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

To analyze this GMM setup, we consider the moment condition g1,t = y_t − α x_t^β. The derivative of g1,t with respect to α is ∂g1,t/∂α = − x_t^β. The population counterpart is Γ11 = E[ ∂g1,t/∂α ] = − E[x_t^β].

Given the model context, β is estimated as 2, so x_t^β = x_t^2. Therefore, at the true parameter values, Γ11 = − E[x_t^2].

From the data, we are provided with ∑ x_t^2 = 4000 and T = 1000 observations. A natural estimator of E[x_t^2] is the sample mean: (1/T) ∑ x_t^2 = 4000 / 1000 = 4. Consequently, the estimated Γ11 is Γ̂11 ≈ − 4.

Now, evaluate the answer choices:
- 𝛤̂11 = −1: This would require E[x_t^2] = 1, which contradicts the given data since ∑ x_t^2 = 4000 implies E[x_t^2] = 4. Therefore this is incorrect.
- 𝛤̂11 = 1000: This is simply T and does not align with the derivative of the moment w.r.t α. It misinterprets the quantity and is incorrect.
- There is not enough information to compute 𝛤̂11.: In fact, we have explicit sufficient statistics for E[x_t^2] via ∑ x_t^2 and T, so we can compute Γ̂11 as −(1/T) ∑ x_t^2. Thus this option is incorrect.
- 𝛤̂11 = 4000: This is the sum of x_t^2, not the average. The derivative uses the expectation, which corresponds to the average, so this overstates the quantity and is incorrect.
- 𝛤̂11 = −4: This matches the computation Γ̂11 ≈ − (1/T) ∑ x_t^2 = − 4000/1000 = −4. This is the correct estimation based on the provided statistics.

In summary, using the derivative of the first moment with respect to α and the data-derived estimate of E[x_t^2], we obtain Γ̂11 = −4, and the other options stem from misapplying sums vs. averages or misidentifying the derivative quantity.

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