A Majorization-Minimization gLASSO Framework for SETAR Models: Theory, Simulation, and Application to PM2.5 Data

This study proposes an optimized estimation approach for Self-Exciting Threshold Autoregressive (SETAR) models by integrating the Majorization-Minimization Group Least Absolute Shrinkage and Selection Operator (MM-gLASSO) algorithm, with a primary focus on improving forecasting performance in complex regime-switching environments. While SETAR models are powerful in capturing non-linear regimes and asymmetric dynamics in time series data, they often suffer from high-dimensional parameter spaces. This typically leads to high-dimensional parameter spaces that cause overfitting and reduced forecasting stability. We address this by employing the MM algorithm to simplify the complex, non-differentiable gLASSO penalty into a more manageable surrogate function. This ensures stable convergence and allows for simultaneous variable selection and parameter estimation across multiple regimes. The gLASSO penalty is specifically utilized to ensure that irrelevant lags are excluded consistently across the threshold structure. We provide a detailed derivation of the estimation procedure and evaluate its performance through extensive simulation studies. The results indicate that the MM-gLASSO framework significantly outperforms traditional methods, particularly in terms of sparsity recognition and parameter consistency. Finally, an empirical application on PM2.5 concentration demonstrates the model's superior forecasting capability and its effectiveness in identifying structural transitions in real-world time series data.