### Prewhitening

If, as is usually the case, an input series is autocorrelated, the direct cross-correlation function between the input and response series gives a misleading indication of the relation between the input and response series.

One solution to this problem is called prewhitening. You first fit an ARIMA model for the input series sufficient to reduce the residuals to white noise; then, filter the input series with this model to get the white noise residual series. You then filter the response series with the same model and cross-correlate the filtered response with the filtered input series.

The ARIMA procedure performs this prewhitening process automatically when you precede the IDENTIFY statement for the response series with IDENTIFY and ESTIMATE statements to fit a model for the input series. If a model with no inputs was previously fit to a variable specified by the CROSSCORR= option, then that model is used to prewhiten both the input series and the response series before the cross-correlations are computed for the input series.

For example,

```   proc arima data=in;
identify var=x;
estimate p=1 q=1;
identify var=y crosscorr=x;
run;
```

Both X and Y are filtered by the ARMA(1,1) model fit to X before the cross-correlations are computed.

Note that prewhitening is done to estimate the cross-correlation function; the unfiltered series are used in any subsequent ESTIMATE or FORECAST statements, and the correlation functions of Y with its own lags are computed from the unfiltered Y series. But initial values in the ESTIMATE statement are obtained with prewhitened data; therefore, the result with prewhitening can be different from the result without prewhitening.

To suppress prewhitening for all input variables, use the CLEAR option in the IDENTIFY statement to make PROC ARIMA disregard all previous models.

#### Prewhitening and Differencing

If the VAR= and CROSSCORR= options specify differencing, the series are differenced before the prewhitening filter is applied. When the differencing lists specified in the VAR= option for an input and in the CROSSCORR= option for that input are not the same, PROC ARIMA combines the two lists so that the differencing operators used for prewhitening include all differences in either list (in the least common multiple sense).