Exponential Smoothing

Study guides: Exam SRM, Exam MAS-II

Exponential smoothing forecasts a Time Series with a weighted average of its past values, the weights decaying geometrically into the past. Simple exponential smoothing has one parameter; Holt adds a trend and Holt-Winters adds seasonality.

$\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha)\hat{Y}_t, \qquad 0 < \alpha < 1$

$\hat{Y}_{t+1} = \alpha\sum_{j=0}^{\infty}(1-\alpha)^{j}Y_{t-j}$

$\alpha$ near 1 tracks the series closely and reacts fast to shocks; $\alpha$ near 0 smooths heavily and reacts slowly. It is chosen by minimizing one-step forecast SSE
Simple smoothing has a flat $h$ -step forecast — it assumes no trend, and is the optimal forecast for an ARIMA(0,1,1)
Holt's linear method adds a level and a slope equation ( $\beta$ ), giving a forecast that extrapolates the trend; Holt-Winters adds a seasonal equation ( $\gamma$ ), additive or multiplicative
The recursion needs no history in storage — only the current level (and slope, and seasonals) — which is why it is standard in high-volume forecasting
It is a model-free method: no Stationarity requirement and no ACF/PACF identification, but also no prediction intervals without an underlying state-space model

Two Steps of Simple Smoothing With $\alpha = 0.3$ and a current forecast $\hat Y_{10} = 420$ , the actual $Y_{10} = 470$ arrives, then $Y_{11} = 440$ . Find $\hat Y_{11}$ and $\hat Y_{12}$ , and the 3-step forecast.

Answer $\hat Y_{11} = 0.3(470) + 0.7(420) = 141 + 294 = 435$ $\hat Y_{12} = 0.3(440) + 0.7(435) = 132 + 304.5 = 436.5$ The 3-step forecast is also 436.5 — simple exponential smoothing gives a flat forecast at every horizon. In error-correction form, $\hat Y_{11} = 420 + 0.3(470-420)$ : the update is $\alpha$ times the last forecast error.