enforces stationarity constraints on the estimation. Unless
cancellation of factors is present (see the next section), it is unlikely for
X13ARIMASEATS’
nonlinear estimation
to approach the boundary of the stationary region, since the loglikelihood approaches
∞
as this boundary is
approached.
If the likelihood is defined conditionally with respect to the AR parameters, stationarity is neither assumed
nor enforced by the
X13ARIMASEATS
software. Model estimation, forecasting, etc., are not compromised by
parameter values outside the stationary region in this case. Inference results, however, are affected, as noted
in Section 4.5.
Special techniques (as in, e.g., Fuller 1976, Section 8.5) are required for inference about AR
parameters outside the stationary region.
5.4
Cancellation (of AR and MA factors) and overdifferencing
Cancellation of AR and MA factors is possible when a model with a mixed ARMA structure is estimated. A
model as in (1) or (3) is said to have a mixed ARMA structure if either
p >
0 and
q >
0, or
P >
0 and
Q >
0. (Technically, a model with
p >
0 and
Q >
0, or with
P >
0 and
q >
0, is also mixed, but such mixed
models are unlikely to lead to cancellation problems.)
The simplest example of cancellation occurs with the
ARMA(1,1) model, (1

φB
)
z
t
= (1

θB
)
a
t
, when
φ
=
θ
. Cancelling the (1

φB
) factor on both sides of the
model (1

φB
)
z
t
= (1

φB
)
a
t
leaves the simplified model,
z
t
=
a
t
. Because of this, the likelihood function
will be nearly constant along the line
φ
=
θ
.
This can lead to difficulties with convergence of the nonlinear
estimation if the MLEs for the ARMA(1,1) model approximately satisfy
ˆ
φ
=
ˆ
θ
. Analogous problems occur in
more complicated mixed models when an AR polynomial and an MA polynomial have a common zero (e.g., the
ARIMA (2,1,2)(0,1,1) model that is used as a candidate model for the
automdl
spec). For a fuller discussion
of this topic, see Box and Jenkins (1976), pp. 248250.
If the
X13ARIMASEATS
program has difficulty in converging when estimating a mixed model, cancellation
of AR and MA factors may be responsible.
In any case, possible cancellation can be checked by computing
zeroes of the AR and MA polynomials (setting
print=roots
in the
estimate
spec), and examining these for
zeroes common to an AR and an MA polynomial.
If a common zero (or zeroes) is found, then the model
should be simplified by cancelling the common factor(s) (reducing the order of the corresponding AR and MA
polynomials), and the model should be reestimated. Cancellation need not be exact, but may be indicated by
zeroes of an AR and an MA polynomial that are approximately the same.
It is also possible for estimated MA polynomials to have factors that cancel with differencing operators.
This occurs when a model has a nonseasonal difference and an estimated nonseasonal MA polynomial contains
a (1

B
) factor, or the model has a seasonal difference and an estimated seasonal MA polynomial contains a
(1

B
s
) factor. For example, the model (1

B
)(1

B
s
)
z
t
= (1

θB
)(1

Θ
B
s
)
a
t
involves such cancellation
44
CHAPTER 5.
POINTS RELATED TO REGARIMA MODEL ESTIMATION
if either
θ
or Θ is estimated to be one.
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 Spring '14
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