题目
题目
单项选择题

Consider the following linear 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 three theoretical moment conditions 𝔼 [ 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 𝑥 𝑖 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 𝑥 𝑖 2 ] = 0 What is the optimal weight matrix 𝑊 to estimate 𝛼 and 𝛽 ?

选项
A.The optimal weight matrix 𝑊 is: 𝑊 = ( 𝔼 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] ) − 1 .
B.There is not enough information to compute the matrix 𝛤 0 .
C.The optimal weight matrix 𝑊 is: 𝑊 = 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 4 ]
D.The optimal weight matrix 𝑊 is: 𝑊 = 𝔼 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .
E.The optimal weight matrix 𝑊 is: 𝑊 = ( 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 4 ] ) − 1
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思路分析
We start by restating the setup and the choices so we can evaluate them clearly. Question recap: We have a linear regression model y_i = α + β x_i + ε_i with E[ε_i | x_i] = 0 and i.i.d. data. To estimate α and β by GMM, we use the three moment conditions E[y_i − α − β x_i] = 0, E[(y_i − α − β x_i) x_i] = 0, and E[(y_i − α − β x_i) x_i^2] = 0. The question asks for the optimal weight matrix W in the GMM estimation of (α, β). Option-by-option analysis: Option 1: “The optimal weight matrix W is: W = ( E[ −1 − x_i ; −x_i ; −x_i^2 ; −x_i^2 ; −x_i^3 ] )^{−1}.” - What it appears to be attempting: some inverse of an expected matrix built from moments or derivatives, but the notation is unclear and inconsistent with the standard GMM form. The true optimal W in GMM with the moment vector g_i(θ) = [y_i − α − β x_i, (y_i − α − β x_i) x_i, (y_i − α − β x_i) x_i^2]ᵀ is the inverse of the variance-covariance matrix of those moment conditions, i.e., W = [E(g_i(θ) g_i(θ)ᵀ)]^{-1} evaluated at the true θ (or its consistent estimate). The representation in Option 1 does not align with E[g gᵀ] nor with the standard form; it seems to mix elements and signs incorrectly and includes terms like −1 and −x_i as if they were a block of a matrix in a way that would not yield a proper 3×3 weight matrix. Thus, as stated, this option misidentifies the fundamental building block of the optimal W and provides a structurally dubious matrix. Option 2: “There is not enou......Login to view full explanation

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类似问题

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 To compute the variance of the estimates, we need to estimate the matrices 𝛤 0 and 𝛷 0 .

Consider the following nonlinear regression model: yi=α+βxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we need at least two moment conditions, and we use 𝔼[yi−α−βxi]=0 𝔼[(yi−α−βxi)xiβxi−1]=0 Chose the correct 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 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 = 3000 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 5000 . We obtain point estimates 𝛼 ̂ = − 3 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:

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 = 100 , ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 200 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 3 = 800 . We obtain point estimates 𝛼 ̂ = − 1 and 𝛽 ̂ = 3 . 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:

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