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 problem setup and the claim to evaluate. The nonlinear regression model is y_i = α + e^{β x_i} + ε_i, with i.i.d. data and E[ε_i | x_i] = 0. We want to form two moment conditions for GMM to estimate α and β.

Option: The two moments are E[y_i − α − e^{β x_i}] = 0 and E[(y_i − α − e^{β x_i}) x_i^2] = 0.

Analysis of the first moment:
- The term y_i − α − e^{β x_i} equals ε_i by rearranging the model equation. Therefore E[y_i − α − e^{β x_i}] = E[ε_i].
- Given E[ε_i | x_i] = 0, taking the unconditional expectation yields E[ε_i] = E[ E[ε_i | x_i] ] = E[0] = 0. Thus the first moment condition is valid: the mean residual is zero.

Analysis of the second moment:
- The second moment uses a function of x_i, namely x_i^2, as the instrument: E[(y_i − α − e^{β x_i}) x_i^2] = E[ε_i x_i^2].
- Because E[ε_i | x_i] = 0, the law of iterated expectations gives E[ε_i x_i^2] = E[ E[ε_i x_i^2 | x_i] ] = E[ x_i^2 E[ε_i | x_i] ] = E[ x_i^2 · 0 ] = 0. So this moment condition is also valid.

Why this combination makes sense for GMM:
- GMM requires as many moment conditions as there are parameters to estimate (here, α and β, so two moments).
- The condition E[ε_i | x_i] = 0 implies that any function of x_i, when multiplied by ε_i, has zero expectation: E[ε_i f(x_i)] = 0 for any suitable f. Therefore choosing f(x_i) = 1 (the constant) and f(x_i) = x_i^2 provides two independent valid moments.

Potential alternative moments (for intuition):
- One could also use E[(y_i − α − e^{β x_i})] = 0 together with E[(y_i − α − e^{β x_i}) x_i] = 0, which would also be valid since x_i is a function of x_i. The choice of x_i^2 merely offers a different instrument function; as long as the instrument is not perfectly collinear with the constant and provides information about x, it remains valid.

Common misconceptions to watch out for:
- It would be incorrect to claim that E[ε_i] = 0 alone is sufficient with only one more moment that is not correlated with the likelihood of the residuals; you need two independent moments for two parameters, and both moments must be valid given the model assumptions.
- Relying on E[ε_i | x_i] = 0 to justify E[ε_i x_i] = 0 is correct, but if one tried to pick an instrument that is perfectly correlated with ε_i or not a function of x_i, the moment condition could fail to be informative. Here, using x_i^2 is a valid function of x_i.

In summary, the proposed two moments E[y_i − α − e^{β x_i}] = 0 and E[(y_i − α − e^{β x_i}) x_i^2] = 0 are valid and appropriate moment conditions for GMM under the given assumptions.

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: Single choice

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