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

The task is to identify the missing entry in the matrix Ω̂0 based on the given asymptotic variance formula for the nonlinear least squares estimator. We begin by restating the available pieces of information and the structure of the problem to ground our analysis.

First, the context: we have a nonlinear model y_t = theta1 x_t^{theta2} + epsilon_t with E(epsilon_t | x_t) = 0, and the NLSE is asymptotically normal with a specific variance formula involving A0 and Omega0. The estimator's asymptotic covariance is A0^{-1} Omega0 A0^{-1} and the problem provides a partial construction of the Omega0 estimator as a T-variate average of terms involving the squared residuals, x_t, and theta-hat components, along with log(x_t).

Now, we evaluate each answer option in light of the standard form: Omega0 is typically built from the outer product of the score or gradient contributions, or from the expectation of the squared moment conditions, which in this nonlinear least squares setting translate into expressions that weight the squared residuals by derivatives with respect to the parameters. The provided pieces suggest a structure of terms like epsilon-hat_t^2 multiplied by functions of x_t and theta-hat, with subcomponents corresponding to theta1, theta2, and possibly log x_t terms arising from derivatives of x_t^{theta2} with respect to theta2 or from a transformation that introduces log x_t into the gradient.

Option-by-option analysis:

Option A: 1/T sum_{t=1}^T epsilon_hat,t^2 theta_hat1 x_t^2 theta_hat2 log(x_t)
- This choice multiplies the squared residual by theta_hat1 and theta_hat2, and by x_t^2 and log(x_t). In typical Omega0 constructions, we expect terms to reflect derivatives with respect to parameters, which would involve x_t, x_t^2, log(x_t) in a way tied to the derivative of x_t^{theta2} with respect to theta2, which brings a log(x_t) factor. However, the direct product with theta_hat1 and theta_hat2 as multipliers lacks a clear justification unless those factors arise from the estimated gradient components themselves. The presence of x_t^2 together with log(x_t) seems somewhat arbitrary and may over-weight large x_t values; in many asymptotic variance derivations, one would expect terms that align more directly with score components rather than including both theta-hat multipliers inside the averaging. This option is plausible in spirit but its exact multiplicative structure appears unusual without a derivation guiding the theta-hat factors.

Option B: 1/T sum_{t=1}^T epsilon_hat,t^2 theta_hat1 x_t log(x_t)
- Here the weight is epsilon_hat,t^2 times theta_hat1 times x_t and log(x_t). This keeps a simpler structure, linking the derivative with respect to theta1 through a factor that resembles the derivative of x_t^{theta2} with respect to theta1, which in a separable form may involve x_t. The log(x_t) factor is commonly associated with derivatives wrt theta2 rather than theta1, so its inclusion here may be less natural unless theta1 multiplies a term that brings in log(x_t). Overall, this option seems more plausible than Option A but still lacks a clear canonical justification for including theta_hat1 with both x_t and log(x_t).

Option C: E(epsilon_hat,t^2 theta_hat1 x_t^2 theta_hat2 log(x_t))
- This option places an expectation around a product of residuals and the same set of regressors as in Option A, but inside an expectation rather than a sample average. In Omega0, we typically see population-level expressions (expectations) or their empirical analogs. If the intended missing entry is an expectation form to be matched by the sample average, this option could reflect the population counterpart of the sample expression in Option A. However, the presence of theta_hat1 and theta_hat2 as multipliers inside the expectation is unconventional, since population moments would involve true parameters theta0 rather than their estimators. This makes Option C less likely as the missing entry.

Option D: 1/T sum_{t=1}^T epsilon_hat,t^2 theta_hat1 2 x_t^2 theta_hat2 log(x_t) (interpreted as 1/T sum ... theta_hat1^2  x_t^2 theta_hat2 log^2(x_t))
- This option multiplies by theta_hat1 squared and log(x_t) squared, introducing a more aggressive scaling. In variance-covariance derivations, squaring the estimated parameters inside the Omega0 component is unusual unless there is a specific delta-method-like expansion. The presence of squared theta_hat1 and squared log(x_t) would typically indicate a second-order adjustment, which is not standard for the basic Omega0 estimation in asymptotic theory. Therefore, this option seems less consistent with the usual form and is likely incorrect unless a very particular expansion is intended.

Option E: 1/T sum_{t=1}^T epsilon_hat,t^2 x_t^2 theta_hat2
- This option removes log(x_t) entirely and ties the weight to x_t^2 and theta_hat2, with no theta_hat1 factor. If the derivative with respect to theta2 involves a term with x_t^2 and the log(x_t) arises from differentiating x_t^{theta2}, omitting log(x_t) would be inconsistent with the standard derivative structure. This option is plausible only if the model's gradient with respect to theta2 yields a factor proportional to x_t^2 without log x_t, which is unlikely given the exponent form y_t = theta1 x_t^{theta2} + error. Hence, this option is questionable.

Option F: E(epsilon_hat,t^2 x_t^2 theta_hat2)
- This is an expected-value form missing any log(x_t) or theta_hat1 factors, and has theta_hat2 multiplying x_t^2. Similar to Option E, the absence of log(x_t) is suspicious given the x_t^{theta2} term in the model; differentiation with respect to theta2 typically introduces a log(x_t). The expectation form also raises questions about whether this is intended as a population quantity rather than the sample estimate. Overall, this option is unlikely to be the correct missing entry.

Summarizing the comparative plausibility: among the options, the one that aligns most closely with the typical structure that connects the gradient components to the variance estimator while incorporating the log(x_t) term required by the derivative of x_t^{theta2} with respect to theta2 is the expression that pairs epsilon_hat,t^2 with theta_hat1, x_t^2, theta_hat2, and log(x_t) inside a 1/T average. The exact proportional factors (whether theta_hat1 and theta_hat2 multiply in this way) would normally come from the detailed gradient calculations, but the presence of x_t^2 and log(x_t) together with residual-squared weighting is the hallmark of a term built from derivatives with respect to theta1 and theta2 in a model with x_t^{theta2} dependence. Therefore, the candidate that best matches this reasoning is the option that writes 1/T sum_{t=1}^T epsilon_hat,t^2 theta_hat1 x_t^2 theta_hat2 log(x_t).

Note: If one wanted to determine the precise justification, it would be helpful to reconstruct the score (gradient) components for theta1 and theta2, derive the outer product or the sandwich form for Omega0, and then compare to each candidate expression to see which exactly matches the derived form. The distinctions above hinge on whether theta-hat factors appear inside the sum as products of gradient components or as separate multipliers, and whether log(x_t) terms are present as dictated by derivatives with respect to theta2.

查看解析

登录即可查看完整答案

类似问题

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单