# Separating common and idiosyncratic components without moment condition and on the weighted $L_1$ minimization path

15 Jun 2020  ·  He Yong, Kong Xin-Bing, Yu Long, Zhang Xinsheng ·

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 weighted $L_1$ minimization on the factor loadings and scores, which amounts to assuming a temporal and cross-sectional quantile 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 and idiosyncratic components for each variable. We obtained the convergence rates of a computationally feasible weighted $L_1$ minimization estimators via iteratively alternating the quantile regression cross-sectionally and serially. Bahardur representations for the estimated factor loadings and scores are provided under some mild conditions. In addition, a robust method is proposed to estimate the number of factors consistently. Simulation experiments checked the validity of the theory. Our analysis on a financial data set shows the superiority of the proposed method over other state-of-the-art methods.

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