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BU.232.630.W6.SP25 Quiz 2 solutions

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Consider the following nonlinear regression model: yt=αx β t +εt Assume i.i.d. data and 𝔼[εt|xt]=0. To estimate α and β by GMM, we use the following moment conditions: 𝔼[yt−αx β t ]=0 𝔼[(yt−αx β t )xt]=0 To compute the variance of the estimates, we need to estimate the matrices Γ0 and Φ0.

Options
A.The estimate of the matrix Γ0 is ˆ Γ 0=[− 1 T ∑ T t=1 x β t − 1 T ∑ T t=1 αx β t log(xt) − 1 T ∑ T t=1 x β+1 t − 1 T ∑ T t=1 αx β+1 t log(xt)]
B.There is not enough information to compute the estimate of the matrix Γ0.
C.The estimate of the matrix Γ0 is ˆ Γ 0=[ 1 T ∑ T t=1 x β t 1 T ∑ T t=1 αx β t log(xt) − 1 T ∑ T t=1 x β+1 t 1 T ∑ T t=1 αx β+1 t log(xt)]
D.The estimate of the matrix Γ0 is ˆ Γ 0=[− 1 T ∑ T t=1 x β t − 1 T ∑ T t=1 αx β t log(xt) − 1 T ∑ T t=1 αx β t log(xt) − 1 T ∑ T t=1 αx β+1 t log(xt)]
E.The estimate of the matrix Γ0 is ˆ Γ 0=[− 1 T ∑ T t=1 x β t − 1 T ∑ T t=1 x β+1 t − 1 T ∑ T t=1 αx β t log(xt) − 1 T ∑ T t=1 αx β+1 t log(xt)]
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Step-by-Step Analysis
We restate the problem setup and list the answer options clearly so you can compare them easily. Question setup: We have the nonlinear regression y_t = α x_t^β_t + ε_t with i.i.d. data and E[ε_t | x_t] = 0. The GMM moment conditions are g_t(θ) = [ y_t − α x_t^β_t , (y_t − α x_t^β_t) x_t ]', with θ = (α, β). The variance of the GMM estimator requires the matrices Γ0 and Φ0, where Γ0 = E[ ∂g_t(θ)/∂θ' ] evaluated at the true θ. The answer options propose sample approximations of Γ0, i.e., a 2×2 matrix whose entries are expectations of the partial derivatives, estimated by their sample analogs. Option analysis: Option A: The estimate of the matrix Γ0 is ˆΓ0 = [ −(1/T) ∑ x_t^β_t −(1/T) ∑ α x_t^β_t log(x_t) −(1/T) ∑ x_t^{β+1} −(1/T) ∑ α x_t^{β+1} log(x_t) ] - Why this matches the theory: You compute ∂g/∂α and ∂g/∂β for each moment. For the first moment m1,t = y_t − α x_t^β, the derivative with respect to α is −x_t^β and with respect to β is −α x_t^β log x_t (since ∂ x_t^β / ∂β = x_t^β log x_t)......Login to view full explanation

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