#==========================================================
# Simulation of an AR(1) Model using arima.sim()
# Code is from p 67 of Z&W (slightly modified)
# Also illustrates par(mfrow(2,2)) and graph labeling 
#=======================================================


#The simulation, using set.seed(101) and start.innov burn-in

set.seed(101)
e<-rnorm(100,sd=1)
e.start=rnorm(25, sd=1)
y.ar1<- 1+arima.sim(model=list(ar=0.75), n=100,innov=e, start.innov=e.start)

# check on the mean

mean(y.ar1)


#Three pictures in one:
#
#(1) the simulated series,
#(2) the theoretical ACF
#(3) the empirical ACF for our simulation

par(mfrow=c(2,2))
tsplot(y.ar1, main="Simulated AR(1)")
abline(h=1)
gamma.j<-rep(.75,10)^seq(10)
tsplot(gamma.j, type="h", main="Theoretical ACF for AR(1)", ylab="Autocorreleation", xlab="Lag")
temp<-acf(y.ar1, lag.max=10, plot=F)
tsplot(temp$acf,type="h", main="Empirical ACF for Simulated AR(1)", ylab="Autocorreleation", xlab="Lag")
 

#One line to make it prettier:
abline(h=0)