Reference no: EM131092190
Econometrics 715 Problem Set 2, Fall 2015
The model has two parts. The first is for the conditional mean:
yi = β1x1i + β2x2i + β3exp(β4x3i) + β5 + ei.
Let β = (β1, β2, β3, β4, β5).
The second is for the error, which states that ei = θ1ui, where ui 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/ni=1∑n(y - β1x1 - β2x2 - β3exp (β4x3) - β5)2.
Given β^ we calculate the residuals
e^i = yi - β^1x1i - β^2x2i - β^3exp (β^4x3i) - β^5
Second, θ^ minimizes the log-likelihood for ei, 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
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