56
The invertibility condition requires that
3_1(B)xt £t
where 8*(B) must converge and the roots of the characteristic equation
8(B) be outside the unit circle.
A moving average process of order one (q=l) has a memory of only
one period. In general, a moving average process of order q has a
memory of exactly q periods and the autocorrelation function is given by
0x(L) -
-8. + 8, 8T ., + ... 8 T 8
ii 1 L+l q-L q
1 3 2
1 + 87 + 87 + ... + B
1 2 q
, L
1,
q
0 (truncated) L < q
Thus, the autocorrelation function for a moving average process has q
non-zero values and is zero for lags greater than q. This can be con
trasted to the exponentially decaying lags for an autoregressive pro
cess. There exists a relationship between moving average and autore
gressive processes such that a finite order moving average process can
be expressed as an infinite order autoregressive process. The converse
is also true. In other words, an autoregressive process can be inverted
into a pure moving average process and vice versa. This requires that
certain invertibility conditions are met. In particular, the roots of
the characteristic equations $(8) and 8(B) must again all be outside the
unit circle (Nelson, 1973).
Integrated Autoregressive Moving Average (ARIMA) Process
Many time series encountered are neither characterized by a pure
moving average or pure autoregressive process. In addition, these time
series are often non-stationary. Thus, time series such as these are