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BU.232.630.W4.SP25 sample_quiz_2

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Consider the following nonlinear regression model: yi=α+eβxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we use the two theoretical moment conditions 𝔼[yi−α−eβxi]=0 𝔼[(yi−α−eβxi)xi]=0 To compute the variance of the GMM estimator we need the matrices Γ0 and Φ0.

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The question asks to identify the matrix Γ0 used in the variance computation for the GMM estimator under the two moment conditions. To approach this, recall that for GMM with moments g_i(θ) where θ = (α, β)', Γ0 is the expected Jacobian of the moment conditions with respect to θ, i.e., Γ0 = E[∂g_i(θ)/∂θ']. Given the two moments g1,i(α,β) = y_i − α − e^{β x_i} g2,i(α,β) = (y_i − α − e^{β x_i}) x_i, we comp......Login to view full explanation

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Consider the following nonlinear regression model: yi=α+eβxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=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 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 = 8000 observations, with ∑ 𝑡 = 1 𝑇 𝑥 𝑡 = 2000 , ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 4000 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 3 = 8000 . We obtain point estimates 𝛼 ̂ = − 5 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:

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 + 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 To compute the variance of the GMM estimator we need the matrices 𝛤 0 and 𝛷 0 .

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