Consider the nonlinear model
yt=θ
xt
1
+θ
zt
2
+εt
where the sample data (y1,x1,z1),...,(yT,xT,zT) are i.i.d. and E(εt|xt,zt)=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
t
ˆ
θ
2(xt−1)
1

1

T
∑
T
t=1
ˆ
ε
2
t
z
2
t
ˆ
θ
2(zt−1)
2
]
What are the missing entry in the matrix
ˆ
Ω
0?

Question

Consider the nonlinear model
yt=θ
xt
1
+θ
zt
2
+εt
where the sample data (y1,x1,z1),...,(yT,xT,zT) are i.i.d. and E(εt|xt,zt)=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
t
ˆ
θ
2(xt−1)
1

1

T
∑
T
t=1
ˆ
ε
2
t
z
2
t
ˆ
θ
2(zt−1)
2
]
What are the missing entry in the matrix 
ˆ
Ω
0?

BlackTom AI · Accepted Answer

The question concerns the missing entry in the matrix that estimates Ω0, which appears in the asymptotic distribution of the nonlinear least squares estimator. We are given the model and the form of the estimator’s asymptotic covariance, and we must identify which term completes the (1 1) block corresponding to the variance contributed by the x-term and the θ-terms.

Option 1: 1

T ∑ T t=1 ˆε 2 t ˆx t ˆθ xt−1 1 z t ˆθ z t−1 2

This option places the squared residual ˆε t^2 multiplied by x t and a cross term involving ˆθ with xt−1, and similarly includes a term with z t and z t−1. The structure resembles a sum of outer products evaluated at the regressor components, but the appearance of ˆε t ˆx t and ˆθ xt−1 is unusual since typical Ω0 entries involve either ε̂ t^2 times a function of the regressors or cross terms between regressors, not mixed terms like ˆx t multiplied by ˆθ xt−1 in a single scalar entry. The placement of 1 z t ˆθ z t−1 2 at the end also seems inconsistent with standard forms where the blocks correspond to derivatives with respect to θ and to the regressors x and z separately. Overall, this option looks structurally questionable for a variance-covariance entry intended to scale the score with the squared residual and regressor values.

Option 2: 1

T ∑ T t=1 ˆθ 2 xt 1 z 2 t ˆθ 2 (zt−1) 2

Here we have ˆθ squared terms multiplying xt and zt, and then another factor involving (zt−1) squared. This mixes parameter estimates with lagged regressors in a way that is atypical for the Ω0-structure, which should reflect the second moment of the score components, i.e., derivatives of the log-likelihood (or objective) with respect to θ, weighted by the regressors, not by the squared θ estimate itself. The presence of ˆθ^2 multiplies Xt and Zt terms in a manner not aligned with the usual sandwich form for Ω0, making this option unlikely to be the correct missing entry.

Option 3: 1

T ∑ T t=1 ˆε 2 t x 2 t + ˆθ 2 (xt−1) 1 ˆθ 2 z t 2

This candidate splits into two additive pieces: one involving ˆε t^2 times x t^2, and another involving squared θ terms with lagged regressors, but it oddly combines them without clear cross-covariance structure between the x- and z-regressor components. The dimensions and units would not match the Ω0 block that couples the residual variance to the x and z regressor terms through the θ-derivative components. Also, the cross-term structure is missing, which is typically needed to capture the influence of both x and z jointly on the asymptotic variance. This makes it suspect as the correct entry.

Option 4: 1

T ∑ T t=1 ˆε 2 t x 2 t ˆθ 2 xt−1 1 z t ˆθ 2 z t−1 2

This expression contains the residual-squared term ˆε t^2 multiplied by x t^2, plus a separate product ˆθ^2 xt−1 multiplying the x-part and a separate z t times ˆθ^2 z t−1 term for the z-part. The combination looks like it tries to assemble a block with x and z, but the mixed appearance of ˆε 2 t x 2 t alongside ˆθ^2 (xt−1) and ˆθ^2 (zt−1) terms without clear cross-products between x and z or between ε and regressors is atypical. The indexing and placement of xt−1 and zt−1 with ˆθ^2 factors do not align with standard Ω0 constructions for nonlinear least squares where the score involves derivatives w.r.t θ times the regressor values, not products of squared θ terms with lagged regressors in this dispersed fashion.

Option 5: 1

T ∑ T t=1 ˆε 2 t xt ˆθ 2 xt−1 1 z t ˆθ 2 z t−1 2

This option explicitly ties the squared residual to the x-regressor with a multiplier ˆθ^2 xt−1, and similarly ties the residual to the z-regressor with ˆθ^2 z t−1. The structure suggests a cross-contribution where the contribution of the t-th observation to Ω0 is weighted by both the residual and the (lagged) squared θ terms for each regressor. The pattern aligns more closely with a block that accounts for the sensitivity of the objective to θ via the derivative terms, combined with the regressor values xt and zt, and scaled by appropriate lagged θ components. While the exact algebra would depend on the detailed derivation, this form at least mirrors the expected dependence on ˆε t^2 and the regressor components with appropriate θ-derivative factors, making it the most plausible among the choices.

Across the options, the common theme is assembling the Ω0 entry from terms that involve ˆε t^2 and regressor-related factors, modulated by θ-derivative terms. Several choices introduce unlikely cross-terms or misaligned indices, which tend to misrepresent how Ω0 is constructed in a nonlinear least squares context. The option that most coherently matches the typical structure—combining ˆε t^2 with the product of xt, zt, and their associated θ-derivative components with lagged terms—appears to be the one that best reflects the intended Ω0 entry.

BU.232.630.W1.SP25 Quiz 1 solutions

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