A common way to select orders parameters (ex: to choose the number of AR terms to be included in the model ) in time series modelling is to rely on some Information Criteria (AIC, BIC, Hannan Quinn..) to measure the relative quality of the model : let’s call it Rule A.
Then in a second time, robustness tests are performed ( Ljung box test , Engle test ..).
However the methodology is not clear to me when I need to choose a model for a serie which has both autocorrelations in the mean and variance process :
I noticed that the model selected (by using rule A) is not always the same if :
- I use a “two steps method ” : First , I select orders parameters of the mean process using rule A , secondly, keeping the parameters obtained in the first step, I use rule A again to select parameters in the variance process.
Example : I fit all ARMA(p,q) to the series with (p,q)=0:2 and select the most parsimonious one. Let’s say the best model is p=1 and q=2. Second step : if fit all ARMA(1,2)-GARCH(s,t) models to the serie with (s,t)=0:2 and I select the "best" s,t parameters using rule A again. If we let p:q to be in the range 0:4 and s,t in the range 0:2 they are $5^2 + 3^2$ models to be estimated .
- OR a “direct way” modelling: I fit directly the full ARMA(p,q)-GARCH (s,t) to the time serie and select the best model (p,q,s,t) using rule A again. However in this case the number of combinations (number of models to be fitted) can be very high :if we let p:q to be in the range 0:4 and s,t in the range 0:2 they are $5^2 \times 3^2$ candidat models (it takes time and CPU..) .
Obviously the second method will evaluate the model selected by the two steps method and it may gives the strongest significant results. I said “may” because it is possible than the model selected by the direct method do not pass the misspecification part ..
My question is : How can I deal with this cost/efficiency problem ? How should I proceed ?