Trend Elimination — Differencing
Backward Shift Operator
$$BX_t = X_{t-1}, \qquad B^k X_t = X_{t-k}$$Difference Operator
$$\nabla X_t = X_t - X_{t-1} = (1 - B)X_t$$Higher-Order
$$\nabla^2 X_t = X_t - 2X_{t-1} + X_{t-2}$$Key Theorem
If $m_t$ is polynomial degree $p$, then $p$ times differencing eliminates the trend.
Example: Linear trend $m_t = a + bt$ → $\nabla m_t = b$ (constant).
Differencing vs Estimation
- Estimation: explicitly estimate $m_t$, subtract
- Differencing: apply $\nabla^p$ directly, no need to know functional form