Wed Oct 1 17:10:57 CEST 2008
AR - ARMA
How to fit an ARMA model? It's been a while.
There are three primary stages in building a Box-Jenkins time series
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
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