题目
题目

BU.232.630.W1.SP25 Quiz 2 solutions

单项选择题

Consider the following nonlinear regression model: yi=α+βxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we need at least two moment conditions, and we use 𝔼[yi−α−βxi]=0 𝔼[(yi−α−βxi)xiβxi−1]=0 Chose the correct answer below.

选项
A.The following equation is also a valid moment condition for estimation of this model and can be added as a third equation, 𝔼[(yi−α−βxi)xi]=0
B.The following equation is also a valid moment condition for estimation of this model and can be added as a third equation, 𝔼[(yi−α−βxi)log(xi)]=0
C.All of the answers are correct.
D.We can substitute the second equation above with the following moment condition 𝔼[(yi−α−βxi)xi]=0
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标准答案
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思路分析
We start by restating the setup and listing the candidate moment conditions that could be used to estimate α and β in the nonlinear regression model yi = α + β xi + εi with E[εi | xi] = 0. Option 1: The following equation is also a valid moment condition for estimation of this model and can be added as a third equation, E[(yi − α − β xi) xi] = 0 - Here, the residual is ui = yi − α − β xi = εi. Since E[εi | xi] = 0, it follows that E[εi xi] = E[ E[εi | xi] xi ] = E[0 · xi] = 0. Therefore this is indeed a valid moment condition of the form E[ui zi] = 0 with zi = xi. - In short, using E[(yi − α − β xi) xi] = 0 exploits the orthogona......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 use the following moment conditions: 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) 𝑥 𝑡 ] = 0 To compute the variance of the estimates, we need to estimate the matrices 𝛤 0 and 𝛷 0 .

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 = 3000 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 5000 . We obtain point estimates 𝛼 ̂ = − 3 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 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 = 100 , ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 = 200 and ∑ 𝑡 = 1 𝑇 𝑥 𝑡 3 = 800 . We obtain point estimates 𝛼 ̂ = − 1 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 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:

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