Track 12 · Time Series and Forecasting · Lesson 5 of 11

Holt and Holt–Winters

Extend exponential smoothing with a trend (Holt) and a seasonal component (Holt–Winters), compute their forecasts, and choose between additive, multiplicative and damped versions on a holdout.

40 minAdvanced7 graded exercisesMembers

Ceiling-fan orders grow every year and triple every summer. A flat forecast is useless for planning next June. Holt–Winters keeps three running estimates — level, slope and a seasonal factor for each month — and updates all three every month.

Holt's linear trend method adds a slope $b_t$ to the level. Each period the level is updated towards the new observation, and the slope towards the latest change in level. Forecasts extend the latest level along the latest slope.

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It has 7 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Update Holt's level and slope and forecast h steps ahead
Use a damped trend for long horizons
Forecast with additive and multiplicative Holt–Winters, using the seasonal state of the right season
Compare smoothing methods with a seasonal naive benchmark on a holdout

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Formula sheet

Holt's linear trend method
$\begin{aligned} \ell_t &= \alpha\, y_t + (1-\alpha)(\ell_{t-1} + b_{t-1}) \\ b_t &= \beta\,(\ell_t - \ell_{t-1}) + (1-\beta)\, b_{t-1} \\ \hat{y}_{T+h\,|\,T} &= \ell_T + h\, b_T \end{aligned}$
Holt–Winters forecasts for h ≤ m (the seasonal state of the same season, one cycle earlier)
$\text{additive: } \hat{y}_{T+h\,|\,T} = \ell_T + h\,b_T + s_{T+h-m} \qquad \text{multiplicative: } \hat{y}_{T+h\,|\,T} = (\ell_T + h\,b_T)\, s_{T+h-m}$

Reference

Holt's linear trend and Holt–Winters seasonal methods
Exponential smoothing extended with a smoothed trend (Holt) and seasonal terms (Holt–Winters, additive or multiplicative), using the error-correction recursions of FPP3.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.