A common and important problem is forecasting. We try to forecast tomorrow's weather or when the next big earthquake will hit. But our efforts are fraught with uncertainty. When there exist historical data and we can find empirical or mechanistic models to fit to these data, we may be able to produce accurate forecasts. For certain problems such as the Dow Jones Industrial Average (DJIA) for earthquakes, good models are not available to fit to the historical data, and forecasting becomes very problematic. In some cases such as the DJIA, there is a good empirical time series model called the random walk. But that model leaves too much unexplained variability to be useful in forecasting.

With the exception of finance that requires more volatility, commonly used forecasting techniques based on historical data without covariates are the Box–Jenkins autoregressive integrated moving average (ARIMA) models and exponential smoothing. The simplest form of exponential smoothing is a particular ARIMA model namely the IMA(1, 1) model. The basic approach for iteratively selecting and fitting an ARIMA model was first developed in Box and Jenkins (1970). The text has been revised twice since 1970 (Box and Jenkins, 1976; Box et al., 1994). The autoregressive models that are an important subclass of the ARIMA models go back to Yule (1927).

The basic exponential smoothing model provides forecasts of future values using exponentially decreasing weights on the ...

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