Over the last decade, nonparametric methods have gained increasing attention for modeling complex data structures due to their flexibility and minimal structural assumptions. In this paper, we study a general multivariate nonparametric regression framework that encompasses a broad class of parametric models commonly used in financial econometrics. Both the response and the covariate processes are allowed to be multivariate with fixed finite dimensions, and the framework accommodates temporal dependence, thereby introducing additional modeling and theoretical hurdles. To address these challenges, we adopt a functional dependence structure which permits flexible dynamic behavior while maintaining tractable asymptotic analysis. Within this setting, we establish strong and weak convergence results for the estimators of the conditional mean and volatility functions. In addition, we investigate conditional geometric quantiles in the multivariate time series context and prove their consistency under mild regularity conditions. The finite sample performance is examined through comprehensive simulation studies, and the methodology is illustrated by modeling the stock returns of Maersk and Lockheed Martin as a nonparametric function of a geopolitical risk index.

翻译：过去十年间，非参数方法因其灵活性和最少的结构假设，在建模复杂数据结构方面日益受到关注。本文研究了一个通用的多元非参数回归框架，该框架涵盖了金融计量经济学中常用的广泛参数模型类别。响应过程和协变量过程均允许为具有固定有限维度的多元过程，且该框架能够容纳时间依赖性，从而引入了额外的建模和理论障碍。为应对这些挑战，我们采用了一种函数依赖结构，该结构允许灵活的动态行为，同时保持可处理的渐近分析。在此设定下，我们建立了条件均值函数和波动率函数估计量的强收敛与弱收敛结果。此外，我们在多元时间序列背景下研究了条件几何分位数，并在温和的正则性条件下证明了其一致性。通过全面的模拟研究检验了有限样本性能，并以马士基和洛克希德·马丁的股票收益作为地缘政治风险指数的非参数函数进行建模，对该方法进行了例证。