Consider the nonlinear model
yt=θ1x
θ2
t
+εt
where the sample data (y1,x1),...,(yT,xT) are i.i.d. and E(εt|xt)=0. We know that the nonlinear least square estimator is asymptotically normal, that is
√
T
(
ˆ
θ
NL−θ0)
d
⤳
N(0,A
−1
0
Ω0A
−1
0
)
To compute the standard errors we need to estimate Ω0,
ˆ
Ω
0=[
1

T
∑
T
t=1
ˆ
ε
2
t
x
2
ˆ
θ
2
t

1

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

T
∑
T
t=1
ˆ
ε
2
t
ˆ
θ
2
1
x
2
ˆ
θ
2
t
log2(xt)]
What is the missing entry in the matrix
ˆ
Ω
0?

Question

Consider the nonlinear model
yt=θ1x
θ2
t
+εt
where the sample data (y1,x1),...,(yT,xT) are i.i.d. and E(εt|xt)=0. We know that the nonlinear least square estimator is asymptotically normal, that is
√
T
(
ˆ
θ
NL−θ0)
d
⤳
N(0,A
−1
0
Ω0A
−1
0
)
To compute the standard errors we need to estimate Ω0,
ˆ
Ω
0=[
1

T
∑
T
t=1
ˆ
ε
2
t
x
2
ˆ
θ
2
t

1

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

T
∑
T
t=1
ˆ
ε
2
t
ˆ
θ
2
1
x
2
ˆ
θ
2
t
log2(xt)]
What is the missing entry in the matrix 
ˆ
Ω
0?

BlackTom AI · Accepted Answer

We start by restating the core components of the problem to ground the discussion. The model is nonlinear: y_t = f(x_t, θ) + ε_t, with E(ε_t | x_t) = 0. The asymptotic covariance Ω0 appears in the sandwich form A^{-1} Ω0 A^{-1}, and the consistent estimator ̅Ω0hat is built by replacing population moments with sample analogs. Specifically, Ω0 is the expectation of the outer product of the score (the gradient of the mean function with respect to θ) weighted by ε_t^2, i.e., it involves terms like ε_t^2 times components of ∂f/∂θ. When forming the plug-in estimator, we replace ε_t by its residual ε̂_t and x_t (and any transformed variables like log x_t) by their observed values, and we assemble the appropriate outer-product structure across θ components. With this in mind, let’s analyze each option one by one.

Option 1: 𝔼(ε̂_t^2 ́θ̂_2 1 x^2 θ̂_2^t log^2(x_t))
- Why this might be tempting: it places ε̂_t^2 in front and includes log(x_t) terms, which often arise from derivatives with respect to θ2 in models featuring exponentiation or log-terms. The presence of log^2(x_t) could reflect a second-order term or a squared derivative component.
- Why it’s likely incorrect: Ω0 is a population second-moment-type object that lives inside a sample-average expression for the estimator rather than a pure expectation with respect to ε̂_t^2 and transformed x_t. In the plug-in estimator, we typically replace the expectation by the empirical average 1/T ∑, not an expectation outside the sum, and we form the product of residual-squared terms with the corresponding derivatives across θ. The appearance of log^2(x_t) inside an expectation, rather than as part of the empirical sum of ε̂_t^2 times a derivative term, signals a mismatch with the standard structure of Ω0 hat for a nonlinear least squares setting.

Option 2: 1/T ∑_{t=1}^T ε̂_t^2 x_t^2 θ̂_2^t log(x_t)
- Why this could look plausible: It resembles the idea of ε̂_t^2 times a derivative component that includes x_t and a log term, which would be natural if the derivative with respect to θ2 involves log(x_t) and x_t raised to some θ2 power, yielding a product form with x_t^2 as an additional scaling.
- Why it’s still likely incorrect: The standard Ω0 hat structure typically uses the outer product of the gradient components, i.e., sums of ε̂_t^2 times the outer product of the partial derivatives ∂f/∂θ. If f has components that lead to ∂f/∂θ1 ∝ x_t^{θ2 t} and ∂f/∂θ2 ∝ x_t^{θ2 t} log x_t, then the (1,1) entry of the outer product would involve (∂f/∂θ1)^2 times ε̂_t^2, and the (2,2) entry would involve (∂f/∂θ2)^2 times ε̂_t^2, with cross-terms as well. Simply stating ε̂_t^2 times a single derivative component with an extra x_t^2 factor does not align with the typical outer-product assembly unless the model’s Jacobian explicitly yields that exact scalar structure. Without that explicit derivative form, this option seems unlikely to match the required entry.

Option 3: 𝔼(ε̂_t^2 x_t^2 ∂f/∂θ2_t)
- Why this could be tempting: If the missing entry is the population moment corresponding to a single derivative component (a row or column of the Jacobian), then encountering ε̂_t^2 times x_t^2 times a derivative with respect to θ2 could match a single column (or row) of the outer-product structure when the first component, for instance, is proportional to x_t^2, and the second derivative involves log x_t. The inclusion of ∂f/∂θ2_t is plausible given that θ2 is the exponent parameter.
- Why it’s probably incorrect: Ω0 is formed as an expectation of the outer product of the entire gradient vector with itself, not as a single moment involving just ∂f/∂θ2 multiplied by ε_t^2 and x_t^2. Unless the question explicitly indicates we are extracting a particular diagonal or a particular cross-term corresponding to ∂f/∂θ2 alone, this option would represent only a slice of the full Ω0 and would omit the necessary cross-covariance structure with other θ components. Hence, as a standalone missing entry, it is unlikely to be the correct full entry.

