What exactly is the Box-Jenkins method for ARIMA processes?

Question

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?

Answer 1

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
San Francisco: Holden-Day

Answer 2

The Box-Jenkins methodology is a strategy or procedure that can be used to build an ARIMA model. The methodology is outlined in the book Time Series Analysis: Forecasting and Control by George E. P. Box and Gwilym M. Jenkins, originally published in 1970 - more recent editions exist.

By opening up SAS, calling proc ARIMA, and supply numbers for p, d, and q, you have merely estimated an ARIMA model. Doing this blindly, that is, by not using any particular recognized methodology to identify the ARIMA model itself, is a bit like playing with matches - the dangers of software!

If you keep repeating this process - estimating lots and lots of ARIMA models - you will eventually be able to select a model with the lowest Akaike Information criterion (from the set of models that you have estimated). In this context, a more systematic approach would be to use an algorithm based on comparing AIC values for a variety of different models to automatically select an ARIMA model for you, such as the one provided by the forecast package in R - the relevant function name is auto.arima().

In any event, the procedure you outlined involved selecting an ARIMA model based on minimizing some information criterion (in this case, AIC, but there are other measures). This is one particular methodology, but it's not the Box-Jenkins methodology; an alternative.

The Box-Jenkins methodology comprises five stages (although sometimes said to involve just three stages):

1. Checking for stationarity or non-stationarity and transforming the data, if necessary;
2. Identification of a suitable ARMA model;
3. Estimation of the parameters of the chosen model;
4. Diagnostic checking of model adequacy; and
5. Forecasting, or repetition of steps two to five.

Notably, it is an iterative process that involves the model builder exercising some judgement - and this is one aspect of the methodology that has been considered a shortcoming. The judgemental part comes into play particularly when interpreting two tools; namely, the (estimated) autocorrelation function (ACF) and partial autocorrelation function (PACF).

If you'd like to become a practitioner of the Box-Jenkins methodology, I'd recommended consulting the original text (you'd be surprised what modern textbooks omit!) alongside whatever modern variations you can find. Alan Pankratz has a couple of excellent textbooks, which I'd highly recommend, too; for example, Forecasting with Univariate Box-Jenkins Models: Concepts and Cases.

Experience suggests to me that the term "Box-Jenkins methodology" is used in a loose manner because I've heard some people use it to simply refer to building ARIMA models in general - and not to the actual process involved in building an ARIMA model - while others use it to refer to a modified version of what was published in 1970. As @Glen_b has pointed out, "there are numerous documents that describe the Box-Jenkins methodology these days that would include the use of AIC or similar quantities".

Q: Are you supposed to use Box-Jenkins methodology to come up with initial estimates of p,d,q?

As already mentioned, there are different model selection strategies so the answer is no it's not necessarily the case that you need to employ the Box-Jenkins methodology, but you could if you wanted to.

Q: Did SAS use it internally somehow?

Highly unlikely unless that software offers a quite sophisticated function! Consult the official SAS documentation for details of what the software does or is capable of doing. If it was R, you could look at the source code, but I doubt that's an option with SAS.