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The ARIMA Procedure |

In addition to past values of the response series and past errors,
you can also model the response series using
the current and past values of other series, called *input series*.
Several different names are used to describe ARIMA models with input series.
*Transfer function model*, *intervention model*,
*interrupted time series model*, *regression model with ARMA errors*,
*Box-Tiao model*, and *ARIMAX model*
are all different names for ARIMA models with input series.
Pankratz (1991) refers to these models as *dynamic regression*.

proc arima data=a; identify var=sales crosscorr=price; estimate input=price; run;

This example performs a simple linear regression of SALES on PRICE, producing the same results as PROC REG or another SAS regression procedure. The mathematical form of the model estimated by these statements is

The parameter estimates table for this example (using simulated data) is shown in Figure 7.19. The intercept parameter is labeled MU. The regression coefficient for PRICE is labeled NUM1. (See the section "Naming of Model Parameters" later in this chapter for information on how parameters for input series are named.)

Any number of input variables can be used in a model. For example, the following statements fit a multiple regression of SALES on PRICE and INCOME:

proc arima data=a; identify var=sales crosscorr=(price income); estimate input=(price income); run;

The mathematical form of the regression model estimated by these statements is

proc arima data=a; identify var=sales(1) crosscorr=price(1); estimate input=( 1 $ price ); run;

These statements estimate the model

proc arima data=a; identify var=sales crosscorr=(price income); estimate p=1 q=1 input=(price income); run;

These statements estimate the model

There is no requirement that the input series be stationary. If the inputs are nonstationary, the response series will be nonstationary, even though the noise process may be stationary.

When nonstationary input series are used, you can fit the input variables first with no ARMA model for the errors and then consider the stationarity of the residuals before identifying an ARMA model for the noise part.

However, if the input series are independent of the noise series, you can use the residuals from the regression model as an estimate of the noise series, then apply the ARIMA modeling identification process to this residual series. This assumes that the noise process is stationary.

The PLOT option on the ESTIMATE statement produces for the model residuals the same plots as the IDENTIFY statement produces for the response series. The PLOT option prints an autocorrelation function plot, an inverse autocorrelation function plot, and a partial autocorrelation function plot for the residual series.

The following statements show how the PLOT option is used to identify the ARMA(1,1) model for the noise process used in the preceding example of regression with ARMA errors:

proc arima data=a; identify var=sales crosscorr=(price income) noprint; estimate input=(price income) plot; run; estimate p=1 q=1 input=(price income) plot; run;

In this example, the IDENTIFY statement includes the NOPRINT option since the autocorrelation plots for the response series are not useful when you know that the response series depends on input series.

The first ESTIMATE statement fits the regression model with no model for the noise process. The PLOT option produces plots of the autocorrelation function, inverse autocorrelation function, and partial autocorrelation function for the residual series of the regression on PRICE and INCOME.

By examining the PLOT option output for the residual series, you verify that the residual series is stationary and identify an ARMA(1,1) model for the noise process. The second ESTIMATE statement fits the final model.

Although this discussion addresses regression models, the same remarks apply to identifying an ARIMA model for the noise process in models that include input series with complex transfer functions.

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