## Intervention Models and Interrupted Time Series

One special kind of ARIMA model with input series
is called an *intervention model* or *interrupted time series* model.
In an intervention model, the input series is an indicator variable
containing discrete values that flag the occurrence of an event affecting
the response series.
This event is an intervention in or an interruption of
the normal evolution of the response time series,
which, in the absence of the intervention, is usually assumed to be
a pure ARIMA process.

Intervention models can be used both to model and forecast the response series
and to analyze the impact of the intervention.
When the focus is on estimating the effect of the intervention,
the process is often called *intervention analysis* or
*interrupted time series analysis*.

*Impulse Interventions*

The intervention can be a one-time event.
For example, you might want to study the effect of a short-term advertising
campaign on the sales of a product.
In this case, the input variable has the value of 1 for the
period during which the advertising campaign took place
and the value 0 for all other periods.
Intervention variables of this kind are sometimes called
*impulse functions* or *pulse functions*.
Suppose that SALES is a monthly series, and a special advertising effort
was made during the month of March 1992.
The following statements estimate the effect of this intervention assuming
an ARMA(1,1) model for SALES.
The model is specified just like the regression model, but the
intervention variable AD is constructed in the DATA step as a zero-one
indicator for the month of the advertising effort.
data a;
set a;
ad = date = '1mar1992'd;
run;
proc arima data=a;
identify var=sales crosscorr=ad;
estimate p=1 q=1 input=ad;
run;

*Continuing Interventions*

Other interventions can be continuing, in which case the input variable
flags periods before and after the intervention.
For example, you might want to study the effect of a change in
tax rates on some economic measure.
Another example is a study of the effect of a change in speed limits
on the rate of traffic fatalities.
In this case, the input variable has the value 1
after the new speed limit went into effect and the value 0 before.
Intervention variables of this kind are called *step functions*.
Another example is the effect of news on product demand.
Suppose it was reported in July 1996 that consumption of the product
prevents heart disease (or causes cancer),
and SALES is consistently higher (or lower) thereafter.
The following statements model the effect of this news intervention:
data a;
set a;
news = date >= '1jul1996'd;
run;
proc arima data=a;
identify var=sales crosscorr=news;
estimate p=1 q=1 input=news;
run;

*Interaction Effects*

You can include any number of intervention variables in the model.
Intervention variables can have any pattern--impulse and continuing
interventions are just two possible cases.
You can mix discrete valued intervention variables and continuous
regressor variables in the same model.
You can also form interaction effects by multiplying input variables
and including the product variable as another input.
Indeed, as long as the dependent measure forms a regular time series,
you can use PROC ARIMA to fit any general linear model
in conjunction with an ARMA model for the error process
by using input variables that correspond to the columns
of the design matrix of the linear model.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.