Factor models have found widespread applications in economics and finance, but the heavy-tailed character of macroeconomic and financial data is often overlooked in the existing literature. To address this issue and achieve robustness, we propose an approach to estimate factor loadings and scores by minimizing the Huber loss function, motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms based on different penalty forms. The first minimizes the $\ell_2$-norm-type Huber loss, performing PCA on the weighted sample covariance matrix and is named Huber PCA. The second version minimizes the element-wise type Huber loss and can be solved by an iterative Huber regression algorithm. We investigate the theoretical minimizer of the element-wise Huber loss function and show that it has the same convergence rate as conventional PCA under finite second-moment conditions on idiosyncratic errors. Additionally, we propose a consistent model selection criterion based on rank minimization to determine the number of factors robustly. We demonstrate the advantages of Huber PCA using a real financial portfolio selection example, and an R package called ``HDRFA" is available on CRAN to conduct robust factor analysis.

翻译：因子模型在经济学和金融学中得到了广泛应用，但现有文献往往忽视了宏观经济和金融数据中的重尾特征。为解决此问题并实现稳健性，我们提出了一种通过最小化Huber损失函数来估计因子载荷和得分的方法，其动机源于传统主成分分析（PCA）与因子模型中约束最小二乘法的等价性。我们基于不同的惩罚形式提供了两种算法。第一种算法最小化ℓ2-范数型Huber损失，对加权样本协方差矩阵进行PCA，并命名为Huber PCA。第二种版本最小化逐元素型Huber损失，可通过迭代Huber回归算法求解。我们研究了逐元素Huber损失函数的理论最小化器，并证明在异质误差仅需有限二阶矩的条件下，其收敛速度与传统PCA相同。此外，我们提出了一种基于秩最小化的一致性模型选择准则，以稳健地确定因子个数。通过一个真实的金融投资组合选择实例，我们展示了Huber PCA的优势，同时CRAN上提供了名为"HDRFA"的R包，用于进行稳健因子分析。