Trend Estimation — Regression
For $X_t = m_t + Y_t$ with $E(Y_t) = 0$:
Linear Regression
$$\hat{m}_t = \hat{a} + \hat{b} \times t$$where $\hat{a}, \hat{b}$ minimize $\sum_{t=1}^n (X_t - a - bt)^2$.
Extends to polynomial: $\hat{m}_t = \hat{a}_0 + \hat{a}_1 t + \cdots + \hat{a}_p t^p$.
Characteristics
- Parametric: must specify functional form
- Can extrapolate → usable for forecasting
- Sensitive to misspecification