Reference no: EM131092190

Econometrics 715 Problem Set 2, Fall 2015

The model has two parts. The first is for the conditional mean:

y_i =  β₁x_1i + β₂x_2i + β₃exp(β₄x_3i) + β₅ + e_i.

Let β = (β₁, β₂, β₃, β₄, β₅).

The second is for the error, which states that e_i = θ₁u_i, where u_i has a student t density with degree-of-freedom θ₂. For reference, u has the density

f(u) = (Γ(θ₂+1/2)/√(θ₂π)Γ(θ₂/2)) (1 + u²/θ₂)^-(θ2+1)/2

Let θ = (θ₁; θ₂):

You consider the following estimator. First, β^{^} minimizes the NLLS criterion, e.g.

β^ = argmin 1/n_i=1∑ⁿ(y - β₁x₁ - β₂x₂ - β₃exp (β₄x₃) - β₅)².

Given β^{^} we calculate the residuals

e^{^}_i = y_i - β^{^}₁x_1i - β^{^}₂x_2i - β^{^}₃exp (β^{^}₄x_3i) - β^{^}₅

Second, θ^{^} minimizes the log-likelihood for e_i, conditional on β^{^} and e^{^}_i. That is, θ^{^} is calculated taking the "data points" e^{^}_i as given.

We are interested in the asymptotic distribution of β^{^} and θ^{^} and calculation of standard errors.

1. Show that β^{^} →_pβ where β is the pseudo-true minimizer of E (y - β₁x₁ - β₂x₂ - β₃exp(β₄x₃) - β₅)².

Try to work out reasonable regularity conditions, do not just state high-level assumptions on the criterion function if possible. Also, try and avoid imposing "correct specification" assumptions or conditional homoskedasticity.

2. Find the asymptotic distribution of √n(β^{^} - β) (similar comments as above).

3. Write θ^{^} as a two-step m-estimator. Show that θ^{^} →_p θ, where θ is the pseudo-true parameter value. Be specific about your regularity conditions.

4. Find the asymptotic distribution of √n(θ^{^} - θ). Use the two-step GMM approach.

5. Numerical application: Using the data, calculate β^{^}, θ^{^}, and standard errors for each.

Attachment:- ps1.dat.rar