# Function to simulate an AR time series, to plot
# the nonparametric density estimate, 

karsim<-function(alpha,n) {
	x<-arima.sim(n,model=list(ar=alpha,))
	par(mfrow=c(3,2))
	tsplot(x,main="Simulated Time Series: AR(1)")
	plot(ksmooth(x,kernel="normal",bandwidth=.5),
		type="l",main="Nonparametric Density")
	acf(x,type="corr")
	acf(x,type="partial")
	lag.plot(x,lags=6,layout=c(3,2),head="Lagged Scatterplots")
	return(x)
	}

x<-karsim(.3,200)
x<-karsim(.9,200)
x<-karsim(-.3,200)
x<-karsim(-.9,200)

x<-karsim(c(0,.8),200)
x<-karsim(c(0,-.8),200)

#EXAMPLE OF PARTIAL AUTOCORRELATIONS for n=200

par(mfrow=c(2,2))

#first partial
plot(x[1:199],x[2:200],main="First PACF")
cor(x[1:199],x[2:200])

#second partial
X1<-matrix(0,nrow=198,ncol=1)
X1[ ,1]<-x[2:199]
rf1<-lsfit(X1,x[3:200],intercept=F) #regression
rf1<-rf1$residuals                  #extract residuals
rb1<-lsfit(X1,x[1:198],intercept=F)
rb1<-rb1$residuals
plot(rf1,rb1,main="Second PACF")
cor(rf1,rb1)

#third partial
X2<-matrix(0,nrow=197,ncol=2)
X2[ ,1]<-c(x[2:198])
X2[ ,2]<-c(x[3:199])
yf2<-c(x[4:200])
rf2<-lsfit(X2,yf2,intercept=F)
rf2<-rf2$residuals
yb2<-c(x[1:197])
rb2<-lsfit(X2,yb2,intercept=F)
rb2<-rb2$residuals
plot(rf2,rb2,main="Third PACF")
cor(rf2,rb2)

#fourth partial
X3<-matrix(0,nrow=196,ncol=3)
X3[ ,1]<-x[2:197]
X3[ ,2]<-x[3:198]
X3[ ,3]<-x[4:199]
rf3<-lsfit(X3,x[5:200],intercept=F)
rf3<-rf3$residuals
rb3<-lsfit(X3,x[1:196],intercept=F)
rb3<-rb3$residuals
plot(rf3,rb3,main="Fourth PACF")
cor(rf3,rb3)