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
=
3000
and
∑
𝑡
=
1
𝑇
𝑥
𝑡
2
=
5000
. We obtain point estimates
𝛼
̂
=
−
3
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
=
3000
 and 
∑
𝑡
=
1
𝑇
𝑥
𝑡
2
=
5000
. We obtain point estimates 
𝛼
̂
=
−
3
 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 need to evaluate each proposed value for Γ̂11 given the moment conditions and the provided sample moments.
Option A: Γ̂11 = −3. The element Γ̂11 corresponds to the expected derivative of the first moment with respect to α, i.e., ∂m1/∂α evaluated at the estimated parameters. If m1,t = y_t − α x_t^β with β known (here β̂ = 2 from the estimation), then ∂m1,t/∂α = − x_t^β = − x_t^2. The GMM cross-derivative Γ̂11 is the average of these derivatives over the sample, i.e., Γ̂11 = − (1/T) ∑ x_t^2. We’re given ∑ x_t^2 = 5000 and T = 1000, so Γ̂11 = −(1/1000) × 5000 = −5, not −3. Therefore A is inconsistent with the data.

Option B: There is not enough information to compute Γ̂11. This would be tempting if we lacked the necessary derivative structure or the exact form of the first moment w.r.t. α. However, given the standard moment m1,t = y_t − α x_t^β and β known, the derivative term ∂m1,t/∂α is simply − x_t^β, which depends only on x_t and β. Since β̂ = 2 is provided and ∑ x_t^2 is given, we can compute the average derivative directly. Thus the necessary information is available, and B is not correct.

Option C: Γ̂11 = 3. As argued above, Γ̂11 equals − (1/T) ∑ x_t^2 when β = 2, which yields Γ̂11 = −5, not +3. The sign and magnitude are determined by the squared x_t terms; a positive 3 would require a positive average of x_t^2, which is incompatible with the derivative structure of m1,t with respect to α. Therefore C is incorrect.

Option D: Γ̂11 = 5000. This value resembles the unscaled sum ∑ x_t^2, not the averaged derivative Γ̂11 = − (1/T) ∑ x_t^2. Since we must take the average by dividing by T, the correct averaged derivative would be −5000/1000 = −5, not 5000. Hence D is not correct.

Option E: Γ̂11 = −5. This matches the calculation Γ̂11 = −(1/T) ∑ x_t^2 with T = 1000 and ∑ x_t^2 = 5000. The negative sign and the magnitude align with the derivative of the first moment with respect to α under β = 2. This is consistent with the provided data and the standard GMM setup. Therefore E is the consistent value given the information.

类似问题

Consider the following nonlinear regression model: yi=α+βxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we need at least two moment conditions, and we use 𝔼[yi−α−βxi]=0 𝔼[(yi−α−βxi)xiβxi−1]=0 Chose the correct answer below.

How did Egyptian culture and governance change during the Ptolemaic period compared to the pharaonic period?

Name the Supreme Court decision that removed financial constraints on PACs.

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