Factor models have been widely used in economics and finance. However, the heavy-tailed nature of macroeconomic and financial data is often neglected 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, which is motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms that use different penalty forms. The first algorithm, which we refer to as Huber PCA, minimizes the $\ell_2$-norm-type Huber loss and performs PCA on the weighted sample covariance matrix. The second algorithm involves an element-wise type Huber loss minimization, which can be solved by an iterative Huber regression algorithm. Our study examines the theoretical minimizer of the element-wise Huber loss function and demonstrates that it has the same convergence rate as conventional PCA when the idiosyncratic errors have bounded second moments. We also derive their asymptotic distributions under mild conditions. Moreover, we suggest a consistent model selection criterion that relies on rank minimization to estimate the number of factors robustly. We showcase the benefits of Huber PCA through extensive numerical experiments and a real financial portfolio selection example. An R package named ``HDRFA" has been developed to implement the proposed 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。然而，现有文献往往忽略了宏观经济与金融数据的重尾特征。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与得分的方法，这一方法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。我们提供了两种采用不同惩罚形式的算法。第一种算法称为Huber PCA，通过最小化ℓ₂范数型Huber损失并对加权样本协方差矩阵执行PCA实现；第二种算法涉及逐元素型Huber损失最小化，可通过迭代Huber回归算法求解。我们研究了逐元素型Huber损失函数的理论最小化器，证明当特质误差具有有界二阶矩时，其收敛速率与传统PCA相同，并在温和条件下推导了其渐近分布。此外，我们提出了一种基于秩最小化的一致模型选择准则，以稳健地估计因子数量。通过大量数值实验及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。本研究开发了R包"HDRFA"用于实现所提出的稳健因子分析。