Run the following code, using R, to simulate 15-day log returns assuming that daily log returns are t-distributed with parameters (0.0002,0.012,4). The number of simulations is 25,000.

set.seed(4886)

library(rugarch)

niter = 25000

N = 15

muTrue = 0.0002

sigmaTrue = 0.012

nuTrue = 4

## Create vector of niter * T daily returns

DailyReturns = rdist(niter*N, distribution="std", mu=muTrue, sigma=sigmaTrue, shape=nuTrue)

## Reshape vector to a niter by T matrix

DailyReturns = matrix(DailyReturns,nrow=niter)

R = sort(rowSums(DailyReturns)) ## order statistics of 15-day log returns

(a) Fit a t-distribution to the 25,000 15-day log returns by maximum likelihood. The MLE of , hat, should be larger than 4. Why?

(b) What is the estimate of ?

(c) Make a qq plot of the 15-day log returns using hat in the theoretical distribution. How well does a t-distribution fit the 15-day returns?

(Include code)