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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 .

Options
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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Step-by-Step Analysis
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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