This paper introduces a novel Mixture Density Network (MDN) tailored for density forecasting of time series exhibiting locally explosive behavior, such as financial bubbles. The model uses skewed t-distributions as mixture components to capture the heavy-tailed, asymmetric, and potentially multimodal predictive densities that arise from mixed causal-noncausal ARMA (MARMA) processes. To address the rarity of extreme events, the authors develop an adaptive weighting scheme that emphasizes tail observations during training, combined with a post-hoc calibration procedure to correct for distribution shift. The MDN is lightweight (two hidden layers of 64 units) and produces near-instantaneous forecasts once trained.

Through extensive Monte Carlo simulations on various MARMA specifications with α-stable innovations, the MDN demonstrates superior performance compared to established methods: nonparametric kernel density estimation, simulation-based approaches, closed-form predictive densities, and general-purpose learning methods (FlexZBoost). The MDN accurately captures the theoretical bimodality of predictive densities in tail regions, provides the most accurate conditional moment estimates, and achieves the lowest KL divergence and integrated squared error when the true density is known (Cauchy case). Two empirical applications—forecasting U.S. natural gas prices and inflation—confirm that the MDN outperforms competitors in both density and point forecast accuracy, while requiring substantially less computational time. The paper highlights that the MDN's ability to generate realistic trajectories and its flexibility in handling multiple conditioning variables make it a powerful tool for risk management and economic forecasting in the presence of explosive dynamics.