Consider the following nonlinear regression model:
𝑦
𝑡
=
𝛼
𝑥
𝑡
𝛽
+
𝜀
𝑡
Assume i.i.d. data and
𝔼
[
𝜀
𝑡
|
𝑥
𝑡
]
=
0
. To estimate
𝛼
and
𝛽
by GMM, we need two moment conditions. Choose the best answer below.

Question

Consider the following nonlinear regression model:
𝑦
𝑡
=
𝛼
𝑥
𝑡
𝛽
+
𝜀
𝑡
Assume i.i.d. data and 
𝔼
[
𝜀
𝑡
|
𝑥
𝑡
]
=
0
. To estimate 
𝛼
 and 
𝛽
 by GMM, we need two moment conditions. Choose the best answer below.

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to ensure understanding: we have the nonlinear regression model y_t = α x_t^β + ε_t with E[ε_t | x_t] = 0, and we want two moment conditions to estimate α and β using GMM.

Option A: The two moments are E[y_t − x_t^β] = 0  and  E[(y_t − α x_t^β) x_t^β] = 0
- The first moment here uses y_t − x_t^β, which omits the parameter α entirely inside the subtraction. Since the model specifies y_t − α x_t^β, using y_t − x_t^β assumes a wrong residual that does not reflect the estimated regression line. This distorts the base moment condition and does not align with the structure of the model.
- The second moment, E[(y_t − α x_t^β) x_t^β] = 0, resembles a plausible score-like condition, but because the first moment is incorrect, this pair cannot deliver consistent estimates of α and β.

Option B: There is not enough information to write two moment conditions.
- In principle, with the given model and the exogeneity assumption E[ε_t | x_t] = 0, we can construct at least two moment conditions by differentiating the implied sample criterion or by using score-type conditions. Therefore, asserting that there is not enough information ignores standard GMM/CS (Christensen, particular for nonlinear least squares) reasoning. This option overlooks the available derivative structure and exogeneity, so it is not the best choice.

Option C: The two moments are E[y_t − α x_t^β] = 0  and  E[(y_t − α x_t^β) x_t^β] = 0
- The first moment, E[y_t − α x_t^β] = 0, corresponds to setting the unconditional mean of the residual to zero, which is not the typical moment for a regression with covariates, and it ignores the information contained in x_t. In a regression with regressors, we generally want moments that weight the residual by the regressor(s) to ensure orthogonality between residuals and the regressors. However, the second moment E[(y_t − α x_t^β) x_t^β] = 0 attempts to address that weighting, and together these two moments form a plausible pair for a nonlinear model when considering the gradient-like structure, though the exact gradient would involve derivatives with respect to both α and β. This option therefore captures a reasonable pair of moments, at least in spirit, and is commonly encountered in some GMM formulations for nonlinear parameters.

Option D: The two moments are E[y_t − α x_t^β] = 0  and  E[y_t − x_t^β] = 0
- This pair uses two distinct residuals, one with α and one without α, inside the expectations. The second moment E[y_t − x_t^β] = 0 again treats the residual as if α = 1, which is inconsistent with the model specification and would not generally hold across samples. Mixing inconsistent residual definitions yields incompatible information about α and β and does not reflect the correct residual structure of the nonlinear model.

Option E: The two moments are E[y_t α x_t^β] = 0  and  E[(y_t − α x_t^β)] = 0
- The first moment E[y_t α x_t^β] = 0 introduces a product of y_t with α and x_t^β inside the expectation, which is not a standard moment condition derived from the orthogonality or score arguments for this model. It conflates parameters with data in a way that typically does not produce consistent estimating equations. The second moment E[(y_t − α x_t^β)] = 0 is again a problematic form for the same reason as in Option D and does not coherently reflect the model's residual structure.

Overall, when evaluating moment conditions for GMM in this nonlinear context, we look for moments that reflect the residual y_t − α x_t^β and how that residual should orthogonalize with respect to the regressors (or their relevant transformations) under the exogeneity assumption. Among the presented options, the combination that aligns with the standard notion of a two-moment system for this setup is E[y_t − α x_t^β] = 0 and E[(y_t − α x_t^β) x_t^β] = 0, which captures both a direct mean-zero condition and a correlation (orthogonality) condition with the regressor-like term x_t^β. However, note that the exact theoretical gradient-based moments for (α, β) would typically involve the derivative of the model with respect to both parameters, yielding moments proportional to x_t^β and α x_t^β log x_t, respectively. The given pair presents a commonly seen, simplified two-moment setup in multiple-choice contexts, and matches the option that lists the two moments as E[y_t − α x_t^β] = 0 and E[(y_t − α x_t^β) x_t^β] = 0.

Consider the following nonlinear regression model: 𝑦 𝑡 = 𝛼 𝑥 𝑡 𝛽 + 𝜀 𝑡 Assume i.i.d. data and 𝔼 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we need two moment conditions. Choose the best answer below.

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类似问题

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:

Consider the following nonlinear regression model: 𝑦 𝑖 = 𝛼 + 𝑒 𝛽 𝑥 𝑖 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we need two moment conditions. Choose the best answer below:

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