Introduction | 1. The Standard HP Filter | 1.1 Problem Formulation | 1.2 Matrix Formulation | 1.3 The Problem with Fixed Lambda | 2. The Modified HP Filter | 2.1 Cross-Validation Approach | 2.2 Generalized Cross-Validation | 2.3 Algorithm | 3. Why Modified HP Filter Works Better | 3.1 Simulation Evidence | 3.2 Empirical Evidence | 4. Theoretical Justification | 4.1 Why Cross-Validation? | 4.2 Relationship to Signal Extraction | 4.3 Connection to Spline Smoothing | 5. Practical Recommendations | 5.1 When to Use Modified HP Filter | 5.2 Choosing max_lambda | 5.3 Interpreting Results | 6. Mathematical Appendix | 6.1 Derivation of HP Filter Solution | 6.2 Structure of Matrix A | 6.3 GCV Approximation | References

Overview | Why Modified HP Filter? | Installation | Quick Start | Example 1: GDP-like Series | Visualization with autoplot | Comparison with Standard HP Filter | Example 2: Why Fixed Lambda Matters | Visual Comparison | Key Insights from Comparison | Batch Processing: Multi-Country Analysis | Example 3: Cross-Country GDP Comparison | Visualizing Cross-Country Cycles | Highlighting Specific Countries | Batch Comparison with Standard HP | Example 4: Real Business Cycle Analysis | Cyclical Properties Visualization | Example 5: Sensitivity to Lambda | Performance Tips | Choosing max_lambda | Batch Processing Efficiency | Summary and Recommendations | When to Use Modified HP Filter | Key Takeaways | References | See Also

Introduction | Province Level | Division Level | District Level

1 Overview | Session Info | References

1 Introduction