题目
题目

BU.232.630.W1.SP25 Quiz 1 solutions

单项选择题

Consider the nonlinear model 𝑦 𝑡 = 𝜃 1 𝑥 𝑡 + 𝜃 2 𝑧 𝑡 + 𝜀 𝑡 where the sample data ( 𝑦 1 , 𝑥 1 , 𝑧 1 ) , . . . , ( 𝑦 𝑇 , 𝑥 𝑇 , 𝑧 𝑇 ) are i.i.d. and 𝐸 ( 𝜀 𝑡 | 𝑥 𝑡 , 𝑧 𝑡 ) = 0 . We know that the nonlinear least square estimator is asymptotically normal, that is ⤳ 𝑇 ( 𝜃 ̂ 𝑁 𝐿 − 𝜃 0 ) ⤳ 𝑑 𝑁 ( 0 , 𝐴 0 − 1 𝛺 0 𝐴 0 − 1 ) To compute the standard errors we need to estimate 𝛺 0 , 𝛺 ̂ 0 = [ 1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑥 𝑡 2 𝜃 ̂ 1 2 ( 𝑥 𝑡 − 1 ) 1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑧 𝑡 2 𝜃 ̂ 2 2 ( 𝑧 𝑡 − 1 ) ] What are the missing entry in the matrix 𝛺 ̂ 0 ?

选项
A.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 𝑥 𝑡 𝑧 𝑡 2 𝜃 ̂ 2 ( 𝑧 𝑡 − 1 )
B.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜃 ̂ 1 2 𝑥 𝑡 𝑧 𝑡 2 𝜃 ̂ 2 2 ( 𝑧 𝑡 − 1 )
C.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑥 𝑡 2 + 𝜃 ̂ 1 2 ( 𝑥 𝑡 − 1 ) 𝜃 ̂ 2 2 𝑧 𝑡
D.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑥 𝑡 𝜃 ̂ 1 2 𝑥 𝑡 − 1 𝑧 𝑡 𝜃 ̂ 2 2 𝑧 𝑡 − 1
E.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑥 𝑡 𝜃 ̂ 1 𝑥 𝑡 − 1 𝑧 𝑡 𝜃 ̂ 2 𝑧 𝑡 − 1
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思路分析
The core idea in this nonlinear least squares setting is that, for y_t = θ1 x_t + θ2 z_t + ε_t with ε_t uncorrelated with (x_t, z_t), the asymptotic covariance involves the variance of the score, which is driven by ε_t^2 times the outer product of the gradient of the mean function with respect to θ. The gradient with respect to θ is [x_t, z_t]'. Therefore, the population version of Ω0 should be the 2x2 matrix with entries: - Ω0(1,1) = E[ε_t^2 x_t^2] - Ω0(1,2) = Ω0(2,1) = E[ε_t^2 x_t z_t] - Ω0(2,2) = E[ε_t^2 z_t^2] In sample form, the estimator Ω̂0 would be (1/T) ∑ ε̂_t^2 times the outer product [x_t; z_t] [x_t, z_t], i.e., [ Ω̂0(1,1) Ω̂0(1,2) ; Ω̂0(2,1) Ω̂0(2,2) ] = (1/T) ∑ ε̂_t^2 [ [x_t^2, x_t z_t], [x_t z_t, z_t^2] ]. Option-by-option reasoning: Option 1: '1/T ∑ ε̂_t^2 θ̂1 x_t z_t^2 θ̂2 (z_t − 1)' - Wh......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 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) 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 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?

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?

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 ?

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