Consider here some model,

We’ve seen in class that stationnarity of that time series, in the sense that and , was valid if the roots (in ) of the characteristic polyonomial – – were outside the unit circle.

To visualize this point, consider the following time series

To generate that time series, we need to generate a bivariate white noise, i.e. (not necessarily a diagonal matrix), and . For instance

> n=500
> r=0.7
> set.seed(1)
> Z1=rnorm(n)
> Z2=rnorm(n)
> E1=Z1
> E2=r*Z1+sqrt(1-r^2)*Z2

To generate now our time series, use

> A=matrix(c(.7,.2,.4,.3),2,2)
> X1=X2=rep(0,n)
> for(t in 2:n){
+   X1[t]=A[1,1]*X1[t-1]+A[1,2]*X2[t-1]+E1[t]
+   X2[t]=A[2,1]*X1[t-1]+A[2,2]*X2[t-1]+E2[t]  
+ }

Here, we have

> plot(X1,type="l",col="red")
> lines(X2,col="blue")

Those two time series seem to be stationnary. And, indeed,

> polyroot(c(1,-sum(diag(A)),det(A)))
[1] 1.18+0i 6.51-0i
> Mod(polyroot(c(1,-sum(diag(A)),det(A))))
[1] 1.18 6.51

The two roots of the characteristic polynomial are outsite the unit circle. Now, what if we consider

> A=matrix(c(.9,.2,.1,.8),2,2)
> A
     [,1] [,2]
[1,]  0.9  0.1
[2,]  0.2  0.8

This time, one of the two roots is exactly on the border of the circle (i.e. there is a unit root here)

> polyroot(c(1,-sum(diag(A)),det(A)))
[1] 1.00+0i 1.43-0i
> Mod(polyroot(c(1,-sum(diag(A)),det(A))))
[1] 1.00 1.43

What our series look like, here?

> X1=X2=rep(0,n)
> for(t in 2:n){
+   X1[t]=A[1,1]*X1[t-1]+A[1,2]*X2[t-1]+E1[t]
+   X2[t]=A[2,1]*X1[t-1]+A[2,2]*X2[t-1]+E2[t]  
+ }
> plot(X1,type="l",col="red")
> lines(X2,col="blue")

Here, the two series are integrated. But observe, further more, that those two series are here cointegrated (as defined in class this morning). Indeed, it seems that there is a common factor

and the remaining series are stationary.

> Z=(X1+X2)/2
> plot(Z,type="l")
> plot(X1-Z,type="l",col="red")
> lines(X2-Z,col="blue")

A very different graph would be

> A=matrix(c(1,0,0,1),2,2)
> A
     [,1] [,2]
[1,]    1    0
[2,]    0    1
> polyroot(c(1,-sum(diag(A)),det(A)))
[1] 1+0i 1-0i
> Mod(polyroot(c(1,-sum(diag(A)),det(A))))
[1] 1 1

Here, the series would be non-stationary, but here, there is no cointegration (even if there might be simultaneous correlation, because of the white noise)

> plot(X1,type="l",col="red",ylim=c(-20,20))
> lines(X2,col="blue")