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.There is not enough information to compute the matrix 𝛀 0 .
B.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .
C.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 ] .
D.The matrix 𝛀 0 is: 𝛀 0 = 𝔌 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .
E.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 ] .
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标准答案
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思路分析
We begin by restating the core question: in a GMM setup for a linear regression with moments g1, g2, g3, what is the form of the Γ0 matrix, i.e., the matrix of expected derivatives of the moment conditions with respect to the parameters (α, β, γ)? The moment conditions are: g1 = y_i − α − β x_i − γ x_i^2 g2 = (y_i − α − β x_i − γ x_i^2) x_i g3 = (y_i − α − β x_i − γ x_i^2) x_i^2 We compute the partial derivatives of each moment condition with respect to each parameter. This yields a 3×3 matrix whose (r,c) entry is ∂g_r / ∂ξ_c, where Ξ = (α, β, γ). For g1, the derivatives are: ∂g1/∂α = −1, ∂g1/∂β = −x_i, ∂g1/∂γ = −x_i^2. For g2, the derivatives are: ∂g2/∂α = −x_i, ∂g2/∂β = −x_i^2, ∂g2/∂γ = −x_i^3. For g3, the derivatives are: ∂g3/∂α = −x_i^2, ∂g3/∂β = ......Login to view full explanation

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类䌌问题

Consider the following nonlinear regression model: 𝑊 𝑡 = 𝛌 𝑥 𝑡 𝛜 + 𝜀 𝑡 Assume i.i.d. data and 𝔌 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛌 and 𝛜 by GMM, we need two moment conditions. Choose the best answer below.

Consider the following nonlinear regression model: 𝑊 𝑡 = 𝛌 𝑥 𝑡 𝛜 + 𝜀 𝑡 Assume i.i.d. data and 𝔌 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛌 and 𝛜 by GMM, we chose among the following moment conditions: 𝔌 [ 𝑊 𝑡 − 𝛌 𝑥 𝑡 𝛜 ] = 0 𝔌 [ ( 𝑊 𝑡 − 𝛌 𝑥 𝑡 𝛜 ) 𝑥 𝑡 ] = 0 𝔌 [ ( 𝑊 𝑡 − 𝛌 𝑥 𝑡 𝛜 ) 1 𝑥 𝑡 ] = 0 Choose the most appropriate answer below:

Consider the following nonlinear regression model: 𝑊 𝑡 = 𝛌 𝑥 𝑡 𝛜 + 𝜀 𝑡 Assume i.i.d. data and 𝔌 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛌 and 𝛜 by GMM, we use the following moment conditions: 𝔌 [ 𝑊 𝑡 − 𝛌 𝑥 𝑡 𝛜 ] = 0 𝔌 [ ( 𝑊 𝑡 − 𝛌 𝑥 𝑡 𝛜 ) 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 𝛌 ̂ = 1 and 𝛜 ̂ = 2 . To compute the variance of the estimates, we need to estimate the matrix 𝛀 0 , 𝛀 ̂ 0 = [ 𝛀 ̂ 11 𝛀 ̂ 12 𝛀 ̂ 21 𝛀 ̂ 22 ] Then, the value 𝛀 ̂ 11 is:

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

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