Question about phi range in AR(1) process


Makridakis Forecasting text book says that phi should be restricted to -1<phi<1. I thought that this restriction should exist only if we want the process to be stationary. Why it should be restricted? If phi>1 or phi<-1 the process would not be stationary and in order to model it we would take a first difference (or more). So why we state that phi in AR(1) processes cannot by >1 or <-1?

Thnaks in advance,



Dark Knight
phi should be restricted to -1<phi<1.

when phi =1 or -1 leads to random walk. if phi >1 or <-1, you can prove that process is not stationary

for eg: \( y_t = \phi y_{t-1} +e_t\)
by substitution,
\( y_t = \phi^2 y_{t-2} +\phi e_{t-1} +e_t\)

By taking the intial value for y0=0 or any random variable, one can prove that V(yt) is changing over time.

Thanks for your answer. I understand that if phi>1 or phi<-1 the process will be non stationary because the varince would be infinite. But why we should restrict phi between (-1, 1)? In the case it is not stationary we can take differences and make it stationary. So we we cannot have phi=e.g. 1.3 or phi=2?

Thnaks again,



Dark Knight
when phi is inside the unit circle we say the time series is causal. when phi>1 ,phi<-1 ,the series doesn't make sense (interpretation). MA representation of these series is in the form of future e_t 's ( Refer Brockwell and davis BOOK).
By taking difference also you may not be able to achieve the stationarity.


Fortran must die
Why would you ever want the process not to be stationary? That is pretty much a requirement for any time series analysis (not to mention finding meaningful answers)?