The semiparametric factor model serves as a vital tool to describe the dependence patterns in the data. It recognizes that the common features observed in the data are actually explained by functions of specific exogenous variables.Unlike traditional factor models, where the focus is on selecting the number of factors, our objective here is to identify the appropriate number of common functions, a crucial parameter in this model. In this paper, we develop a novel data-driven method to determine the number of functional factors using cross validation (CV). Our proposed method employs a two-step CV process that ensures the orthogonality of functional factors, which we refer to as Functional Twice Cross-Validation (FTCV). Extensive simulations demonstrate that FTCV accurately selects the number of common functions and outperforms existing methods in most cases.Furthermore, by specifying market volatility as the exogenous force, we provide real data examples that illustrate the interpretability of selected common functions in characterizing the influence on U.S. Treasury Yields and the cross correlations between Dow30 returns.

翻译：半参数因子模型是描述数据中依赖模式的重要工具，该模型认识到数据中观测到的共同特征实际上是由特定外生变量的函数所解释。与传统因子模型关注因子数目选择不同，本文旨在识别该模型中的关键参数——公共函数的适当数量。我们提出了一种基于交叉验证的新型数据驱动方法来确定函数因子数量。该方法采用两步交叉验证过程确保函数因子的正交性，我们将其称为"函数型双重交叉验证"。大量仿真实验表明，FTCV能准确选择公共函数数量，且在多数情况下优于现有方法。此外，通过将市场波动率设定为外生变量，我们提供了真实数据示例，展示了所选公共函数在刻画美国国债收益率影响及道琼斯30指数收益交叉相关性方面的可解释性。