|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 that contains 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 also 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.
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 by 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;
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;
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 is continuous and 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.