This paper proposes a Mixture Density Network for forecasting time series that exhibit locally explosive behavior. By incorporating skewed t-distributions as mixture components, our approach offers enhanced flexibility in capturing the skewed, heavy-tailed, and potentially multimodal nature of predictive densities associated with bubble dynamics modeled by mixed causal-noncausal ARMA processes. In addition, we implement an adaptive weighting scheme that emphasizes tail observations during training and hence leads to accurate density estimation in the extreme regions most relevant for financial applications. Equally important, once trained, the MDN produces near-instantaneous density forecasts. Through extensive Monte Carlo simulations and an empirical application on the natural gas price, we show that the proposed MDN-based framework delivers superior forecasting performance relative to existing approaches.

翻译：本文提出一种混合密度网络用于预测呈现局部爆炸性行为的时间序列。通过引入偏斜t分布作为混合分量，该方法能更灵活地捕捉由混合因果-非因果ARMA过程建模的泡沫动态所关联的预测密度所具有的偏斜、厚尾及潜在多峰特性。此外，我们设计了一种自适应加权方案，在训练过程中重点强调尾部观测值，从而在最相关于金融应用的极端区域实现精确的密度估计。同样重要的是，训练完成后该混合密度网络可生成近乎即时的密度预测。通过大量蒙特卡洛模拟及对天然气价格的实证应用，我们证明所提出的基于混合密度网络的框架相较于现有方法具有更优越的预测性能。