Trend Estimation — Moving Average Filter

Nonseasonal (2q+1)-point MA

$$\hat{m}_t = \frac{X_{t-q} + \cdots + X_t + \cdots + X_{t+q}}{2q+1}, \qquad q+1 \leq t \leq n-q$$

For Seasonal Data (period $d$)

$d$ even, $d = 2q$:

$$\hat{m}_t = \frac{0.5\,x_{t-q} + x_{t-q+1} + \cdots + x_{t+q-1} + 0.5\,x_{t+q}}{d}$$

$d$ odd, $d = 2q+1$:

$$\hat{m}_t = \frac{x_{t-q} + \cdots + x_t + \cdots + x_{t+q}}{d}$$

Limitations

  1. Cannot estimate near endpoints (need $q$ obs on each side)
  2. Cannot forecast (no extrapolation)
  3. Equal weights for all observations

These motivate Trend Estimation — Exponential Smoothing.