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

To begin, recall the standard setup for a GMM estimation with a regression model yi = α + e^{β xi} + εi and the assumption that E[εi | xi] = 0. This conditional mean restriction implies certain unconditional moment conditions by multiplying εi with functions of xi and taking expectations.

Option A: E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi})] = E(εi).
- The first moment E[yi − α − e^{β xi}] = 0 is correct in principle, since E[εi] = 0 given the model and the conditional mean restriction reduces to the unconditional mean of ε being zero. However, the second proposed moment E[(yi − α − e^{β xi})] = E(εi) is circular: it restates the same quantity on both sides and does not provide a second, informative moment that links εi to an instrument or function of xi. In GMM you need a second independent moment function, not just a restatement, so this option is not a valid pair of moments.

Option B: E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi}) log(xi)] = 0.
- This pair aligns with the conditional moment restriction E[εi | xi] = 0: multiplying εi by any function of xi and taking expectations yields zero, so E[ (εi) log(xi) ] = 0. Since yi − α − e^{β xi} = εi, the second moment becomes E[εi log(xi)] = 0, which is valid for instrumenting with log(xi). Together these two moments provide a valid, informative GMM system.

Option C: E[yi − α − e^{β xi}] = 0 and E[yi − e^{β xi}] = 0.
- The second moment E[yi − e^{β xi}] = 0 would imply E[α] = E[yi], which is not a valid moment condition derived from E[εi | xi] = 0. It ignores the α parameter correctly and does not correspond to any sensible transformation of εi that leverages the conditional mean assumption. This makes the pair inappropriate for identification in the GMM sense.

Option D: E[yi − α − e^{β xi}] = 0 and E[xi e^{β xi}] = 0.
- The second moment E[xi e^{β xi}] = 0 involves xi and the nonlinear term e^{β xi} but does not incorporate the error term εi in any way. It does not follow from E[εi | xi] = 0 and, in fact, imposes a separate, unrelated restriction that would not identify α and β from the model. Hence this is not a valid moment condition for this estimation problem.

Option E: There is not enough information to write two moment conditions.
- Given E[εi | xi] = 0 and the model form yi = α + e^{β xi} + εi, we can derive at least one moment condition E[yi − α − e^{β xi}] = 0, and by multiplying εi by any function of xi (such as log xi) we obtain a second valid moment E[(yi − α − e^{β xi}) g(xi)] = 0. Therefore, two moment conditions can indeed be written; this option is incorrect.

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