题目
题目

BU.232.630.W5.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 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?

选项
A.1 T ∑ T t=1 ˆ ε 2 t ˆ θ 2 1 x 2 ˆ θ 2 t log2(xt)
B.𝔼( ˆ ε 2 t ˆ θ 2 1 x 2 ˆ θ 2 t log2(xt))
C.𝔼( ˆ ε 2 t x 2 ˆ θ 2 t )
D.1 T ∑ T t=1 ˆ ε 2 t x 2 ˆ θ 2 t
E.𝔼( ˆ ε 2 t ˆ θ 1x 2 ˆ θ 2 t log(xt))
F.1 T ∑ T t=1 ˆ ε 2 t ˆ θ 1x 2 ˆ θ 2 t log(xt)
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思路分析
We begin by restating the problem setup in our own words to ensure clarity: we have a nonlinear model y_t = θ1 x_t^{θ2} + ε_t with E(ε_t | x_t) = 0, and the asymptotic variance Ω0̂ needs to be estimated. The missing entry in Ω0̂ is a term that appears in the empirical sandwich form, combining the squared residuals, derivatives of the model with respect to the parameters, and a transformation of x_t (specifically log terms). Now let's examine each candidate: Option 1: 1 T ∑ T t=1 ˆε_t^2 ˆθ2 1x^2 ˆθ2_t log2(xt) This option mirrors the structure seen in the output for Ω0̂: it includes a sum over t of squared residuals, multiplied by the squared derivative term with respect to θ2 (here denoted as ˆθ2, which represents the derivative of the model with respect to θ2 given x_t). It also includes a log2(xt) term, consistent with the presence of a log transformation in the described Ω0̂ expression. The placement of the 1/x^2 term and the s......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 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 ?

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 𝐴 0 , 𝐴 ̂ 0 = [ 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 𝜃 ̂ 1 2 ( 𝑥 𝑡 − 1 ) 𝜃 ̂ 2 2 𝑧 𝑡 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝜃 ̂ 1 2 𝑥 𝑡 − 1 𝑧 𝑡 𝜃 ̂ 2 2 𝑧 𝑡 − 1 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝜃 ̂ 1 2 𝑥 𝑡 − 1 𝑧 𝑡 𝜃 ̂ 2 2 𝑧 𝑡 − 1 ] What is the missing entry in the matrix 𝐴 ̂ 0 ?

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