# 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] $