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:

Question

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:

BlackTom AI · Accepted Answer

Analyzing the moment conditions for a nonlinear regression y_i = α + β x_i + ε_i with E[ε_i | x_i] = 0, we want two orthogonality constraints that identify α and β in a GMM framework.

Option A: This option proposes E[y_i − α − εβ x_i] = 0 and E[(y_i − α − εβ x_i) x_i^2] = 0. Here, the first moment resembles the usual mean-zero error condition, but the second moment uses x_i^2 instead of x_i. In standard linear/affine models with E[ε_i | x_i] = 0, the natural second moment is with x_i, not x_i^2. Using x_i^2 would generally not yield the necessary orthogonality to identify β in the usual GMM sense, making this option inconsistent with the standard moment structure.

Option B: It states E[y_i − α − εβ x_i] = 0 and E[(y_i − α − εβ x_i)] = 0. The second condition collapses to E[y_i − α − εβ x_i] = 0 again, which is not providing a second, independent moment condition. In other words, both moments end up the same, so they do not supply two distinct equations needed to identify α and β. This makes option B invalid as a pair of moment conditions for GMM.

Option C: It presents E[y_i − α − εβ x_i] = 0 and E[x_i ε_i] = 0 (where ε_i stands for the error term). The first moment is the standard mean-zero condition for the regression residue, and the second is the standard exogeneity/orthogonality condition: the regressor x_i should be orthogonal to the error term ε_i, yielding E[x_i ε_i] = 0. This pair of moment conditions is the conventional and correct choice for GMM estimation of α and β in this setup, aligning with E[ε_i | x_i] = 0 and the definition ε_i = y_i − α − β x_i.

Option D: It states E[y_i − α − εβ x_i] = 0 and E[y_i − εβ x_i] = 0. The second equation is problematic because it omits the intercept α from the residual, effectively imposing a constraint that may not be valid given the model specification. It does not represent a proper orthogonality condition consistent with the regression error structure, so this choice is incorrect.

Option E: This option claims there is not enough information to write two moment conditions. In this context, we do have two natural and standard moment conditions derived from the model and the exogeneity assumption: E[y_i − α − β x_i] = 0 and E[x_i (y_i − α − β x_i)] = 0. Therefore, stating there is insufficient information is inaccurate.

In summary, the intended two moment conditions for GMM estimation are the mean-zero residual condition together with the orthogonality of the residual to the regressor: E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0. Among the given options, Option C mirrors this structure by combining E[y_i − α − ε_i x_i] = 0 with E[x_i ε_i] = 0, where ε_i denotes the error term in the model. The other options either misuse the regressor (e.g., x_i^2), collapse the two moments into the same equation, omit α in the second moment, or claim insufficient information, making them invalid choices.

BU.232.630.W4.SP25 sample_quiz_2

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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类似问题

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.

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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