Computer Lab: Assignment training exercises

This lab is a variation on Lab 5.

Web page: http://www.ma.hw.ac.uk/~andrewc/erm

Go to the module web page and, if you have not already done so
copy the following files to your directory \FRM
functions10A.r
functions10B.r		# This is a new set of functions
			# some of which replace ones in functions10A.r
sp500.r
nasdaq.r

Load each one into your R workspace, if you have not already done so:
for example 

source("functions10A.r")
source("functions10B.r")
source("sp500.r")
source("nasdaq.r")

### Note some graphical results with discussion will appear later
### on the module web page.

x=sp500$adjclose
tv=sp500$t
n=length(x)
tv=tv[2:n]
d=diff(log(x))
nd=length(d)

# Note that tv is the same length as d

nv=(1:nd)[tv > 1980] 	# nv is a set of index numbers
tv=tv[nv]			# tv[nv] extracts the dates which are all after 1980
d=d[nv]			# This line extracts the daily log returns after 1980
nd=length(d)		# recalculate the length of d

plot(tv,d,pch=20)

# The next few lines take a while to run!
# They output the current log-likelihood on the screen so you can watch progress
# Don't worry about warning messages that appear at the end. These occur if the 
# optimisation algorithm steps outside the admissible region for parameters
# at any stage (e.g. it tries a value for nu that gives infinite variance.

res1=fit.garch11.normal(d)	# Fit GARCH(1,1) assuming Z(t)~N(0,1)
				# This is the same as the quasi maximum likelihood
				# which assumes the Z(t) are normal
res2=fit.garch11.t(d)		# Fit GARCH(1,1); Z(t)~t_nu with E[Z]=0, Var[Z]=1
res3=fit.garch11.nct(d)		# Fit GARCH(1,1); Z(t)~NCT with E[Z]=0, Var[Z]=1


####################################
# Simulation
####################################

# functions10B.r contains functions that can be used for simulation
# assuming normally distributed Z(t)
#
# Students need to develop their own code for the t and NCT

n.sim=1000		# The number of independent sample paths that we want to generate
x=sp500$adjclose	# reset this vector in case anything has been done to it in the meantime
X0=x[length(x)]		# Final value of the index = starting point for simulations
T1=50			# How many days forward do you want to simulate
res=res1		# Use the output from the fit.garch11.normal routine
Sres1=simulate.garch11.normal.R(n.sim,X0,T1,res)

names(Sres1)	# Check what is output in Sres1

# t in a time vector running from time 0 (index value X0) to T1 (i.e. T1+1 values)
# X is an array of dimension n.sim x (T1+1) and it gives the sample paths of the index, X(t)
### The first column of X is time 0 and is therefore constant and equal to X0
### The second column of X is time 1 and gives the simulated values of the index X(t)
# d is an array of dimension n.sim x T1 and gives the corresponding daily log returns
# sigma is an array of dimension n.sim x T1 and gives the corresponding daily volatilities
### The first column of sigma will be constant and equal to sigma(n+1) derived
### from the GARCH(1,1) fitting exercise
# Z is an array of dimension n.sim x T1 and gives the corresponding daily i.i.d. innovations Z(t)

# Thus, try e.g.

dim(Sres1$X)

plot(Sres1$t,Sres1$X[1,],type="l")	# Plot sample path number 1 of X(t)
lines(Sres1$t,Sres1$X[2,],col=2)	# Add sample path number 2

# Now plot a bundle of sample paths of X(t)
setaxes(0,50,1000,1400)
for(i in 1:1000){ lines(Sres1$t,Sres1$X[i,],col=i) }



#######################################################
# Fitting GARCH(1,1) models with time-dependent mu(t)
#######################################################

res1=fit.garch11.normal(d)	# Fit GARCH(1,1) assuming Z(t)~N(0,1)
				# and mu(t)=mu constant

res1B=fit.garch11.normal2(d)	# Fit GARCH(1,1) assuming Z(t)~N(0,1)
				# and mu(t)=mu0 + mu1.(delta(t-1) - mu0)

# Compare the log-likelihoods
res1$ll
res1B$ll

# Under the AIC would you prefer the basic model or the model with mu(t)=mu0 + mu1.(delta(t-1) - mu0)?
# What about the BIC?