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
ˆ
θ
1x
2
ˆ
θ
2
t
log What is the missing entry in the matrix
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
ˆ
θ
1x
2
ˆ
θ
2
t
log What is the missing entry in the matrix 
𝛺
̂
0
?

BlackTom AI · Accepted Answer

We start by restating the problem setup and the object to be computed. The nonlinear least squares estimator is asymptotically normal with a sandwich-type asymptotic variance. The matrix Ω0 hat is the consistent estimator of the second piece in the sandwich, given by the outer product of the score (gradient of the nonlinear mean with respect to θ) weighted by squared residuals, averaged over T observations: Ω̂0 = (1/T) ∑ ε̂_t^2 ∂f_t(θ̂)/∂θ ∂f_t(θ̂)' where f_t(θ) is the mean part of y_t as a function of θ. In the typical linear-in-parameters or additive model y_t = θ1 x_t + θ2 t + ε_t, the gradient ∂f_t/∂θ is the vector [x_t, t]′. Therefore, the (1,1) entry of Ω̂0 is (1/T) ∑ ε̂_t^2 x_t^2, the (1,2) and (2,1) entries are (1/T) ∑ ε̂_t^2 x_t t, and the (2,2) entry is (1/T) ∑ ε̂_t^2 t^2. With this in mind, we analyze the given option.

Option under consideration: "1
𝑇
∑
𝑡
=
1
𝑇
𝜀
̂
𝑡
2
𝑥
𝑡
2
𝜃
̂
2" (as written, the entry seems to be proposing a term that includes ε̂_t^2, x_t^2, and θ̂_2 in a single expression, and it also appears to mix summation indices in a way that isn’t consistent with the standard (1/T) ∑ ε̂_t^2 x_t^2 structure for the (1,1) element of Ω̂0).

Why this is not correct as the missing entry for Ω̂0:
- The correct (1,1) element of Ω̂0 should be (1/T) ∑ ε̂_t^2 x_t^2. It is a sum of squared residuals times x_t^2, averaged, with no θ̂_2 inside the sum. Including θ̂_2 in the same term is inconsistent with the standard derivation, because θ̂ appears only as the argument for computing the residuals ε̂_t and does not multiply the derivative terms inside the variance estimator.
- The presence of θ̂_2 inside the (1,1) entry would imply a dependence on the estimated slope parameter inside the variance weighting in a way that the usual sandwich form does not prescribe. The element should be purely a sample average of ε̂_t^2 times the product of the corresponding gradient components, here x_t with itself, i.e., x_t^2, not a product involving θ̂_2.
- Even if we try to interpret the given line as attempting to express a matrix entry, the structure 1/T ∑ ε̂_t^2 x_t^2 (something) would still need to be x_t^2 alone for the (1,1) element; the θ̂_2 factor would belong to the mean function evaluation, not to the weight for the Ω part.

What the correct reasoning shows: the missing entry for the (1,1) position of Ω̂0 should be 1/T ∑ ε̂_t^2 x_t^2. The other potential entries in Ω̂0 would be 1/T ∑ ε̂_t^2 x_t t for the (1,2) and (2,1) positions, and 1/T ∑ ε̂_t^2 t^2 for the (2,2) position. Since the option mixes in θ̂_2 and does not clearly align with the standard (1,1) form, it does not match the correct expression for the (1,1) entry of Ω̂0.

In summary, the key idea is that Ω̂0 is built from the outer product of the gradient [x_t, t] with itself, weighted by ε̂_t^2 and averaged over T, which yields the (1,1) term as (1/T) ∑ ε̂_t^2 x_t^2; the given option’s inclusion of θ̂_2 makes it inconsistent with that derivation.

BU.232.630.W6.SP25

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