Consider the nonlinear model
𝑦
𝑡
=
𝜃
1
𝑥
𝑡
𝜃
2
+
𝜀
𝑡
where the sample data
(
𝑦
1
,
𝑥
1
)
,
.
.
.
,
(
𝑦
𝑇
,
𝑥
𝑇
)
are i.i.d. and
𝐸
(
𝜀
𝑡
|
𝑥
𝑡
)
=
0
. We know that the nonlinear least square estimator is asymptotically normal, that is
⤳
𝑇
(
𝜃
̂
𝑁
𝐿
−
𝜃
0
)
⤳
𝑑
𝑁
(
0
,
𝐴
0
−
1
𝛺
0
𝐴
0
−
1
)
To compute the standard errors we need to estimate
𝛺
0
,
𝛺
̂
0
=
[
1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝑥
𝑡
2
𝜃
̂
2

1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝜃
̂
1
𝑥
𝑡
2
𝜃
̂
2
log
(
𝑥
𝑡
)

1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝜃
̂
1
𝑥
𝑡
2
𝜃
̂
2
log
(
𝑥
𝑡
)

]
What is the missing entry in the matrix
𝛺
̂
0
?

Question

Consider the nonlinear model
𝑦
𝑡
=
𝜃
1
𝑥
𝑡
𝜃
2
+
𝜀
𝑡
where the sample data 
(
𝑦
1
,
𝑥
1
)
,
.
.
.
,
(
𝑦
𝑇
,
𝑥
𝑇
)
 are i.i.d. and 
𝐸
(
𝜀
𝑡
|
𝑥
𝑡
)
=
0
. We know that the nonlinear least square estimator is asymptotically normal, that is
⤳
𝑇
(
𝜃
̂
𝑁
𝐿
−
𝜃
0
)
⤳
𝑑
𝑁
(
0
,
𝐴
0
−
1
𝛺
0
𝐴
0
−
1
)
To compute the standard errors we need to estimate 
𝛺
0
,
𝛺
̂
0
=
[
1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝑥
𝑡
2
𝜃
̂
2
	
1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝜃
̂
1
𝑥
𝑡
2
𝜃
̂
2
log
(
𝑥
𝑡
)

1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝜃
̂
1
𝑥
𝑡
2
𝜃
̂
2
log
(
𝑥
𝑡
)
	
]
What is the missing entry in the matrix 
𝛺
̂
0
?

BlackTom AI · Accepted Answer

We begin by restating the problem setup and the options to ensure clarity for each choice.
Question: What is the missing entry in the matrix Ω̂0 for the nonlinear least squares estimator, given that the asymptotic distribution is ⤳T(θ̂N L − θ0) ⤳d N(0, A0−1 Ω0 A0−1)? The entries shown for Ω̂0 involve terms with 1/T sums of ε̂t^2, x_t^2, θ̂1, θ̂2, and log(x_t).
Answer options:
- 𝔼(ε̂t^2 x_t^2 θ̂2) // Option A
- 1/T ∑t=1^T ε̂t^2 θ̂1 2 x_t^2 θ̂2 log(x_t) // Option B
- 𝔼(ε̂t^2 θ̂1 x_t^2 θ̂2 log(x_t)) // Option C
- 1/T ∑t=1^T ε̂t^2 x_t^2 θ̂2 // Option D
- 1/T ∑t=1^T ε̂t^2 θ̂1 2 x_t^2 θ̂2 log(x_t) // Option E
- 𝔼(ε̂t^2 θ̂1 2 x_t^2 θ̂2 log^2(x_t)) // Option F

Option-by-option reasoning:
- Option A: 𝔼(ε̂t^2 x_t^2 θ̂2). This form mixes an expectation with the random quantity θ̂2 without incorporating the log(x_t) term that appears in the estimated information matrix for the nonlinear least squares with a log transformation on x_t. Since the given construction of Ω̂0 contains products of ε̂t^2 with θ̂1, x_t^2, θ̂2, and log(x_t) in a specific combination, this option omits θ̂1, log(x_t), and any dependence pattern that is present in the problem, making it inconsistent with the stated structure.
- Option B: 1/T ∑t ε̂t^2 θ̂1 2 x_t^2 θ̂2 log(x_t). This expression multiplies ε̂t^2 by θ̂1, x_t^2, θ̂2, and log(x_t) and includes a 2 visible as θ̂12, which is likely a misprint or misplacement of a squared term. The presence of θ̂1 2 (as a product) and the mixing of terms in a single sum without a clean expectation form makes this option suspect; moreover, the problem’s provided correct-looking term uses θ̂1 x_t^2 and θ̂2 log(x_t) in a linear fashion within the sum, not θ̂1^2 or log(x_t) squared, so this option is unlikely.
- Option C: 𝔼(ε̂t^2 θ̂1 x_t^2 θ̂2 log(x_t)). This option has an expectation of a product that includes ε̂t^2, θ̂1, x_t^2, θ̂2, and log(x_t) all inside an expectation. The Ω̂0 entries in the problem statement are expressed as sums over t of terms like ε̂t^2 multiplied by predictor components (x_t^2, log(x_t)) and parameter estimates (θ̂1, θ̂2), but typically the missing entry in Ω̂0 is a sum, not an expectation, and the structure given shows a sum with 1/T multiplying components, not a plain 𝔼(·). Thus, this option diverges from the intended sample-average form.
- Option D: 1/T ∑t ε̂t^2 x_t^2 θ̂2. Here the term lacks log(x_t) and θ̂1; it includes ε̂t^2 multiplied by x_t^2 and θ̂2 only. Since the problem’s expression explicitly involves log(x_t) and θ̂1 in combination with x_t^2 and ε̂t^2, this trimmed product excludes important components and does not match the expected pattern of Ω̂0 as indicated.
- Option E: 1/T ∑t ε̂t^2 θ̂1 2 x_t^2 θ̂2 log(x_t). This option resembles Option B but without the squared term in the 1/T sum’s factor; it still includes θ̂1 2 (printed as θ̂12), x_t^2, θ̂2, and log(x_t). The appearance of θ̂1 2 (instead of θ̂1) is unusual and would imply a misinterpretation of the original, and overall the combination may not align with the standard form given (which uses θ̂1 x_t^2 and θ̂2 log(x_t) multiplied with ε̂t^2 in the sum).
- Option F: 𝔼(ε̂t^2 θ̂1 2 x_t^2 θ̂2 log^2(x_t)). This option introduces log^2(x_t) inside the expectation, which is inconsistent with the provided structures that contain log(x_t), not its square, as part of the product inside the sum. The squared log term would alter the asymptotic variance form and is not aligned with the stated Ω̂0 construction.

Synthesis and common themes across the options: each candidate tries to place combinations of ε̂t^2, θ̂1, θ̂2, x_t^2, and log(x_t) into a 1/T sum, but only certain mixtures preserve the exact linear structure described in the problem statement’s Ω̂0, which combines ε̂t^2 with θ̂1 x_t^2 and θ̂2 log(x_t) in a specific way. Several options introduce extraneous squares or omit essential components like log(x_t) or θ̂1, leading to misalignment with the given reference form.

Putting it all together, the option that mirrors the intended multiplicative structure—ε̂t^2 multiplied by θ̂1 x_t^2 and θ̂2 log(x_t) within a 1/T sum—without extraneous squaring or missing factors, is the one that aligns with the described missing entry in Ω̂0. The remaining options either drop necessary components (log(x_t) or θ̂1), introduce squared terms inconsistently, or place expectations where a sample average is expected, all of which contradict the stated estimation framework.

BU.232.630.F3.SP25 QUIZ 1 2025

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