A Brief Tutorial On Exponential Smoothing Models Major Observation Exponential smoothing models are special cases of Box Jenkins models When adopting exponential smoothing models for forecasting you are implicitly claiming that the same Box Jenkins model fits all time series equally well In many instances exponential smoothing methods may forecast well but there may also be many instances where building a general Box Jenkins model for forecasting would do even better If one only has a few observations to begin with say 10 or less then an ad hoc choice of the smoothing parameter say 0 3 can at least generate forecasts whereas having less than 10 observations makes it difficult to use the Box Jenkins method for building an adequate forecasting model One has to wait on more observations before progressing from the exponential smoothing models to Box Jenkins models 1 The Single Exponential Smoothing SES Model is equivalent to a ARIMA 0 1 1 Box Jenkins The Single Exponential Smoothing SES Model can be written as Ft At 1 1 Ft 1 1 where Ft exponentially smoothed forecast for period t At 1 actual value in prior period Ft 1 exponentially smoothed forecast for period t 1 smoothing constant If is close to zero the forecast relies heavily on past observations i e is smoothed heavily If 1 the evidence of past data is ignored completely and the forecast is given by the value of the current observation But model 1 is equivalent to the ARIMA 0 1 1 model At at 1 at 1 2 where at represents a white noise error term and the correspondence between model 1 and model 2 is 1 1 Therefore model 1 the single exponential smoothing model is a special case of the ARIMA 0 1 1 Box Jenkins model The above point was made by A C Harvey in his book Time Series Models 1981 Philip Allan Publishers p 168 This brings up the following Point Why should one attempt to fit all time series using one particular BoxJenkins model In model 2 there is no drift term therefore the single exponential smoothing model should not be applied to data with trend A na ve user of the SES model however is not likely to know of this subtle point One can use MLE on the ARIMA 0 1 1 model of 2 and obtain an estimate of the smoothing parameter as 1 1 where 1 is the MLE of the moving average order one parameter 1 Alternatively one could do a grid search over the interval 0 1 and apply 1 to determine the RMSEs of one step forecasts for different values of and choose the value that minimizes RMSE This is essentially the same as using MLE to estimate the ARIMA 0 1 1 model 2 2 The Seasonal Single Exponential Smoothing SSES Model is equivalent to a ARIMA 0 0 0 x 0 1 1 s Multiplicative Seasonal Box Jenkins model The seasonal single exponential smoothing SSES model is of the form Ft At s 1 Ft s 3 This model is equivalent to the ARIMA 0 0 0 x 0 1 1 Multiplicative Seasonal Box Jenkins Model s At a t 1 at s 4 We have the correspondence 1 1 Therefore one can estimate as where is the MLE of in model 4 above 1 1 1 1 Again the question arises as to why one would restrict the search for a good time series model to a special case of the Multiplicative Seasonal Box Jenkins model Are all seasonal time series equally well characterized by the same special case Box Jenkins model Notice 4 does not allow for trend in the data after taking the seasonal span difference which certainly doesn t apply to all time series data In many time series the more appropriate differencing of seasonal time series is 1 s One can now appreciate the inflexibility built into the SSES model