QUESTION:

How can I extend ARIMA to include regressors ?

ANSWER:

The problem you refer to is handled by Transfer Functions. Box and Jenkins wrote a text, but most people

have only read the first half of it. The first half (roughly speaking) dealt with ARIMA or auto-projective

methods and the issues involved in identifying, estimating and forecasting. The second half dealt with

Transfer Functions and augmented the auto-projective model with eXgoneous or X input series. These

input series were modelled as right-hand side variables and their respective lag structures were to be

optimized.

This is then a super-set of multiple regression where Y is set upon by X and its lags and an error process

that may have some color ( i.e. an ARIMA process). Unfortunately, most readers and software developers

have ignored the opportunities in teaching and implementing these single equation, multivariable time

series methods. The input series may either be stochastic (probabalistic) or deterministic (fixed).

Identification strategies and model refinements differ for both stochastic and deterministic inputs.

Pre-whitened cross correlations are used to identify possible structures. Note that the differencing

involved in these identification steps are not necessarily part and parcel of the Transfer Model. Both of

these tools (ARIMA and TRANSFER FUNCTIONS) are NOT ROBUST to certain violations of the

Gaussian assumptions. Specifically, lack of normality doesn't seem to be a problem, although it could be.

Discrete data as opposed to continuous data can be a problem at low levels of response. Interventions, i.e. a

non-constant mean of the errors, has a significant effect on each step of the modeling process. The analog

to ROBUST REGRESSION when dealing with time series data is INTERVENTION DETECTION.

Interventions i.e. changes in the mean of the residuals can arise in at least four ways: 1. Pulse (one-time

unusual values) 2. Step or Level (consecutive unusual values .... thus locally usual) 3. Seasonal Pulses

(consecutive unusual values, , s periods apart. .. thus usual) 4. Time trends (trends do not necessarily

begin at time period 1 and end at time period t).

Changes in the second moment of the residuals (the variance) can have a significant effect. Work on

detecting relationships between central tendency (the mean) and variability (the variance) led to

developing optimum transformations (log, square root etc.). Recent work has identified regime changes

in the variance leading to Weighted Least Squares Estimators. The good news is that AUTOBOX

provides an integration of these opportunities even providing a "productivity aid" by automatically

identifying and building these models. AUTOBOX shows each and every step of the process and users

can set up their own procedures, models and simulations and fine tune the analysis.