Robust Factor Analysis without Moment Constraint
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In large-dimensional factor analysis, existing methods, such as principal component analysis (PCA), assumed finite fourth moment of the idiosyncratic components, in order to derive the convergence rates of the estimated factor loadings and scores. However, in many areas, such as finance and macroeconomics, many variables are heavy-tailed. In this case, PCA-based estimators and their variations are not theoretically underpinned. In this paper, we investigate into the L_1 minimization on the factor loadings and scores, which amounts to assuming a temporal and cross-sectional median structure for panel observations instead of the mean pattern in L_2 minimization. Without any moment constraint on the idiosyncratic errors, we correctly identify the common components for each variable. We obtained the convergence rates of a computationally feasible L_1 minimization estimators via iteratively alternating the median regression cross-sectionally and serially. Bahardur representations for the estimated factor loadings and scores are provided under some mild conditions. Simulation experiments checked the validity of the theory. In addition, a Robust Information Criterion (RIC) is proposed to select the factor number. Our analysis on a financial asset returns data set shows the superiority of the proposed method over other state-of-the-art methods.
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