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Wed Oct 1 17:10:57 CEST 2008

AR - ARMA

How to fit an ARMA model?  It's been a while.

http://en.wikipedia.org/wiki/Box-Jenkins

There are three primary stages in building a Box-Jenkins time series
model. 

1. Model identification

* Detecting stationarity: can be done by inspecting the
  autocorrelation.  Slow decay (a flat spower spectrum without
  isolated peaks) can indicate non-stationarity.

* Detecting seasonality: if there is significant periodicity this can
  be removed (modeled separately), or included in the model order
  estimation.  (The idea being that periodicity comes from external
  inputs, something which the ARMA model doesn't accomodate.)

* MA order (q) selection from autocorrelation plot.  For an exact
  model, this becomes zero after lag = q.

* AR order selection (p) from partial autocorrelation plot. (CHECK THIS).

This could be automated using information-based criteria such as FPE
(Final Prediction Error) and AIC (Aikake Information Criterion).

2. Model estimation

Once a suitable model order is found, use a NL-LS or ML method to
estimate the model parameters.

3. Model validation

The error term is assumed to follow the assumptions for a stationary
univariate process.  


For ARMAX the approach is similar?

So, what's the difference between using NL-LS or ML methods?  Linear
least squares corresponds to maximum likelihood if the errors have a
normal distribution.  It looks like this is no longer the case for
NL-LS.



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