Re: Maximization of ARMA-GARCH models

David Jones wrote:
monnomiznogoud@xxxxxxxxxxxx wrote:
Thank you for your answer.
You're right, the question was about any model (hence the "similar"
i
used.)

So what you're saying is that using a different distriburion for
the
innovations/errors makes another model out of the first one ?

Yes they would be separate models, unless that is you find amodel
for
the residual that includes the Student-t distribution and the
Hyperbolic distribution as special cases. The theory I mentioned
provides one general way of doing this, but there are probably
others.


It would have been great if i could compare directly the ML
function
of an ARMA(1,1)-GARCH(1,1) with Student-t distribution with an
ARMA(1,1)-GARCH(1,1) with a GED distribution.

It is true that comparing ARMA(1,1)-GARCH(1,1) with
ARMA(1,1)-APARCH(1,1) might not be evident, but i can't see why the
first example could not be compared directly (thus preferring the
one
with the maximum value)

The problem is that of finding the valid distribution for the
test-statistic. Standard maximum likelihood theory does not provide
this. If you really do want to do a test using the difference of the
log-likelihoods you can't easily do the obvious thing of deriving
the
critical region for a test by doing simulations since this would
entail you finding a pair of models from each class that have the
same
"true" likelihood. However it is possible you could make use of the
theory in my long-ago paper to help you construct a reasonable test
of
the difference in log-likelihoods: see Jones DA (1983) Statistical
analysis of empirical models fitted by optimisation, Bimometrika,
70(1), 67-88. It may need some consideable adaptation, but Section 5
seems to relate.

Note that the "testing separate families of hypotheses" approach I
mentioned previously does not do a direct test of one model being
better than another. Instead it works by supplying two tests: taking
each model in turn it asks whether there is evidence that the given
model needs to be expanded into the combined model.

David Jones

A subsequent thought .... if both the following are true, you may be
able to do something simpler and more intuitively reasonable. If:

(i) the sample size is reasonably large so that both (a) the effect
of parameter estimation errors is small and (b) you can divide the
data set into a reasonably large number of subsets (say 20);

(ii) the models are such that you can pre-preocess the data with any
difference operators to lead to series with fairly short temporal
dependence.

Then:

(i) fit the two models to the full (prepocessed) data set;

(ii) Specificy subsets of the data .. the first m observations, the
next m observations etc.

(iii) calculate log-likelihoods for each subset of data (for the
parameter values fitted to the complete data-set)... giving L1i, L2i.
If you work with conditional likelihoods this should be
straightforward to define (ie likelihoods of each observation given
past observations).

(iv) Calculate differences Di=L1i-l2i.

(v) Do a test that the mean of the Di's is zero, on the basis of the
sample mean and sample variance of the Di's and assuming that the Di's
are effectively uncorrelated. For the last to be reasonable you need
both the assumption about short range dependence and the choice of m
above in (ii) to be reasonably matched. The idea here is closely
related to the suggestion of Moran (1975) (Biometrika 69 19-27).

David Jones


.

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