美联储-线性状态空间模型中增强鲁棒滤波和预测的缺失数据替换（英）-2025.1_38页_2mb

Missing Data Substitution for Enhanced Robust Filtering and Forecasting

Background

• State-space models are widely used in time-series analysis.
• Existing robust filters, like the Huberized Robust Kalman Filter (RobKF), perform optimally for independent, identically distributed (i.i.d.) outliers but lose robustness when outliers cluster in time or are highly correlated.
• This paper introduces two complementary methods for improving outlier-robust filtering and forecasting: supervised missing data substitution (MD) and unsupervised missing data substitution via exogenous randomization (RMDX).

Supervised Missing Data Substitution (MD)

• Purpose: Eliminate large outliers above the Huber detection threshold.
• Mechanism: Replaces outlier measurements exceeding a Huber threshold with missing data, improving performance when outliers of the same sign cluster.
• Example Filter: MD-RobKF, which outperforms standard RobKF when optimality conditions are violated.

Unsupervised Missing Data Substitution via Randomization (RMDX)

• Purpose: Suppress smaller outliers below the Huber detection threshold.
• Mechanism: Averages filtered or forecasted targets over measurement series with randomly induced missing data at a predefined randomization rate, creating regularization through a bias-variance trade-off.
• Example Filters: RMDX-KF, RMDX-RobKF, and RMDX-MD-RobKF, which combine with MD to enhance overall robustness.

Key Theoretical Result

• The randomization rate in RMDX acts as a regularization parameter controlling the bias-variance trade-off.
• Bias-Variance Trade-off: Bias decreases with lower randomization rates (β), while variance increases.
• Optimal Randomization: Can be determined via cross-validation to minimize root mean squared error (RMSE).

Empirical Validation

Monte Carlo Simulations

1. i.i.d. Outliers: RMDX improves MD-RobKF, especially for mid-sized outliers. Combinations show the highest performance gains.
2. Clustered Outliers: Supervised MD and unsupervised RMDX significantly improve performance, with the combined RMDX-MD-RobKF outperforming all other filters in clustered outlier scenarios.

Real-World Application (Inflation Forecasting)

• Data: Observed quarterly PCE inflation data (1960Q1-2015Q1) using state-space models (UC, ARMF, AR).
• Findings: The RMDX-MD-RobKF filter consistently produces the lowest mean squared forecasting errors (MSFEs) across all models and forecast horizons, especially for longer horizons. It provides performance comparable to established benchmark models (UC-T, UCSVO) and is more robust to clustered outliers.

Conclusion

• The proposed methods (MD and RMDX) are easy to implement and significantly enhance outlier-robust filtering and forecasting.
• Combining supervised and unsupervised methods (RMDX-MD-RobKF) yields the highest performance improvements.
• Missing data randomization offers a time-series extension of bagging and has promising applications in various fields.