hp-filter

Hodrick-Prescott Filter

$$\underset{\{X_t^{\text{trend}}{\}}_{t=1}^T}{\min }\sum _{t=1}^T{\left(X_t^{\text{raw}}-X_t^{\text{trend}}\right)}^2+\lambda \sum _{t=2}^{T-1}\left[\left(X_{t+1}^{\text{trend}}-X_t^{\text{trend}}\right)-\left(X_t^{\text{trend}}-X_{t-1}^{\text{trend}}\right)\right]$$