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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  - causal - process,&#62; graphar1=function(phi){+ nf &#60;- 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.&#62; graphar1(.8)or&#62; graphar1(-.7)(with, on the bottom, the root of the characteristic polynomial, the value of the parameter , the autocorrelation function  and the partial autocorrelation function ).Of course, it is possible to do something similar with  processes,&#62; graphar2=function(phi1,phi2){+ nf &#60;- 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,&#62; graphar2(.65,.3)or&#62; graphar2(-1.4,-.7)