Option 4: 1/T ∑ ε̂_t^2 θ̂1 x_t^2 log(x_t)
- Why this could seem reasonable: The term includes ε̂_t^2 and a product involving x_t^2 log(x_t), which aligns with derivatives that produce log(x_t) factors when differentiating with respect to a log-parameter or exponent. If ∂f/∂θ1 contains a factor proportional to x_t^{θ2 t}, and after some simplification the relevant component could resemble x_t^2 log(x_t), one might imagine this appearing in the entry for θ1’s influence.
- Why it’s unlikely to be correct: Similar to the reasoning above, Ω0 hat is formed from ε_t^2 times the outer product of the gradient components, which should involve both θ1 and θ2 derivatives in a coupled fashion (and their cross-products). A single term proportional to θ1 without including the corresponding derivative-magnitude factor for θ2 (or the cross-term) does not reflect the full structure of the outer-product matrix unless the model’s Jacobian collapses to a diagonal form, which is not generally the case here. Therefore, this option would generally be missing essential components to constitute a valid entry of Ω0 hat.

Option 5: 1/T ∑ ε̂_t^2 ∂f/∂θ1 ∂f/∂θ2
- Why this could be compelling: This form directly mirrors the outer-product construction, where an off-diagonal entry of Ω0 (or a cross-term in the sandwich) is proportional to the average of the product of the two gradient components, scaled by ε̂_t^2. If the question’s missing entry is the off-diagonal element corresponding to Cov(∂f/∂θ1, ∂f/∂θ2) weighted by ε_t^2, this would be a natural candidate.
- Why it might be correct in principle: In many GMM/NLS settings, the Ω0 matrix comprises expectations of ε_t^2 times the outer product g_t g_t′, where g_t is the gradient ∂f/∂θ. The (1,2) or (2,1) entry would then be E[ε_t^2 (∂f/∂θ1)(∂f/∂θ2)]. If the question is asking for one specific off-diagonal or a particular cross-term, this form could be the correct entry.
- Why it’s not guaranteed to be the correct single entry: Without the explicit indexing (which block of Ω0 is missing) or the precise definition of the gradient components for this nonlinear specification, we cannot confirm this is the exact missing piece. The option’s structure is plausible but may or may not match the intended entry depending on the exact parameterization of f and how the gradient components are defined in the problem.

Option 6: 𝔼(ε̂_t^2 ∂f/∂θ1 x_t ∂ f/∂θ2)
- Why this could be plausible: This resembles a cross-moment between the two gradient components weighted by ε_t^2, a common form in Ω0 when considering cross-terms in the outer-product of the gradient vector. If the problem’s missing entry is the population cross-moment between ∂f/∂θ1 and ∂f/∂θ2, this presentation would be natural.
- Why it’s not definite: The expression mixes an expectation with actual derivatives in a way that may not align with the standard Ω0hat construction, which would typically be an empirical average of ε̂_t^2 times the product (∂f/∂θ1)(∂f/∂θ2). The presence of x_t (without an accompanying derivative term for θ1 or a squared X factor) and the exact placement of ∂f/∂θ1 and ∂f/∂θ2 together makes it ambiguous whether this is the intended missing entry. Without the precise model-specific gradient expressions, this option remains speculative.

Synthesis and guidance for choosing among options:
- The correct missing entry should align with the plug-in estimator for Ω0, which is formed as the empirical average of ε̂_t^2 times the outer product of the gradient g_t = ∂f/∂θ evaluated at θ̂, i.e., Ω0hat ≈ 1/T ∑ ε̂_t^2 g_t g_t′. That means: (i) the entry reshaped into a scalar should reflect either a squared gradient component (diagonal) or a product of two gradient components (off-diagonal), (ii) no extraneous population expectations like 𝔼(·) should be present in the final empirical sum, and (iii) all x_t- and log(x_t)-dependent terms in the gradient should appear exactly as they arise from ∂f/∂θ1 and ∂f/∂θ2.
- Therefore, the most plausible correct form among the listed candidates would be the one that mirrors an empirical average of ε̂_t^2 times the product (∂f/∂θi)(∂f/∂θj) for the relevant i and j, without introducing extraneous expectation operators or unmotivated single-derivative scalings. Options that contain 𝔼(·) outside the empirical sum are unlikely to be the intended Ω0hat entry, and options that present only a single derivative term without the corresponding counterpart (for the cross-term) are also less likely unless the context specifies a diagonal simplification.

Note: Without the explicit expressions for ∂f/∂θ1 and ∂f/∂θ2 from the model y_t = f(x_t, θ) being used here (which appears to involve x_t raised to a θ2 exponent and potentially logs), we cannot definitively pick a single option as correct. The key takeaway is to match the structure to the gradient-based outer product form, avoid extraneous population expectations, and ensure the terms reflect the actual derivatives of f with respect to θ1 and θ2, evaluated at θ̂, multiplied by ε̂_t^2 and averaged over t.

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