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The ARIMA Procedure

Forecasting Log Transformed Data

The log transformation is often used to convert time series that are nonstationary with respect to the innovation variance into stationary time series. The usual approach is to take the log of the series in a DATA step and then apply PROC ARIMA to the transformed data. A DATA step is then used to transform the forecasts of the logs back to the original units of measurement. The confidence limits are also transformed by using the exponential function.

As one alternative, you can simply exponentiate the forecast series. This procedure gives a forecast for the median of the series, but the antilog of the forecast log series underpredicts the mean of the original series. If you want to predict the expected value of the series, you need to take into account the standard error of the forecast, as shown in the following example, which uses an AR(2) model to forecast the log of a series Y:

   data in;
      set in;
      ylog = log( y );

   proc arima data=in;
      identify var=ylog;
      estimate p=2;
      forecast lead=10 out=out;

   data out;
      set out;
      y   = exp( ylog );
      l95 = exp( l95 );
      u95 = exp( u95 );
      forecast = exp( forecast + std*std/2 );
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