[Gretl-devel] ARMA developments
I'm pleased to announce that our code for exact ML estimation of
ARMA models is now (in git and snapshots) MUCH faster than it used
to be (like, two orders of magnitude faster in many cases).
This is due to two main factors. First, we now use by default G.
Melard's algorithm AS 197 (Applied Statistics, 1984). This employs a
very compact recursion similar to but distinct from the Kalman
filter. Second, Jack developed an elegant means of "flipping" an MA
polynomial that has roots inside the unit circle: this means that
our Hannan-Rissanen initializer should never be unusable, and also
that BFGS has less work to do. (We never have to tell it, "You're
out of bounds, try again.")
One qualification: we can't use AS 197 when there are missing values
within the sample range; computing the MLE in that case requires the
Kalman filter. However, our first resort is now the Kalman algorithm
AS 154 (Gardner, Harvey and Phillips, 1980). This is not as fast as
AS 197 but, being specialized for ARMA, it's still a good deal
faster than using our general-purpose Kalman code. The most
difficult case is ARIMA (i.e. with differencing) in the presence of
missing values: that demands an extended state-space representation
incorporating levels, for which we fall back on our original Kalman
code.
At present there are two "secret" (undocumented) options for the
arma/arima command, which we've used for testing. We could document
these and make them generally available but I'm not sure that would
be worthwhile. Anyway, in case anyone wishes to do some testing,
here they are:
First, there's an option to force use of AS 154 when AS 197 would be
used by default:
arma ... --as=154
Second, there's an option to use our original Kalman code when AS
197 or AS 154 would be used by default:
arma ... --kalman
To summarize, the default algorithm is AS 197 unless there are
missing values, in which case we use AS 154 if the specification is
plain ARMA or our full Kalman code if it's ARIMA.
Thanks go to Oleh for pushing us in this direction!
Allin