Questions
BU.232.630.W1.SP25 Quiz 2 solutions
Single choice
Consider the following nonlinear regression model: yt=αx β t +εt Assume i.i.d. data and 𝔼[εt|xt]=0. To estimate α and β by GMM, we chose among the following moment conditions: 𝔼[yt−αx β t ]=0 𝔼[(yt−αx β t )xt]=0 𝔼[(yt−αx β t ) 1 xt ]=0 Choose the most appropriate answer below:
Options
A.Only the first and third equations are valid moment conditions to estimate α and β by GMM.
B.Only the second and third equations are valid moment conditions to estimate α and β by GMM.
C.All equations are valid moment conditions to estimate α and β by GMM.
D.Only the first and second equations are valid moment conditions to estimate α and β by GMM.
E.None of the equations above are valid moment conditions to estimate α and β by GMM.
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Step-by-Step Analysis
We begin by restating the setup and the available moment conditions to be evaluated for GMM estimation of α and β in the nonlinear regression y_t = α x_t^β_t + ε_t with E[ε_t | x_t] = 0 and i.i.d. data.
Option A: Only the first and third equations are valid moment conditions.
- The first condition E[y_t − α x_t^β_t] = 0 leverages the standard moment condition arising from the mean-zero regression error given x_t, which is valid under E[ε_t | x_t] = 0. In words, the average misfit must vanish, which is a classic unconditional moment.
- The third condition E[(y_t − α x_t^β_t) (1/x_t)] = 0 uses a function of x_t as a weighting in the moment. Since E[ε_t | x_t] = 0, multiplying ε_t by any function of x_t and taking expectation yiel......Login to view full explanationLog in for full answers
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