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

BlackTom AI · Accepted Answer

The question as provided appears garbled, and the answer options section is empty. To proceed meaningfully, I will restate what is given, then analyze the structure of the missing entry in Â0 in a general way, and finally explain why the specific option shown (if interpreted as the missing entry) cannot be correct given the usual form of A0 in nonlinear least squares.

Restating the problem and what’s typically asked:
- We have a nonlinear model y_t = f(x_t; θ) + ε_t, with θ = (θ1, θ2)′, and i.i.d. sample (y_t, x_t) with E(ε_t | x_t) = 0.
- The nonlinear least squares estimator θ̂_NL is asymptotically normal with a sandwich-like asymptotic covariance matrix involving A0 and Ω0, where A0 = plim (1/T) ∑_t ∂f(x_t; θ)/∂θ ∂f(x_t; θ)′ and Ω0 = plim (1/T) ∑_t Var(ε_t ∂f(x_t; θ)/∂θ).
- The question asks for the missing entry in the matrix Â0, which would be the sample analogue of A0 evaluated at θ̂.
- The provided “answer” string in the data appears to be a scalar expression involving T^{-1} ∑ x_t^2 θ̂2, and the answer options block is empty. We will critique this on the basis of the standard form of A0 and the dimensionality involved.

General analysis of what A0 should look like:
- A0 is the expected outer product of the gradient of the mean function with respect to θ, evaluated at the true θ0. That is, A0 = E[ g_t g_t′ ] where g_t = ∂f(x_t; θ)/∂θ |_{θ=θ0}.
- Since θ has two components (θ1 and θ2), g_t is a 2×1 vector, and thus A0 is a 2×2 matrix. The sample analogue Â0 is (1/T) ∑_t ĝ_t ĝ_t′, where ĝ_t = ∂f(x_t; θ)/∂θ |_{θ=θ̂}.
- Therefore, any proposed entry of Â0 must contribute to a 2×2 matrix, i.e., there should be two diagonal entries and two off-diagonal entries in Â0. A single scalar such as (1/T) ∑ x_t^2 θ̂2 cannot itself be the entire Â0; at best it could be one scalar entry if appropriately placed, but the complete Â0 must be a 2×2 matrix constructed from the gradients of the two components of f with respect to θ1 and θ2.

Why the scalar expression (1/T) ∑ x_t^2 θ̂2 is unlikely correct as the missing entry:
- Dimensionality mismatch: Â0 is intended to be a 2×2 matrix, but the expression (1/T) ∑ x_t^2 θ̂2 is a single scalar. A single scalar cannot fill a 2×2 matrix unless it’s intended to stand for a single entry in a larger constructed matrix, which would require the rest of the entries to be specified in a consistent way. With the information given, there is no basis to believe this scalar alone constitutes Â0.
- Invariance under θ̂: Â0 should be computed from gradients and be a function of data and the estimated parameters in a way that, in large samples, converges to the population A0. A scalar ∑ x_t^2 θ̂2 does not reflect the outer product structure of g_t g_t′ unless the model’s gradient itself collapses to a single direction, which is not generally the case for a two-parameter nonlinear model.
- Missing gradient structure: Without the explicit form of f(x_t; θ), we cannot credibly reduce ∂f/∂θ to a scalar times x_t^2 or any other simple form. The standard approach is to compute ∂f/∂θ1 and ∂f/∂θ2, form g_t = [∂f/∂θ1, ∂f/∂θ2]′, and then Â0 = (1/T) ∑ g_t g_t′. Any claimed entry should be interpretable as an entry of that 2×2 matrix.

What this implies for answering the question, given the provided data:
- Since the answer options are missing, and the typical correct form of Â0 is a 2×2 matrix derived from the outer product of the gradient, we cannot validate the scalar expression shown as the missing entry.
- If a candidate answer attempted to claim that Â0 equals a scalar (1/T) ∑ x_t^2 θ̂2, that would conflict with the dimensionality and structure of Â0. The proper entry(s) of Â0 should involve both components of the gradient with respect to θ1 and θ2 and would form a matrix, not a single scalar.

In summary, without the explicit answer options, the reasonable conclusion is that the proposed scalar (1/T) ∑ x_t^2 θ̂2 cannot correctly represent the missing entry of Â0 in a two-parameter nonlinear least squares setting, because Â0 should be a 2×2 matrix given the gradient vector g_t = [∂f/∂θ1, ∂f/∂θ2]′. Any valid solution would present all four entries of Â0, each arising from the appropriate cross-products of the two gradient components, evaluated at θ̂. If you can provide the explicit gradient form ∂f/∂θ1 and ∂f/∂θ2 or supply the complete set of answer options, I can compute and explain the correct Â0 entry precisely.

BU.232.630.W4.SP25 sample_quiz_1

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