Tail-Aware Density Forecasting of Locally Explosive Time Series: A Neural Network Approach
Mixed causal--noncausal (anticipative) models capture locally explosive dynamics and provide economically meaningful probabilities of continuation and collapse, but forecasting with them remains computationally difficult. We develop a two-stage framework that estimates such an ARMA model and then learns its predictive density using a Mixture Density Network with skewed-t components, tail-aware training weights, and post-hoc calibration. The approach accommodates the heavy tails, asymmetry, and multimodality characteristic of non-causal forecasts while remaining computationally tractable. Monte Carlo experiments and a real-time natural-gas application show substantial improvements over existing density-forecasting methods.