#  In this program we are going to use the function "ts.sim" to simulate 200 observations of
#  some simple AR models, plot them and then plot their sample ACF's and sample PACF's.
#  Note that, due to sampling variation, the sample ACF's and sample PACF's will not exactly
#  coincide with their theoretical (population) ACF's and PACF's even though this would be
#  the case as the sample size goes to infinity.  The purpose here is to show the white noise,
#  AR(0) model, the AR(1) model with positive autocorrelation (ar=0.7) and the AR(1) model
#  with negative autocorrelation.  

# Here we simulate an AR(1) model with ar1 = 0.0.
# Notice the lack of autocorrelation at any lag.
# The reversion to the mean is, in essence, immediate.

ts.sim.1<-arima.sim(list(order=c(1,0,1), ar=0.0, ma=0.0), n=200)
ts.plot(ts.sim.1)
windows()
par(mfrow=c(1,2))
acf(ts.sim.1)
pacf(ts.sim.1)

# Here we simulate a AR(1) model with ar1 = 0.7.
# Notice the autocorrelation being positive.
# The series reverts to the mean but slowly.

ts.sim.2<-arima.sim(list(order=c(1,0,1), ar=0.7, ma=0.0), n=200)
windows()
par(mfrow=c(1,1))
ts.plot(ts.sim.2)
windows()
par(mfrow=c(1,2))
acf(ts.sim.2)
pacf(ts.sim.2)

# Here we simulate an AR(1) model with ar1 = -0.7.
# Notice the autocorrelation being negative for lag 1
# and positive for lag 2, continuing to switch back and
# forth across the mean though exhititing mean reversion.
# This is a time series with a "saw-tooth" pattern. 

ts.sim.3<-arima.sim(list(order=c(1,0,1), ar=-0.7, ma=0.0), n=200)
windows()
par(mfrow=c(1,1))
ts.plot(ts.sim.3)
windows()
par(mfrow=c(1,2))
acf(ts.sim.3)
pacf(ts.sim.3)