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In the MAT8181 graduate course on Time Series, we started discussing autoregressive models. Just to illustrate, here is some code to plot $AR(1)$ – causal – process,

> graphar1=function(phi){
+ nf <- layout(matrix(c(1,1,1,1,2,3,4,5), 2, 4, byrow=TRUE), respect=TRUE)
+ e=rnorm(n)
+ X=rep(0,n)
+ for(t in 2:n) X[t]=phi*X[t-1]+e[t]
+ plot(X[1:6000],type="l",ylab="")
+ abline(h=mean(X),lwd=2,col="red")
+ abline(h=mean(X)+2*sd(X),lty=2,col="red")
+ abline(h=mean(X)-2*sd(X),lty=2,col="red")
+ u=seq(-1,1,by=.001)
+ plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,ylim=c(-2,2),xlim=c(-2.5,2.5))
+ polygon(c(u,rev(u)),c(sqrt(1-u^2),rev(-sqrt(1-u^2))),col="light yellow")
+ abline(v=0,col="grey")
+ abline(h=0,col="grey")
+ points(1/phi,0,pch=19,col="red",cex=1.3)
+ plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,ylim=c(-.2,.2),xlim=c(-1,1))
+ axis(1)
+ points(phi,0,pch=19,col="red",cex=1.3)
+ acf(X,lwd=3,col="blue",main="",ylim=c(-1,1))
+ pacf(X,lwd=3,col="blue",main="",ylim=c(-1,1),xlim=c(0,16))}

e.g.

> graphar1(.8)

or

> graphar1(-.7)

(with, on the bottom, the root of the characteristic polynomial, the value of the parameter $\phi_{1}$, the autocorrelation function $h\mapsto\rho(h)$ and the partial autocorrelation function $h\mapsto\psi(h)$).

Of course, it is possible to do something similar with $AR(2)$ processes,

> graphar2=function(phi1,phi2){
+ nf <- layout(matrix(c(1,1,1,1,2,3,4,5), 2, 4, byrow=TRUE), respect=TRUE)
+ e=rnorm(n)
+ X=rep(0,n)
+ for(t in 3:n) X[t]=phi1*X[t-1]+phi2*X[t-2]+e[t]
+ plot(X[1:6000],type="l",ylab="")
+ abline(h=mean(X),lwd=2,col="red")
+ abline(h=mean(X)+2*sd(X),lty=2,col="red")
+ abline(h=mean(X)-2*sd(X),lty=2,col="red")
+ P=polyroot(c(1,-phi1,-phi2))
+ u=seq(-1,1,by=.001)
+ plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,ylim=c(-2,2),xlim=c(-2.5,2.5))
+ polygon(c(u,rev(u)),c(sqrt(1-u^2),rev(-sqrt(1-u^2))),col="light yellow")
+ abline(v=0,col="grey")
+ abline(h=0,col="grey")
+ points(P,pch=19,col="red",cex=1.3)
+ plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,xlim=c(-2.1,2.1),ylim=c(-1.2,1.2))
+ polygon(c(-2,0,2,-2),c(-1,1,-1,-1),col="light green")
+ u=seq(-2,2,by=.001)
+ lines(u,-u^2/4)
+ abline(v=seq(-2,2,by=.2),col="grey",lty=2)
+ abline(h=seq(-1,1,by=.2),col="grey",lty=2)
+ segments(0,-1,0,1)
+ axis(1)
+ axis(2)
+ points(phi1,phi2,pch=19,col="red",cex=1.3)
+ acf(X,lwd=3,col="blue",main="",ylim=c(-1,1))
+ pacf(X,lwd=3,col="blue",main="",ylim=c(-1,1),xlim=c(0,16))}

For example,

> graphar2(.65,.3)

or

> graphar2(-1.4,-.7)