We propose a flexible framework for modeling the predictive distributions of nonlinear, possibly multivariate time series. Our approach expresses a general predictive distribution in an appropriate generative representation that is based on a folklore result from measure theoretic probability. This representation provides a direct simulation-based approximation to the predictive distribution, enabling straightforward computation of forecasts for the conditional mean and variance, fan charts, value at risk, expected shortfall, joint tail risks, and other quantities of interest. We estimate this generative representation using a version of conditional generative adversarial networks and provide a formal statistical analysis of estimation under weak temporal dependence. Specifically, estimation is expressed as a particular minimax problem and we establish consistency of its approximate solutions in Hausdorff distance. The empirical relevance of the approach is illustrated using applications to equity returns, realized variance, and realized covariances. The proposed method is also computationally manageable, with estimation in our applications taking approximately one minute on a standard laptop.

翻译：我们提出了一种灵活框架，用于对非线性、可能为多元的时间序列的预测分布进行建模。该方法基于测度论概率中的一个经典结果，将一般预测分布表达为适当的生成式表示。该表示通过直接基于模拟的方式逼近预测分布，从而能够简便地计算条件均值和方差的预测、扇形图、风险价值、预期亏损、联合尾部风险及其他感兴趣的指标。我们利用条件生成对抗网络的一种变体来估计该生成式表示，并提供了弱时间依赖条件下估计的正式统计分析。具体而言，估计被表述为一个特定的极小极大问题，我们证明了其近似解在豪斯多夫距离下的一致性。通过应用于股票收益率、已实现方差和已实现协方差的实例，展示了该方法的实证相关性。所提出的方法在计算上也可行，在我们的应用中，估计过程在标准笔记本电脑上仅需约一分钟。