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Consider the nonlinear model 𝑦 𝑡 = 𝜃 1 𝑥 𝑡 𝜃 2 + 𝜀 𝑡 where the sample data ( 𝑦 1 , 𝑥 1 ) , . . . , ( 𝑦 𝑇 , 𝑥 𝑇 ) are i.i.d. and 𝐸 ( 𝜀 𝑡 | 𝑥 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 ̂ 2 1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 𝑥 𝑡 2 𝜃 ̂ 2 log ( 𝑥 𝑡 ) 1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 𝑥 𝑡 2 𝜃 ̂ 2 log ( 𝑥 𝑡 ) ] What is the missing entry in the matrix 𝛺 ̂ 0 ?

Options
A.𝔼 ( 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 2 𝑥 𝑡 2 𝜃 ̂ 2 log 2 ( 𝑥 𝑡 ) )
B.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 𝑥 𝑡 2 𝜃 ̂ 2 log ( 𝑥 𝑡 )
C.𝔼 ( 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 𝑥 𝑡 2 𝜃 ̂ 2 log ( 𝑥 𝑡 ) )
D.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝜃 ̂ 1 2 𝑥 𝑡 2 𝜃 ̂ 2 log 2 ( 𝑥 𝑡 )
E.1 𝑇 ∑ 𝑡 = 1 𝑇 𝜀 ̂ 𝑡 2 𝑥 𝑡 2 𝜃 ̂ 2
F.𝔼 ( 𝜀 ̂ 𝑡 2 𝑥 𝑡 2 𝜃 ̂ 2 )
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Step-by-Step Analysis
The task is to identify the missing entry in the matrix Ω̂0 based on the given asymptotic variance formula for the nonlinear least squares estimator. We begin by restating the available pieces of information and the structure of the problem to ground our analysis. First, the context: we have a nonlinear model y_t = theta1 x_t^{theta2} + epsilon_t with E(epsilon_t | x_t) = 0, and the NLSE is asymptotically normal with a specific variance formula involving A0 and Omega0. The estimator's asymptotic covariance is A0^{-1} Omega0 A0^{-1} and the problem provides a partial construction of the Omega0 estimator as a T-variate average of terms involving the squared residuals, x_t, and theta-hat components, along with log(x_t). Now, we evaluate each answer option in light of the standard form: Omega0 is typically built from the outer product of the score or gradient contributions, or from the expectation of the squared moment conditions, which in this nonlinear least squares setting translate into expressions that weight the squared residuals by derivatives with respect to the parameters. The provided pieces suggest a structure of terms like epsilon-hat_t^2 multiplied by functions of x_t and theta-hat, with subcomponents corresponding to theta1, theta2, and possibly log x_t terms arising from derivatives of x_t^{theta2} with respect to theta2 or from a transformation that introduces log x_t into the gradient. Option-by-option analysis: Option A: ......Login to view full explanation

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