The ARMA(
) model is defined as
 |
(12.16) |
or as
with the moving average lag-polynomial
and the autoregressive lag-polynomial
. So that the
process (11.16) has an explicit parameterization, it is
required that the characteristic polynomials
and
do not have any common roots. The process
(11.16) is stationary when all the roots of the
characteristic equation (11.6) lie outside of the unit
circle. In this case (11.16) has the MA(
)
representation
The process
in (11.16) is invertible when all the
roots of the characteristic equation (11.4) lie outside
of the unit circle. In this case (11.16) can be written
as
that is an AR(
) process. Thus we can approximate every
stationary, invertible ARMA(
) process with a pure AR or MA
process of sufficiently large order. Going the other direction, an
ARMA(
) process offers the possibility of parsimonious
parameterization.