Questions
Questions

BU.232.630.W1.SP25 Quiz 2

Single choice

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 chose among the following moment conditions: 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 𝑥 𝑡 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 1 𝑥 𝑡 ] = 0 Choose the most appropriate answer below:

Options
A.Only the second and third equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
B.Only the first and third equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
C.Only the first and second equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
D.None of the equations above are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
E.All equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
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Step-by-Step Analysis
Question restatement: Consider the nonlinear regression model y_t = α x_t^β + ε_t with i.i.d. data and E[ε_t | x_t] = 0. To estimate α and β by GMM, we consider these moment conditions: E[y_t − α x_t^β] = 0, E[(y_t − α x_t^β) x_t] = 0, and E[(y_t − α x_t^β) (1/x_t)] = 0. The multiple-choice options propose different subsets of these three as valid moment conditions for GMM estimation. Option-by-option analysis: Option 1: "Only the second and third equations are valid moment conditions to estimate α and β by GMM." - Why this might be tempting: It relies on the idea that multiplying the residual by regressor(s) or its reciprocal can create orthogonality conditions. - Why this is likely incorrect: The first moment condition, E[y_t − α x_t^β] = 0, arises from E[ε_t] = 0, which follows from E[ε_t | x_t] = 0 if E[ε_t] exists. Since ε_t is mean-zero conditional on ......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 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.

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 .

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