题目
题目
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 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 ?

选项
A.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜃 ̂ 1 2 ( 𝑥 𝑡 − 1 ) 𝜃 ̂ 2 2 𝑧 𝑡
B.1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝜃 ̂ 1 2 𝑥 𝑡 − 1 𝑧 𝑡 𝜃 ̂ 2 2 𝑧 𝑡 − 1
C.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜃 ̂ 1 2 𝑥 𝑡 𝑧 𝑡 2 𝜃 ̂ 2 2 ( 𝑧 𝑡 − 1 )
D.1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 𝜃 ̂ 1 ( 𝑥 𝑡 − 1 ) 𝜃 ̂ 2 𝑧 𝑡
E.1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 2 𝜃 ̂ 1 2 ( 𝑥 𝑡 − 1 ) 𝜃 ̂ 2 2 𝑧 𝑡
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思路分析
The question presents a nonlinear least squares estimation setup and asks for the missing entry in the estimated information matrix Â0. To approach this, we first restate the given pieces and then inspect each candidate for consistency with the expressions shown. Restating the problem: - We have a model y_t = θ1 x_t + θ2 z_t + ε_t with ε_t conditional on (x_t, z_t) having mean zero. - The asymptotic distribution of the NLSE θ̂_NL is given, and the information matrix involves A0 and A0−1/Ω0 A0−1, etc. - The entries shown in the text for Â0 include terms that mix x_t, z_t, and the estimated parameters θ̂1, θ̂2, with factors like 1/T ∑ t and 1/T ∑ t z_t, (x_t − 1), (z_t − 1), and θ̂1, θ̂2. The missing entry must complete the empirical mixed second-derivative or outer-product-of-gradients form consistent with these terms. Option-by-option analysis: - Option 1: 1/T ∑ t = 1/T θ̂1/2 x_t z_t 2 θ̂2/2 (z_t − 1) This candidate suggests a term that combines the impulse of x_t and z_t scaled by θ̂1 and θ̂2, with a (z_t − 1) factor. The appearance of both x_t and z_t inside a single sum with a product is characteristic of cross-d......Login to view full explanation

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类似问题

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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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