Tail-Aware Density Forecasting of Locally Explosive Time Series: A Neural Network Approach: CRED WORKING PAPER 2026-02

This paper proposes a Mixture Density Network specifically designed 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 two empirical applications, on the natural gas price and inflation, we show that the proposed MDN-based framework delivers superior forecasting performance relative to existing approaches.