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 . Single choice

A

The estimate of the matrix 𝛤 0 is 𝛤 ̂ 0 = [ − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 + 1 − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 log ( 𝑥 𝑡 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 + 1 log ( 𝑥 𝑡 ) ]

B

The estimate of the matrix 𝛤 0 is 𝛤 ̂ 0 = [ 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 log ( 𝑥 𝑡 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 + 1 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 + 1 log ( 𝑥 𝑡 ) ]

C

The estimate of the matrix 𝛤 0 is 𝛤 ̂ 0 = [ − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 log ( 𝑥 𝑡 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 + 1 − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 + 1 log ( 𝑥 𝑡 ) ]

D

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

E

The estimate of the matrix 𝛤 0 is 𝛤 ̂ 0 = [ − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝛽 − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 log ( 𝑥 𝑡 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 log ( 𝑥 𝑡 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 𝛼 𝑥 𝑡 𝛽 + 1 log ( 𝑥 𝑡 ) ]

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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:

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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