This paper examines the performance of nonlinear short‐term forecasts of noncausal processes from closed‐form functional predictive density estimators. The processes considered have mixed causal–noncausal MAR(1, 1) dynamics and non‐Gaussian distributions with either finite or infinite variance. The quality of point forecasts is affected by spikes and bubbles in the trajectories of these processes, which also characterize many financial and economic time series. This is due to deformations of estimated predictive densities from multimodality during explosive episodes. We show that two‐step‐ahead predictive densities of future trajectories based on the MAR(1, 1) Cauchy process can be used as a new graphical tool for early detection of bubble outsets and bursts. The method is applied to the Bitcoin/US dollar exchange rates and commodity futures.

中文翻译：

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预测非因果MAR（1，1）流程中的性能和气泡分析

本文研究了封闭形式的功能预测密度估计器对非因果过程的非线性短期预测的性能。所考虑的过程具有因果关系-非因果MAR（1，1）动力学和具有有限或无限方差的非高斯分布。这些过程的轨迹中的尖峰和气泡会影响点预测的质量，这也是许多金融和经济时间序列的特征。这是由于爆炸事件期间多模式估计的预测密度发生了变形。我们表明，基于MAR（1，1）Cauchy过程的未来轨迹的两步式预测密度可以用作早期检测气泡开始和破裂的新图形工具。该方法适用于比特币/美元汇率和商品期货。