  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.
