On Mon, 26 Mar 2012, Daniel Bencik wrote:
Hello everybody,
first off, thank you all for suggestions.
Riccardo, I know that anything bigger than arma(2,1) is going to be a crappy
model. Im forecating volatility and arma is one of the options, the obviously
sloppy one but nevertheless the benchmark and I can prove nowehere else than
on out of sample that its great fit is just an illusion. That's actually the
point...
Well, the first thing I see is that your parameter "gc" is very very small
and at "convergence" the corresponding gradient is not even close to zero.
This, if I may, is the consequence of a poor modelling choice, since
"4.39767E-07" for a parameter when you use numerical derivatives is
definitely asking for trouble and it's a consequence of the scaling you
chose for your dependent variable: try multiplying it by, say, 100 or 1000
and see what happens. When using numerical derivatives it's always a good
idea to parametrise your model such that the order of magnitude of your
estimates is as close to 1 as possible.
Moreover, the output from eviews shows quite clearly that you have two
pairs of common roots in your arma specification. What this means in
practice is that with your data an arma(2,1) model is indentifiable;
anything over that is going to display a loglikelihood function that is
virtually flat in one or more directions (4 in your case). The
consequences of the above are:
1) from a numerical point of view, your function is very, very flat for a
large range of values. Since machine precision is limited anyway, you'll
probably end on a "maximum" which turns out to be such out of
machine-precision noise and little else.
2) From a statistical viewpoint, you have a model for whch no sensible
estimator of the covariance matrix of the parameters exists (because of
course if the loglikelihood is perfectly flat in at least one direction
both the Hessian and the OPG are singular). In a finite-sample,
finite-precision world, whatever standard errors you get from your
algorithm is, again, machine-precision noise (also known as "rubbish")
Finally, our algorithm does not include, as probably Eviews' does, some
form of provision for what to do in the case when the MA polynomial has
roots inside the unit circle (or whose absolute values smaller than one,
if you prefer). So you end up with an estimate of the coefficients for the
MA polynomial whose inverse roots are
-1.2736
-0.40100 + 0.93150i
-0.40100 - 0.93150i
0.63538 + 0.36511i
0.63538 - 0.36511i
Which clearly don't match what eviews gives you, but most importantly the
last conjugate pair is smugly inside the unit circle (and the first one is
dangerously close): so in practice your MA representation is not
invertible.
Riccardo (Jack) Lucchetti
Dipartimento di Economia
Università Politecnica delle Marche
(formerly known as Università di Ancona)
r.lucchetti(a)univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti