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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 need two moment conditions. Choose the best answer below:
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We start by restating the problem in our own words to ensure clarity. The nonlinear regression model is yi = α + e^{β xi} + εi, with i.i.d. data and E[εi | xi] = 0. To estimate α and β by GMM, we need two moment conditions. The given option proposes two moments: E[ yi − α − e^{β xi} ] = 0 and E[ ( yi − α − e^{β xi} ) log(xi) ] = 0.
Option analysis step by step:
First moment option: E[ yi − α − e^{β xi} ] = 0
- This is a standard moment implied by E[εi | xi] = 0. Since E[εi] = E[ E[εi | xi] ] = 0, the residual yi − α − e^{β xi} has mean zero. In GMM terms, using a constant instrument 1 yields the first moment condition. This choice is typical and valid for identifying α and ......Login to view full explanationLog in for full answers
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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.
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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 + 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 To compute the variance of the GMM estimator we need the matrices 𝛤 0 and 𝛷 0 .
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