Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
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 use the two theoretical moment conditions
𝔼
[
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
)
𝑥
𝑖
]
=
0
To compute the variance of the GMM estimator we need the matrices
𝛤
0
and
𝛷
0
.

Question

Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
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 use the two theoretical moment conditions
𝔼
[
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
)
𝑥
𝑖
]
=
0
To compute the variance of the GMM estimator we need the matrices 
𝛤
0
 and 
𝛷
0
.

BlackTom AI · Accepted Answer

The prompt asks about computing the variance of the GMM estimator and the required matrices Γ0 and Λ0, given a nonlinear regression with two moment conditions. We need to evaluate each proposed form of Γ̂0 and the claim about information sufficiency.
Option A: 'There is not enough information to compute the estimate of the matrix Γ0.' This statement would be plausible if the data or the exact form of the moment conditions or the weighting matrix were insufficient to identify Γ0. In GMM theory, Γ0 is typically the limit of the Jacobian (the derivative of the moment conditions with respect to the parameters) times the variance of the instruments, evaluated at the true parameter values. Since we are given the two theoretical moment conditions and a model with observable x_i and y_i, and assuming standard regularity conditions (finite moments, identifiability), Γ0 can be computed from the model's derivatives and the chosen instruments. Therefore, this option would be claiming insufficiency where, given the stated setup, Γ0 is typically computable. This makes the statement potentially incorrect under standard assumptions.
Option B: 'The estimate of the matrix Γ0 is: Γ̂0 = [ 0  − 1^T ∑_{i=1}^T x_i β x_i − 1  − 1^T ∑_{i=1}^T x_i β x_i − 1 ]' This option presents a specific numeric or symbolic structure for Γ̂0, trying to assemble blocks from sums involving x_i and β x_i. The correctness depends on the exact derivation: Γ0 usually involves the expectation of the outer product of the moment conditions with themselves minus the cross-terms arising from the derivative with respect to the parameters. Writing a matrix with blocks that include −1 and sums of x_i and β x_i seems ad hoc and likely inconsistent with the standard form unless β x_i terms appear correctly in both the gradient and the moment variance. Without a clear derivation, this form appears dubious because it does not neatly reflect the standard Jacobian of the moment functions m_i(θ) = [ y_i − α − β x_i, (y_i − α − β x_i) x_i ] with respect to θ = (α, β) and the variance of moments E[m_i m_i^T]. Hence, this option is likely incorrect or at least requires more context to be validated.
Option C: 'The estimate of the matrix Γ0 is: Γ̂0 = [ −1  −1^T ∑_{i=1}^T x_i β x_i −1  −1^T ∑_{i=1}^T x_i  −1^T ∑_{i=1}^T x_i^2 2β x_i −1 ]' This structure again mixes constants with sums involving x_i and β x_i in a nonstandard arrangement. The presence of terms like −1^T ∑ x_i β x_i and −1^T ∑ x_i^2 2β x_i suggests an inconsistent combination of the derivative with respect to α and β and the variance components. Without a coherent derivation verifying dimensions and the exact arrangement of blocks, this option is unlikely to be correct for Γ0 in the usual GMM setup with the given moment conditions. It reads as a mis-specified block matrix.
Option D: 'There is not enough information to compute the estimate of the matrix Γ0.' This repeats the claim in Option A, but here it is stated as a separate multiple-choice option, perhaps to test whether the student recognizes insufficiency. However, as explained earlier, under typical regularity conditions and with the explicit moment conditions provided, Γ0 can be computed from the derivatives of the moment conditions with respect to (α, β) and the variance of the moments evaluated at the true parameter values. Therefore, this option also seems unsupported given the standard theoretical framework for GMM in this model.
Option E: 'The estimate of the matrix Γ0 is: Γ̂0 = [ −1  −1^T ∑_{i=1}^T x_i β x_i −1  ∑_{i=1}^T x_i  β x_i −1  −1^T ∑_{i=1}^T x_i 2 β x_i −1 ]' This option, like B and C, provides a particular block structure with mixed terms such as ∑ x_i β x_i and ∑ x_i^2 2β x_i, which seems dimensionally inconsistent with Γ0, and the arrangement does not align with the standard calculation of the gradient of the two moment conditions m_i with respect to θ and the covariance of m_i. Without a clear derivation, this form is unlikely to be correct.

In summary, the core idea behind evaluating these options is to align them with the typical GMM machinery: Γ0 should reflect the Jacobian of the moment conditions with respect to the parameters and the variance-covariance structure of the moments. The proposed matrices in options B, C, E appear to be ad hoc constructions that do not match the conventional derivation, while the assertion of insufficient information (Options A and D) is generally not consistent with the given problem setup, where the moment conditions are specified and standard regularity conditions are assumed. The critical thinking here involves checking dimensional consistency, the proper placement of sums, and whether the blocks correspond to derivatives w.r.t. α and β and the corresponding moment variances.

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:

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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