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

We start by restating the model and the assumptions to identify appropriate moment conditions.
The nonlinear regression model is yi = α + e^{β xi} + εi, with E[εi | xi] = 0. This implies that the residual εi = yi − α − e^{β xi} has conditional mean zero given xi, so its unconditional mean is also zero: E[εi] = 0. Since εi is uncorrelated with any function of xi due to the conditional mean zero, we can form moment conditions by taking expectations of εi and εi multiplied by xi and setting them to zero.

Option analysis:
- Option: E[yi − α − e^{β xi}] = 0. This directly corresponds to E[εi] = 0, which follows from E[εi | xi] = 0. It is a valid moment condition because the average residual across observations must be zero if the model is correctly specified.
- Option: E[(yi − α − e^{β xi}) xi] = 0. This captures E[εi xi] = 0. Given E[εi | xi] = 0, we have E[εi xi] = E[ E[εi | xi] xi ] = 0, so this is also a valid moment condition. It leverages the exogeneity of xi to provide a second equation that helps identify the two parameters α and β in GMM.

Why these choices are appropriate: with two parameters to estimate, we typically need two independent moment conditions. The pair above uses the unconditional mean of the residual and the residual’s covariance with xi, both derived from the key assumption E[εi | xi] = 0 and the structure of the model. They do not rely on higher-order moments or external instruments, and they respect the fact that ξi = yi − α − e^{β xi} is the fitting error under the model.

Why alternatives (if any) would be problematic: any moment that uses a function of yi alone without involving the residual structure would not necessarily be zero under the model, and any moment that leverages a function of xi that is not uncorrelated with εi could violate exogeneity. However, the two moments presented in the option align with the standard GMM setup for a model with exogenous regressor xi and a regressor e^{β xi} that depends on β, making them the natural and valid choice.

In summary, the proposed two moments—E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi}) xi] = 0—arise directly from E[εi | xi] = 0 and provide the necessary conditions to identify α and β in a GMM framework.

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