To forecast a response series by using an ARIMA model with inputs, you need values of the input series for the forecast periods. You can supply values for the input variables for the forecast periods in the DATA= data set, or you can have PROC ARIMA forecast the input variables.

If you do not have future values of the input variables in the input data set used by the FORECAST statement, the input series must be forecast before the ARIMA procedure can forecast the response series. If you fit an ARIMA model to each of the input series for which you need forecasts before fitting the model for the response series, the FORECAST statement automatically uses the ARIMA models for the input series to generate the needed forecasts of the inputs.

For example, suppose you want to forecast SALES for the next 12 months. In this example, the change in SALES is predicted as a function of the change in PRICE, plus an ARMA(1,1) noise process. To forecast SALES by using PRICE as an input, you also need to fit an ARIMA model for PRICE.

The following statements fit an AR(2) model to the change in PRICE before fitting and forecasting the model for SALES. The FORECAST statement automatically forecasts PRICE using this AR(2) model to get the future inputs needed to produce the forecast of SALES.

   proc arima data=a;
      identify var=price(1);
      estimate p=2;
      identify var=sales(1) crosscorr=price(1);
      estimate p=1 q=1 input=price;
      forecast lead=12 interval=month id=date out=results;
   run;

Fitting a model to the input series is also important for identifying transfer functions. (See the section Prewhitening for more information.)

Input values from the DATA= data set and input values forecast by PROC ARIMA can be combined. For example, a model for SALES might have three input series: PRICE, INCOME, and TAXRATE. For the forecast, you assume that the tax rate will be unchanged. You have a forecast for INCOME from another source but only for the first few periods of the SALES forecast you want to make. You have no future values for PRICE, which needs to be forecast as in the preceding example.

In this situation, you include observations in the input data set for all forecast periods, with SALES and PRICE set to a missing value, with TAXRATE set to its last actual value, and with INCOME set to forecast values for the periods you have forecasts for and set to missing values for later periods. In the PROC ARIMA step, you estimate ARIMA models for PRICE and INCOME before you estimate the model for SALES, as shown in the following statements:

   proc arima data=a;
      identify var=price(1);
      estimate p=2;
      identify var=income(1);
      estimate p=2;
      identify var=sales(1) crosscorr=( price(1) income(1) taxrate );
      estimate p=1 q=1 input=( price income taxrate );
      forecast lead=12 interval=month id=date out=results;
   run;

In forecasting SALES, the ARIMA procedure uses as inputs the value of PRICE forecast by its ARIMA model, the value of TAXRATE found in the DATA= data set, and the value of INCOME found in the DATA= data set, or, when the INCOME variable is missing, the value of INCOME forecast by its ARIMA model. (Because SALES is missing for future time periods, the estimation of model parameters is not affected by the forecast values for PRICE, INCOME, or TAXRATE.)