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 use the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0
𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0
We have an i.i.d. sample with
𝑇
=
1000
observations, with
∑
𝑡
=
1
𝑇
𝑥
𝑡
=
1000
and
∑
𝑡
=
1
𝑇
𝑥
𝑡
2
=
4000
. We obtain point estimates
𝛼
̂
=
1
and
𝛽
̂
=
2
. To compute the variance of the estimates, we need to estimate the matrix
𝛤
0
,
𝛤
̂
0
=
[
𝛤
̂
11

𝛤
̂
12

𝛤
̂
21

𝛤
̂
22
]
Then, the value
𝛤
̂
11
is:

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 use the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0
𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0
We have an i.i.d. sample with 
𝑇
=
1000
 observations, with 
∑
𝑡
=
1
𝑇
𝑥
𝑡
=
1000
 and 
∑
𝑡
=
1
𝑇
𝑥
𝑡
2
=
4000
. We obtain point estimates 
𝛼
̂
=
1
 and 
𝛽
̂
=
2
. To compute the variance of the estimates, we need to estimate the matrix 
𝛤
0
,
𝛤
̂
0
=
[
𝛤
̂
11
	
𝛤
̂
12

𝛤
̂
21
	
𝛤
̂
22
]
Then, the value 
𝛤
̂
11
 is:

BlackTom AI · Accepted Answer

We start by restating the problem setup and the moment conditions used for GMM estimation. The model is y_t = α x_t β + ε_t with i.i.d. data and E[ε_t | x_t] = 0. The moment conditions are:
1) E[y_t − α x_t β] = 0
2) E[(y_t − α x_t β) x_t] = 0
We are given T = 1000, ∑ x_t = 1000, and ∑ x_t^2 = 4000, with the point estimates α̂ = 1 and β̂ = 2.

To form the Γ0 matrix, which is the expected Jacobian of the moment conditions with respect to the parameters θ = (α, β) evaluated at the true values, we compute the derivatives of the moments:
- For g1_t = y_t − α x_t β, the partial derivatives are:
  ∂g1_t/∂α = − x_t β, and ∂g1_t/∂β = − α x_t.
- For g2_t = (y_t − α x_t β) x_t, the partial derivatives are:
  ∂g2_t/∂α = − x_t^2 β, and ∂g2_t/∂β = − α x_t^2.
Thus, the Jacobian matrix Γ0 (the expectation of these derivatives) evaluated at (α, β) is
Γ0 = E[ [−β x_t, −α x_t], [−β x_t^2, −α x_t^2] ].
Therefore, the (1,1) entry of this matrix, Γ11, is −β E[x_t].

Now plug in the available sample information to approximate E[x_t]. The average of x_t is (1/T) ∑ x_t = 1000 / 1000 = 1. Using β̂ = 2 as the estimate of β, we obtain
Γ̂11 ≈ − β̂ × E[x_t] = − 2 × 1 = −2.

Let’s examine the provided answer choices in light of this result:
- Γ̂11 = −4: This would require E[x_t] to be 2 or β to be 4, neither of which is consistent with the given data (β̂ = 2 and ∑ x_t / T = 1).
- Γ̂11 = 4000: This would imply a magnitude far larger than any plausible Jacobian entry here; it cannot be correct given the derivative structure and the data moments.
- There is not enough information to compute Γ̂11: Given we have β̂ and the sample mean of x_t, we can form Γ̂11 ≈ −β̂ E[x_t], so in principle we do have enough information from the provided data to compute it.
- Γ̂11 = −1: This would require E[x_t] = 0.5, which contradicts ∑ x_t / T = 1 from the data.
- Γ̂11 = 1000: This would require an unrealistically large magnitude for the derivative, inconsistent with the model and the moment structure.

From the derivation, the natural plug-in estimate yields Γ̂11 = −2, but none of the given options matches −2. The closest numerical magnitude among the options is −4, which would correspond to a misalignment between the required E[x_t] and the supplied summary statistics, or a mismatch in how the derivative was formed in the presented choices. In a careful calculation with the provided moments, however, the correct functional form is Γ11 = −β E[x_t], giving Γ̂11 = −2 when E[x_t] = 1 and β̂ = 2.

In summary, the correct theoretical expression for Γ11 is −β E[x_t], which using the provided data yields −2. None of the listed options exactly match this computed value, and among the options, none is correct under the stated derivation. The discrepancy highlights the importance of aligning the derivative calculations with the exact moment definitions and the sample statistics supplied.

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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