## non-stationary AR(10)

**I**n the revision of * Bayesian Core* on which Jean-Michel Marin and I worked together most of last week, having missed our CIRM break last summer (!), we have now included an illustration of what happens to an AR(p) time series when the customary stationarity+causality condition on the roots of the associated polynomial is not satisfied. More specifically, we generated several time-series with the same underlying white noise and random coefficients that have a fair chance of providing non-stationary series and then plotted the 260 next steps of the series by the R code

p=10 T=260 dat=seri=rnorm(T) #white noise par(mfrow=c(2,2),mar=c(2,2,1,1)) for (i in 1:4){ coef=runif(p,min=-.5,max=.5) for (t in ((p+1):T)) seri[t]=sum(coef*seri[(t-p):(t-1)])+dat[t] plot(seri,ty="l",lwd=2,ylab="") }

leading to outputs like the following one

January 19, 2012 at 10:37 pm

Can you make a picture where it escapes to infinity in a monotone way? Because there should be a parameter regime where that is true (convert it to a system of 1st order AR processes and make sure the eigenvalue of the transition matrix that is >1 is real), and these pictures make it look like AR(p) processes always oscillate (which I have trouble believing is true).

January 20, 2012 at 6:27 am

Dan: Sure! It is of course possible to pick such parameter values. In fact, if you run my R code a few times you should spot a monotone departure to infinity occurring now and then. I picked a realisation of a graph with oscillations because it was more photogenic!