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

The question asks for the two moment conditions to estimate α and β in the nonlinear regression model yi = α + e^{β xi} + εi (with E[εi | xi] = 0) using GMM.

Option provided: "The two moments are E[yi − α − eβ xi] = 0  E[(yi − α − eβ xi) log]" is incomplete and problematic. Here is why:
- The expression yi − α − eβ xi uses the fitted nonlinear term e^{β xi} (assuming e^{β xi} means exp(β xi)); the associated mean-zero condition under E[εi | xi] = 0 would be E[yi − α − e^{β xi} | xi] = 0. However, a single unconditional moment E[yi − α − e^{β xi}] = 0 would not, in general, identify α and β when xi carries information about εi through the nonlinear term; we typically need at least two moments that are valid instruments or functions of xi that produce zero conditional expectation.
- The second proposed moment, E[(yi − α − eβ xi) log], is clearly ill-posed as written. It lacks a specified argument to the logarithm and does not specify a valid instrument or function of xi. Without a well-defined second instrument or function g(xi) such that E[(yi − α − e^{β xi}) g(xi)] = 0, this cannot serve as a legitimate second moment condition.
- In GMM with two parameters (α, β), you typically require two valid moment conditions. Given E[εi | xi] = 0, a correct approach would use at least two instruments or functions of xi, say z1(xi) and z2(xi), with E[(yi − α − e^{β xi}) z_j(xi)] = 0 for j = 1, 2. A common choice would be E[yi − α − e^{β xi}] = 0 together with E[(yi − α − e^{β xi}) xi] = 0 or with some other proper instrument functions, provided they are uncorrelated with εi and have nonzero variance.
- Therefore, this option fails to provide a complete, well-defined pair of moment conditions. It also misstates the second moment by not specifying a valid second instrument or function, and it uses an undefined or inappropriate log term.

If we were to correct it, a plausible two-moment specification could be:
- E[yi − α − e^{β xi}] = 0
- E[(yi − α − e^{β xi}) h(xi)] = 0 for some valid instrument function h with E[h(xi)] ≠ 0 and E[(εi) h(xi)] = 0, such as h(xi) = xi or another exogenous function of xi, depending on modeling assumptions.

In summary, the given statement is not a correct or complete description of the two moment conditions needed for GMM in this model; the second moment is not properly specified and the first one alone is insufficient to identify α and β.

BU.232.630.W6.SP25 sample_quiz_2

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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