题目
题目

BU.232.630.W6.SP25 sample_quiz_1

单项选择题

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

选项
A.𝔼( ˆ θ 2 1 x 2 ˆ θ 2 t log2(xt))
B.1 T ∑ T t=1 x 2 ˆ θ 2 t
C.𝔼(x 2 ˆ θ 2 t )
D.1 T ∑ T t=1 ˆ θ 2 1 x 2 ˆ θ 2 t log2(xt)
E.𝔼( ˆ θ 1x 2 ˆ θ 2 t log(xt))
F.1 T ∑ T t=1 ˆ θ 1x 2 ˆ θ 2 t log(xt)
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标准答案
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思路分析
To determine the missing entry in the estimated information matrix A0, we need to inspect how A0 is formed in nonlinear least squares for the given model. Option by option analysis: Option 1: 𝔼(ˆθ2^2 1/x^2 ˆθ2 t log^2(xt)) — This mixes various terms in a way that does not align with the standard construction of A0, which uses derivatives of the nonlinear function with respect to θ evaluated at the data and typically involves powers of x_t and log(x_t) separately rather than combining them inside an expectation with 1/x^2 factors. This does not match the conventio......Login to view full explanation

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类似问题

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?

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 ?

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?

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