题目
题目

BU.232.630.W1.SP25 Quiz 2 solutions

单项选择题

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:

选项
A.Only the first and third equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
B.Only the second and third equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
C.All equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
D.Only the first and second equations are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
E.None of the equations above are valid moment conditions to estimate 𝛼 and 𝛽 by GMM.
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标准答案
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思路分析
To tackle the question, I’ll lay out the context and then inspect each proposed moment condition in turn. Option 1: "Only the first and third equations are valid moment conditions to estimate α and β by GMM." - The first moment condition is E[y_t − α x_t β] = 0. This corresponds to setting the mean of the error term e_t = y_t − α x_t β to zero, which is a standard starting point for GMM if e_t is exogenous with respect to the instruments used. - The third moment condition is E[(y_t − α x_t β) · (1/x_t)] = 0. Since 1/x_t is a function of x_t and if E[e_t | x_t] = 0, then E[e_t · (1/x_t)] = 0 as long as x_t ≠ 0 with non-negligible probability. This is a valid moment condition as it uses an instrument (1/x_t) that is uncorrelated with the error term under the exogeneity assumption. - However, the claim that the second condition is not valid contradicts the logic that any funct......Login to view full explanation

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类似问题

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 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 = 1000 observations, with ∑ 𝑡 = 1 𝑇 𝑥 𝑡 = 1000 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 4000 . We obtain point estimates 𝛼 ̂ = 1 and 𝛽 ̂ = 2 . 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 + 𝛾 𝑥 𝑖 2 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 , 𝛽 and 𝛾 by GMM, we use the three theoretical moment conditions 𝔼 [ 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ) 𝑥 𝑖 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 − 𝛾 𝑥 𝑖 2 ) 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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