SAS/Ets User's Guide 28. Provides detailed reference material for using SAS/ETS software and guides you through the analysis and forecasting of features such as univariate and multivariate time series, cross-sectional time series, seasonal adjustments, multiequational nonlinear models, discrete choice models, limited dependent variable models, portfolio analysis, and generation of financial reports, with introductory and advanced examples for each procedure. You can also find complete information about two easy-to-use point-and-click applications: the Time Series Forecasting System, for automatic and interactive time series modeling and forecasting, and the Investment Analysis System, for time-value of money analysis of a variety of investments | 262 F Chapter 7 The ARIMA Procedure That is the -step forecast of xtck given x1 - xt-1 is xck Ck tVt-1 xi --- xt-1 0 where Ck t is the covariance of xt k and x1 - xt _1 and Vt is the covariance matrix of the vector x1 - xt-1 . Ck t and Vt are derived from the estimated parameters. Finite memory forecasts minimize the mean squared error of prediction if the parameters of the ARMA model are known exactly. In most cases the parameters of the ARMA model are estimated so the predictors are not true best linear forecasts. If the response series is differenced the final forecast is produced by summing the forecast of the differenced series. This summation and the forecast are conditional on the initial values of the series. Thus when the response series is differenced the final forecasts are not true finite memory forecasts because they are derived by assuming that the differenced series begins in a steady-state condition. Thus they fall somewhere between finite memory and infinite memory forecasts. In practice there is seldom any practical difference between these forecasts and true finite memory forecasts. 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
