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 What is the optimal weight matrix 𝑊 to estimate 𝛼 and 𝛽 ? 单项选择题

A

The optimal weight matrix 𝑊 is: 𝑊 = ( 𝔼 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] ) − 1 .

B

There is not enough information to compute the matrix 𝛤 0 .

C

The optimal weight matrix 𝑊 is: 𝑊 = 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 4 ]

D

The optimal weight matrix 𝑊 is: 𝑊 = 𝔼 [ − 1 − 𝑥 𝑖 − 𝑥 𝑖 − 𝑥 𝑖 2 − 𝑥 𝑖 2 − 𝑥 𝑖 3 ] .

E

The optimal weight matrix 𝑊 is: 𝑊 = ( 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 2 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 3 ( 𝑦 𝑖 − 𝛼 − 𝛽 𝑥 𝑖 ) 2 𝑥 𝑖 4 ] ) − 1

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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: 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.

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:

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