The Wikipedia page says that Box-Jenkins is a method of fitting an ARIMA model to a time series. Now, if I want to fit an ARIMA model to a time series, I will open up SAS, call proc ARIMA, supply the parameters $p,d,q$ and SAS will give me AR and MA coefficients. Now, I can try different combinations of $p,d,q$ and SAS will give me a set of coefficients in each case. I select the set with the lowest Akaike information criterion.
My question is: where in the above procedure did I use Box-Jenkins? Am I supposed to use Box-Jenkins to come up with initial estimates of $p,d,q$? Or did SAS use it internally somehow?
Best Answer
Box and Jenkins themselves didn't use AIC. Their book came out in 1970 based on methodology developed previously, while Akaike's papers on AIC came (not long) after the book was published.
Their methodology is outlined in their book [1], but what's today included under the mantle of "Box-Jenkins" is a bit broader and varies from person to person.
Box and Jenkins themselves give a simple flowchart on model identification which might be regarded as a useful summary of the process they used to identify models. (I'd suggest looking at the book if you can - most decent university libraries should have a copy.)
They incorporated stages of model identification, estimation and diagnostic checking/validation (including a return to the first stage if the model is inadequate), and then once an adequate model is identified, the model may be forecasted.
The wikipedia page here gives an outline of the sort of thing that's involved, but it contains a number of things that have been added in to what people tend to so since the book came out. Indeed, numerous documents that describe Box-Jenkins methodology these days would include the use of AIC or similar quantities.
See also the discussion here.
More recent books (e.g. see the above wikipedia page) give a more 'modern' version of the general approach.
In the end, if you want to find out what Box-Jenkins methodology really "is", I would say "start with their book". Failing that, a number of more recent treatments of ARIMA models cover broadly similar methodology -- try any number of reasonably decent time series books that cover ARIMA models.
[1]: Box, George; Jenkins, Gwilym (1970),
Time series analysis: Forecasting and control
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