ETC3400 - Principles of Econometrics - Probability Density Function - Vector of Parameters - Economics Assignment Help

ETC3400: Principles of Econometrics Probability Density Function Vector of Parameters Economics Assignment Help

Assignment Task:

Perform the following tasks:
(a) Given the MLE of θ0, θb = (μ, b σ, b νb) 0
, define the MLE of ψ(θ0), ψ(θb).
(b) Use the delta method to state the form of the limiting distribution of √n(ψ(θb) −ψ(θ0)). (Note - I do not expect you to derive the precise form of the Fisher information
matrix!)

(c) Using the template program from lectures (mc simulation 1 lecture 2017) (or anything subsequently used in tutorials), simulate M = 5000 repeated random samples of size n = 50 from Y ∼ Student t(μ0 = 5, σ0 = 2, ν0 = 4). Hint: Use the following EVIEWS command to generate y = (y1, y2, ..., yn)0: series y = !truemu + !truesigma*@rtdist(!truenu), where: truemu = 5, truesigma = 2and truenu = 4.

(d) For each simulated sample, produce and save the MLE of ψ(θ0), ψ(θb). Hint: To do this, use the following type of logl commands at the appropriate point in
the program: logl ll1 ll1.append @logl logf stt (This is giving the log likelihood function a name) ll1.append @param c(1) !truemu c(2) !truesigma c(3) !truenu (This is setting the starting values for the MLE numerical optimization algorithm at the true values!) ll1.append logf stt =..... (you need to insert the appropriate code here)
smpl 1 !n ll1.ml(showopts, maxit = 1000, convcrit = 1e-5) ML estimates(!i) = (c(2)ˆ2)*(c(3)/(c(3)-2)) where you have dimensioned the vector ‘ML estimates’ appropriately at the top of the program.

Use the simulation-based estimate (i.e. as given in (3)) to produce an estimate of the variance of this limiting distribution.....note this is simple but you need to think
carefully!)

(f) Keeping M = 5000, now make appropriate modifications to your program - and record the appropriate plots - to demonstrate numerically the theoretical distributional result stated in Q1b. Note - to keep things simple, at each stage of this demonstration you can use var d(ψ(θb)) to produce an estimate of the true asymptotic variance of ψ(θb).

Notes:
• This is analogous to the result that if a scalar a is positive, so is a −1 , and that a −1 can be expressed as a −1 = b 2 where b −1 exists.

• The actual form of P is very simple in this case given that V is diagonal. However, you don’t need to exploit that particular form to answer the following questions.
Now, using this matrix P, we can transform the linear model as: Py = PXβ + Pu which we then write as y ∗ = X∗β + u ∗

(7) where: y ∗ = Py X∗ = PX u ∗ = Pu

Perform the following tasks:

(a) Using (6) and the properties of P, show that u ∗ ∼ N(0, In).

Now:
i. Show that: E(b) = β

ii. Show that: varcov(b) = (X0X) −1X0VX(X0X) −1
iii. Argue that the matrix difference: (X0X) −1X0VX(X0X) −1 − (X0V−1X) −1 is positive semi-definite. Note: This question can be answered without any computations.

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