Question about phi range in AR(1) process

#1
Hello,

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,

Andreas
 

vinux

Dark Knight
#2
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.
 
#3
Hello,

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,


Andreas
 

vinux

Dark Knight
#4
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.
 

noetsi

Fortran must die
#5
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)?