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BU.232.630.W1.SP25 Quiz 2 solutions

Single choice

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.

Options
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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Step-by-Step Analysis
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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