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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 need two moment conditions. Choose the best answer below:

Options
A.The two moments are 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ 𝑦 𝑡 − 𝑥 𝑡 𝛽 ] = 0
B.The two moments are 𝔼 [ 𝑦 𝑡 − 𝑥 𝑡 𝛽 ] = 0 𝔼 [ 𝑥 𝑡 𝛼 𝑥 𝑡 𝛽 ] = 0
C.The two moments are 𝔼 [ 𝑦 𝑡 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) ] = 𝔼 ( 𝜀 𝑡 )
D.There is not enough information to write two moment conditions.
E.The two moments are 𝔼 [ 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑡 − 𝛼 𝑥 𝑡 𝛽 ) log ( 𝑥 𝑡 ) ] = 0
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Question restatement: We are looking for two moment conditions to use in GMM to estimate α and β in the nonlinear regression y_t = α x_t^β + ε_t with E[ε_t | x_t] = 0. Option A: The two moments are E[y_t − α x_t^β] = 0 and E[(y_t − x_t^β α) log(x_t)] = 0. This option uses the residual r_t = y_t − α x_t^β and combines it with log(x_t) as an instrument in the second moment. The first moment imposes that the mean of the residual is zero, which aligns with the exogeneity assumption E[ε_t | x_t] = 0 when taking unconditional expectation of the residual. The second moment uses log(x_t) as a regressor for the residual, which is a nonlinear transformation of x_t that can help identify both α and β since y_t − α x_t^β sc......Login to view full explanation

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