Questions
Questions

BU.232.630.W5.SP25 sample_quiz_1

Single choice

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?

Options
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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Step-by-Step Analysis
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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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?

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