the GMM estimation: - g1(Ξ) = y_i − α − β x_i − γ x_i^2 - g2(Ξ) = (y_i − α − β x_i − γ x_i^2) x_i - g3(Ξ) = (y_i − α − β x_i − γ x_i^2) x_i^2 Here Ξ = (α, β, γ). To compute the variance of the GMM estimator, we need Γ0, which is the expected Jacobian matrix of the moment functions with respect to Ξ, i.e., Γ0 = E[ ∂g(Ξ)/∂ξ' ], where g(Ξ) stacks g1, g2, g3. Now we differentiate each moment with respect to α, β, γ: - For g1: ∂g1/∂α = −1, ∂g1/∂β = −x_i, ∂g1/∂γ = −x_i^2. - Fo......Login to view full explanation

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类䌌问题

Consider the following nonlinear regression model: yt=αx β t +εt Assume i.i.d. data and 𝔌[εt|xt]=0. To estimate α and β by GMM, we use the following moment conditions: 𝔌[yt−αx β t ]=0 𝔌[(yt−αx β t )xt]=0 To compute the variance of the estimates, we need to estimate the matrices Γ0 and Ί0.

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