The nine Holt-Winters models, on real data
A three by three taxonomy of trend and seasonality. Pick a real series, pick one of the nine ETS variants, read a verdict that tells you whether this is the textbook fit, a workable approximation, or the wrong tool for the job, and what to use instead.
The megaphone test, additive versus multiplicative seasonality
If the seasonal swings widen as the level rises, the seasonality is multiplicative. If the swings stay roughly the same width regardless of level, additive seasonality fits.
Pick a data shape, pick a model
Nine real and synthetic series, each chosen because it puts a different ETS variant under the spotlight. The verdict then names one combination that fits and warns about the ones that do not.
Pick a scenario and a model
Pick a scenario and a model variant to see the verdict.
Six things worth knowing
Click to reveal real-world reasoning behind these data shapes.
Why does AirPassengers grow exponentially?
Post-war commercial aviation expanded as a percentage of a growing market each year, not by a fixed number of seats. Compounding growth produces exponential trajectories. Use multiplicative trend with multiplicative seasonality, the textbook ETS(M,M) shape.
What is the megaphone test?
Plot the series and watch the seasonal swings. If they widen as the level rises, like a megaphone opening to the right, the seasonality is multiplicative. If the swings hold a roughly constant width, additive fits. This is the single fastest way to choose between additive and multiplicative seasonal components.
When does additive trend beat multiplicative?
Additive trend assumes a fixed unit increase per period. UK electricity demand or a regulated utility load grows by roughly the same megawatt-hours each year, not by a percentage. Multiplicative trend would over-extrapolate. Use additive trend when growth is roughly linear in the unit you measure.
Why do Wikipedia pageviews show level shifts?
Pageviews jump on news events, viral moments, or topic shifts, then settle into a new baseline. ETS treats this as noise and bleeds the shift slowly into the level via the smoothing parameter. If level shifts dominate, ETS will lag. A change-point model or a regression with event indicators is a better tool.
What is a damped trend?
The trend component is multiplied by a damping parameter phi smaller than one each period, so projected growth shrinks toward zero in the long run. This prevents implausible long-horizon forecasts. Damping is almost always wise for forecasts more than a few periods ahead.
Why does the verdict for white noise say to use the naive forecaster instead?
White noise has no exploitable structure, by definition. Any model that smooths past values cannot beat the mean as a forecaster, and ETS just adds parameters that overfit the noise. Use the mean, the naive last value, or, honestly, do not bother forecasting and budget for the variance.