Trend Estimation — Exponential Smoothing

$$\hat{m}_t = \alpha X_t + (1 - \alpha)\hat{m}_{t-1}, \quad t = 2, \ldots, n; \qquad \hat{m}_1 = X_1$$

Notation: $\alpha \in [0, 1]$ — smoothing parameter.

$\hat{m}_t = \hat{m}_{t-1} + \alpha(X_t - \hat{m}_{t-1})$: adjust old estimate by fraction $\alpha$ of forecast error.

For $t \geq 2$:

$$\hat{m}_t = \sum_{j=0}^{t-2} \alpha(1-\alpha)^j X_{t-j} + (1-\alpha)^{t-1} X_1$$

Weights decay geometrically → recent observations get exponentially more weight.

Minimize $\sum_{t=1}^n (x_{t+1} - \hat{m}_t)^2$. No distributional assumption required.

Should not be used with trend or seasonality present (basic version assumes constant level)
Holt-Winters generalizes to trend + seasonality

Trend Estimation — Exponential Smoothing#