We introduce a data-driven dynamic factor framework for modeling the joint evolution of high-dimensional covariates and responses without parametric assumptions. Standard factor models applied to covariates alone often lose explanatory power for responses. Our approach uses anisotropic diffusion maps, a manifold learning technique, to learn low-dimensional embeddings that preserve both the intrinsic geometry of the covariates and the predictive relationship with responses. For time series arising from Langevin diffusions in Euclidean space, we show that the associated graph Laplacian converges to the generator of the underlying diffusion. We further establish a bound on the approximation error between the diffusion map coordinates and linear diffusion processes, and we show that ergodic averages in the embedding space converge under standard spectral assumptions. These results justify using Kalman filtering in diffusion-map coordinates for predicting joint covariate-response evolution. We apply this methodology to equity-portfolio stress testing using macroeconomic and financial variables from Federal Reserve supervisory scenarios, achieving mean absolute error improvements of up to 55% over classical scenario analysis and 39% over principal component analysis benchmarks.

翻译：我们提出了一种数据驱动的动态因子框架，用于建模高维协变量与响应变量的联合演化过程，无需参数化假设。仅应用于协变量的标准因子模型通常对响应变量的解释力不足。我们的方法采用各向异性扩散映射这一流形学习技术，通过学习低维嵌入来同时保留协变量的内在几何结构及其与响应变量的预测关系。对于欧几里得空间中朗之万扩散产生的时间序列，我们证明了关联图拉普拉斯算子收敛于底层扩散的生成元。我们进一步建立了扩散映射坐标与线性扩散过程之间近似误差的界限，并证明了在标准谱假设下嵌入空间中的遍历平均值具有收敛性。这些结果为在扩散映射坐标中使用卡尔曼滤波预测协变量-响应联合演化提供了理论依据。我们将该方法应用于基于美联储监管情景中宏观经济与金融变量的股票投资组合压力测试，相比经典情景分析实现了高达55%的平均绝对误差改进，较主成分分析基准模型提升了39%。