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 A0,
ˆ
A
0=[
1

T
∑
T
t=1
x
2
ˆ
θ
2
t

1

T
∑
T
t=1
ˆ
θ
1x
2
ˆ
θ
2
t
log(xt)
1

T
∑
T
t=1
ˆ
θ
2
1
x
2
ˆ
θ
2
t
log2(xt)]
What is the missing entry in the matrix
ˆ
A
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 A0,
ˆ
A
0=[
1

T
∑
T
t=1
x
2
ˆ
θ
2
t

1

T
∑
T
t=1
ˆ
θ
1x
2
ˆ
θ
2
t
log(xt)	
1

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

BlackTom AI · Accepted Answer

The question asks for the missing entry in the estimated information matrix Â0 used in the asymptotic covariance of the nonlinear least squares estimator.
Option under consideration: '1

T
∑
T
t=1
ˆ
θ
1x
2
ˆ
θ
2
t
log(xt)'. In this context, Â0 is built from the outer product of the gradient of the nonlinear model with respect to the parameter θ, evaluated at θ̂, averaged over observations.
To see what goes into A0, recall that for a nonlinear regression model y_t = θ1 x_t^{θ2 t} + ε_t (with log transformations implied in the notation), the gradient components with respect to θ1 and θ2 typically involve terms such as x_t^2 and x_t^2 log(x_t), as well as cross-products with the implied derivative structure. Specifically, the elements of A0 are expectations (or sample analogues) of products of these gradient components across time.
The missing entry in Â0 should pair the appropriate gradient terms so that the (1,1), (2,2), and (1,2) (or (2,1)) entries reflect the variance-covariance structure of the score vector. The candidate term shown in the option, which involves ˆθ1 x_t^2 ˆθ2 t log(x_t) summed over t and scaled by T, corresponds to a cross-term that combines the θ1-related gradient (through x_t^2) and the θ2-related gradient (through log(x_t)).
Intuitively, this cross-term arises because the gradient of the log-likelihood (or sum of squared residuals in nonlinear least squares) with respect to θ1 and θ2 often includes products of x_t^2 and log(x_t) components when the model includes exponents or log-transform-like structures. Therefore, including a term proportional to ∑ ˆθ1 x_t^2 · ˆθ2 log(x_t) (properly scaled by 1/T) aligns with forming the mixed partial contribution to Â0 from the cross-derivative ∂g/∂θ1 · ∂g/∂θ2, where g(θ, x_t) is the model’s mean function.
What about potential alternatives or structural checks? If the matrix A0 is partitioned as [A11 A12; A21 A22], then A11 typically collects terms like x_t^2, A22 collects terms like (log x_t)^2 scaled by θ2-related factors, and A12 (or A21) collects cross-products involving both θ1 and θ2 such as x_t^2 log(x_t). The presence of a term with x_t^2 and log(x_t) multiplied by the estimated parameters matches what one would expect for the cross-block entry, and the 1/T normalization is standard for a sample average approximation to the population expectation.
If one considers the possibility of alternative cross-terms, such as a pure log(x_t) term or a term without the log, those would correspond to different gradient components and would not correctly reflect the interaction between θ1 through x_t^2 and θ2 through log(x_t). Hence such alternatives would be inconsistent with the mixed-derivative structure implied by the model.
In short, the given option captures the cross-product structure between the θ1-related gradient piece (involving x_t^2) and the θ2-related gradient piece (involving log(x_t)), which is exactly what appears in the off-diagonal (1,2) (or (2,1)) entry of Â0 in a nonlinear least squares setting with this model. The other potential entries (had they been listed) would need to match the appropriate gradient components such as x_t^2, (log x_t)^2, or constant terms, but those are not shown here.

BU.232.630.W6.SP25 sample_quiz_1

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