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 question asks about computing the variance of the GMM estimator and the necessary matrices Γ0 and Ω0, given a nonlinear regression model with moment conditions. We are to evaluate the provided options for the estimate of Γ0.

Option 1: 'The estimate of the matrix Γ0 is: Γ̂0 = [[−1, −1]ᵀ ∑i=1^T β x_i, [−1, ∑i=1^T x_i]ᵀ −1ᵀ 𝑥_i ]' This presentation of Γ̂0 is unclear and inconsistent with the usual definition of Γ0 in GMM, which involves the expectation of the Jacobian of the moment conditions with respect to the parameters evaluated at the true values. The expression here mixes sums and vector notation in a way that does not correspond to a standard, well-defined Γ0. Moreover, it does not clearly separate dimensions or show how each block is formed, making it highly suspect as a correct estimator.

Option 2: 'The estimate of the matrix Γ0 is: Γ̂0 = [[−1, −1]ᵀ ∑i=1^T x_i β x_i, [−1, 1ᵀ ∑i=1^T x_i β x_i − 1] ]' This option attempts to assemble Γ0 in a block form but again mixes terms like β x_i inside the matrix entry without a clear derivation. The way β x_i appears inside the Γ0 entry is inappropriate because Γ0 should be derived from the gradient of the moment conditions with respect to (α, β) and should not treat β x_i as an entry inside the Γ0 estimator in this ad hoc fashion. The dimensions and the placement of −1 and β x_i terms do not align with the standard GMM theory.

Option 3: 'The estimate of the matrix Γ0 is: Γ̂0 = [[0, −1]ᵀ ∑i=1^T x_i β x_i − 1, [−1, ∑i=1^T β x_i]ᵀ −1ᵀ ∑i=1^T x_i^2 β x_i − 1]' This option starts with a zero in the top-left block, which is not justified by the derivative structure of the moment conditions. The remaining entries mix multiple sums and cross-products of x_i and β x_i in a way that lacks a coherent derivation from the Jacobian ∂g_i/∂θ. The inconsistent placement of terms and the presence of terms like ∑ x_i^2 β x_i suggest a mis-specification of Γ0 rather than a correct estimator.

Option 4: 'There is not enough information to compute the estimate of the matrix Γ0.' This option asserts that, given the problem setup, one cannot determine Γ0 without additional information. In GMM, Γ0 is the probability limit of the average Jacobian of the moment conditions with respect to the parameters, evaluated at the true parameters (or at a consistent estimate). To compute it, one typically needs either the explicit form of the moment conditions and the true parameter values or a plug-in estimate using sample data and an estimate of the true parameters. Since the question only states the model and the moment conditions but does not provide explicit expressions for the derivatives or the true parameter values, it is indeed not possible to produce a unique, correct numerical form for Γ0 from the information given. Therefore, this statement correctly identifies a lack of sufficient information to uniquely determine Γ0.

Option 5: 'The estimate of the matrix Γ0 is: Γ̂0 = [[−1, −1]ᵀ ∑i=1^T x_i β x_i − 1, [−1, ∑i=1^T x_i]ᵀ 2 β x_i − 1]' This option again asserts a specific numeric form with a highly implausible structure, including a term like ' [−1, ∑i x_i]ᵀ 2 β x_i − 1', which does not arise from the standard Jacobian of the moment conditions with respect to (α, β). The presence of inconsistent products and the lack of a coherent matrix layout indicate that this is not a correct specification of Γ0.

Across these options, several common pitfalls stand out: the incorrect and ambiguous construction of Γ0 blocks, mixing parameter terms inside Γ0 without a derivation from the gradient of the moments, and a lack of a clear, dimensionally consistent Jacobian. In the GMM framework, Γ0 should come from the expected Jacobian ∂g_i(θ)/∂θ evaluated at θ0, and without explicit moment form and parameter values, a precise Γ0 cannot be uniquely written. Given the problem statement, the only defensible position is that there is not enough information to compute the estimate of Γ0, since the explicit form of the moment conditions and their derivatives are not provided.

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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:

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