55
P
2 . < 1
i-1 1
The sufficient condition is that roots of the characteristic equation
<ยก>(B) 0
lie outside the unit circle.
In addition, Fuller (1976) shows that when a time series is a
stationary autoregressive process, the autocorrelation function 9X(L) is
a monotonically declining function of L that decays exponentially to
zero. An autoregressive process possesses infinite memory where the
current value of xt depends on all past values.
Moving Average (MA) Process
Some time series can be defined as a moving average of order q
where x^. is a weighted average of random disturbances lagged back q
periods. This series xt can be denoted as
xt .Yj 5t-j+ s
where is the weight on each lagged disturbance q is the maximum
lag, and S is the mean of the process. Here we assume (as in the case
of autoregressive model) that the random disturbance is generated by a
white noise process. Thus, the mean S is invariant with t. In addi
tion, by assuming stationarity, a moving average is characterized by
Z 0? < -
i-1
However, this is only a necessary condition. Rewriting xt in backshift
notation and letting S=0 yields
WB)5t