  This paper describes a family of seasonal and non-seasonal time series models
that can be viewed as generalisations of additive and multiplicative
exponential smoothing models, to model series that grow faster than linear but
slower than exponential. Their development is motivated by fast-growing,
volatile time series. In particular, our models have a global trend that can
smoothly change from additive to multiplicative, and is combined with a linear
local trend. Seasonality when used is multiplicative in our models, and the
error is always additive but is heteroscedastic and can grow through a
parameter sigma. We leverage state-of-the-art Bayesian fitting techniques to
accurately fit these models that are more complex and flexible than standard
exponential smoothing models. When applied to the M3 competition data set, our
models outperform the best algorithms in the competition as well as other
benchmarks, thus achieving to the best of our knowledge the best results of
per-series univariate methods on this dataset in the literature. An open-source
software package of our method is available.
