I am hoping to use Akaike's Information Criterion (AIC) to choose
between different structural state-space time series models for time
series.

Chatfield (The Analysis of Time Series, 6th ed., Chatman & Hall) gives
the following definition:
AIC = -2(max.likelihood) + 2r

where r denotes the number of independent parameters.

The models are being formulated using the software package, STAMP,
which reports the number of parameters & restrictions, also -2LogL.
Some of the models include explanatory variables, as well as trend,
level, irregular & seasonal components.

I was hoping to estimate the AIC very simply by adding 2r to the
-2LogL reported by STAMP.

My main question is: how do I calculate r (taking all the structural
components & explanatory variables into account)?

If you have time over, my supplementary question is:
there seem to be alternative definitions of the AIC, e.g.
AIC = N.ln(weighted sum of squares) + 2.M
where N is the number of observations.
Are all these definitions equally valid alternative expressions?

---

Clarification of Question by sandracr-ga on 20 Aug 2004 05:41 PDT

Okay, could I make the question more general?

The AIC is used to compare different models with regard to
goodness-of-fit, but generally any explanations seem to relate to
ARIMA models only, I think. How do you compute the AIC for state-space
models that include explanatory variables? And for Holt-Winters
models?

(If I'm asking daft questions, please let me know.)

It is not clear to me how to calculate r, but an alternative method
might be cross validation. No clue if it is more sensitive then the
AIC though, but it's use a very straight forward.


regarding the supplementary question, the weighted sum of squares
assumes that the underlying prob. function. is a gaussian.
have a look at:
http://www-ssc.igpp.ucla.edu/personnel/russell/ESS265/Ch9/autoreg/node15.html#SECTION00043000000000000000
  
Hope this helps