Forecasting time series with increasing seasonal variation

Four options for modeling and forecasting time series data containing increasing seasonal variation are discussed, including data transformations, double seasonal difference models and two kinds of transfer function-type ARIMA models employing seasonal dummy variables. An explanation is given for the typical ARIMA model identification analysis failing to identify double seasonal difference models for this kind of data. A logical process of selecting one option for a particular case is outlined, focusing on issues of linear versus non-linear increasing seasonal variation, and the level of stochastic versus deterministic behavior in a time series. Example models for the various options are presented for six time series, with point forecast and interval forecast comparisons. Interval forecasts from data-transformation models are found to generally be too wide and sometimes illogical in the dependence of their width on the point forecast level. Suspicion that maximum likelihood estimation of ARIMA models leads to excessive indications of unit roots in seasonal moving-average operators is reported.