Derive the conditional density for y given x and the

Question

Econometrics 715 Problem Set 1, Fall 2015

The file ps1.dat is a text file with 1000 rows and 4 columns. Each row is an independent observation. The four variables are (y_i; x_1i; x_2i; x_3i). We will be estimating the parameters of the following equation

yi = θ₁x_1i + θ₂x_2i + θ₃exp (θ₄x_3i) + θ₅ + e_i

Each estimator takes the form

θ^{^} = argmin Sn(θ)

where

S_n(θ) = 1/n _i=1Σⁿm(y_i; x_i; θ)

for a different functions m(y; x; θ).

Do your numerical optimzation in Matlab, R, or your package of choice. Use the standard optimation software, but donít use packages speciÖc for the problems. In each case, report the estimates θ^{^} obtained, and (briefly) the method used for optimization. Include your code.

1. Nonlinear-least-squares (NLLS).

m(y; x; θ) = (y - θ₁x₁ - θ₂x₂ - θ₃exp (θ₄x₃) - θ₅)²

2. Huber criterion

m(y; x; θ) = h (y - θ₁x₁ - θ₂x₂ -  θ₃exp (θ₄x₃) - θ₅)

3. L⁴

m(y; x; θ) = (y - θ₁x₁  - θ₂x₂ - θ₃exp (θ₄x₃) - θ5)⁴

4. Student t likelihood

For this application, suppose that the error e_i takes the form e_i = θ₆u_i where u_i has a student t density with degree-of-freedom θ₇. Thus u has the density

f(u) = (Γ(θ₇+1/2)/√(θ₇π)Γ(θ₇/2)) (1 + u²/θ₇)^-(θ7+1)/2

Derive the conditional density for y given x and the conditional log-likelihood function for y. Write out m(y; x; θ) (the negative log-density). The parameters are θ = (θ₁, θ₂, ..., θ₇).

Attachment:- ps1.dat.rar