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:

Question

BlackTom AI · Accepted Answer

Question restatement: We have a nonlinear regression model yi = α + e^{β xi} + εi with i.i.d. data and E[εi|xi] = 0. To estimate α and β by GMM, two moment conditions are needed. We are given five candidate sets of two moments to evaluate which is best.

Option 1: The two moments are E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi}) log(xi)] = 0.
- Why this could be valid: The first moment E[yi − α − e^{β xi}] equals E[εi], since yi − α − e^{β xi} = εi. By the law of total expectation, E[εi] = E[E[εi|xi]] = E[0] = 0, so the first moment is correct under the given assumption. For the second moment, E[(yi − α − e^{β xi}) log(xi)] = E[εi log(xi)]. Given E[εi|xi] = 0, the conditional expectation of εi given xi is zero, so E[εi log(xi)] = E[E[εi|xi] log(xi)] = E[0 · log(xi)] = 0. Thus both moments are valid and satisfy the population moment conditions.
- Potential caveat: The choice of instruments here is log(xi). This is legitimate as long as log(xi) is a valid instrument in the sense that the moment conditions reflect E[εi | xi] = 0 and the function log(xi) is an observable that yields zero when multiplied by the residual. In this setup, the math supports the two zero-mean moments.

Option 2: The two moments are E[yi − α − e^{β xi}] = 0 and E[yi − e^{β xi}] = 0.
- The first moment is the same as in Option 1 and is valid because E[yi − α − e^{β xi}] = E[εi] = 0 under E[εi|xi] = 0. 
- The second moment, E[yi − e^{β xi}] = 0, simplifies to E[α + εi] = α + E[εi] = α since E[εi] = 0. This would require α = 0 for the moment to be zero, which is not generally true. Therefore this pair of moments is not valid in the general case and would lead to bias unless α happens to be zero.

Option 3: The two moments are E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi})] = E(εi).
- The first equation is the standard valid moment as discussed in Option 1. 
- The second equation is not a proper moment condition for GMM. It states that the expected residual equals E(εi), which itself equals 0 by the law of iterated expectations. However, using a moment statement of the form E[residual] = E(εi) is tautological and does not impose a zero-mean condition in the population that is directly testable or usable for identification in GMM. In practice, you would want a deterministic zero, not an expression involving E(εi). This makes the pair ill-posed as a distinct moment condition and not a proper second moment for GMM.

Option 4: The two moments are E[yi − α − e^{β xi}] = 0 and E[xi e^{β xi}] = 0.
- The first moment is valid as before. 
- The second moment does not involve the regression residual, and thus cannot serve as a valid orthogonality condition for estimating α and β. The moment E[xi e^{β xi}] = 0 would impose a constraint on the data and parameters unrelated to the residual, which is inconsistent with the typical GMM setup where the instrumented moment is the product of the residual and an instrument. Therefore this option is incorrect.

Option 5: There is not enough information to write two moment conditions.
- Given E[εi|xi] = 0, we can construct at least two valid moment conditions by multiplying the residual with functions of xi, such as 1 and log(xi) (i.e., E[(yi − α − e^{β xi}) · 1] = E[εi] = 0 and E[(yi − α − e^{β xi}) log(xi)] = E[εi log(xi)] = 0). Thus, there is enough information to specify two moment conditions. This makes Option 5 incorrect.

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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