Sparse Asymptotic PCA: Identifying Sparse Latent Factors Across Time Horizon in High-Dimensional Time Series

TL;DR

This paper proposes a sparse asymptotic PCA framework to identify sparse latent factors in high-dimensional time series, especially in financial data, allowing for non-sparse loadings and providing a data-driven method to detect factor sparsity over time.

Contribution

It introduces a novel sparse APCA method with a truncated power approach and a cross-validation technique for identifying sparse factors in high-dimensional time series.

Findings

Successfully applied to S&P 500 data, identifying nine key risk factors.

Demonstrates consistency of estimators as data dimensions grow.

Performs well in finite-sample simulations.

Abstract

This paper introduces a novel sparse latent factor modeling framework using sparse asymptotic Principal Component Analysis (APCA) to analyze the co-movements of high-dimensional panel data over time. Unlike existing methods based on sparse PCA, which assume sparsity in the loading matrices, our approach posits sparsity in the factor processes while allowing non-sparse loadings. This is motivated by the fact that financial returns typically exhibit universal and non-sparse exposure to market factors. Unlike the commonly used $\ell_1$-relaxation in sparse PCA, the proposed sparse APCA employs a truncated power method to estimate the leading sparse factor and a sequential deflation method for multi-factor cases under $\ell_0$-constraints. Furthermore, we develop a data-driven approach to identify the sparsity of risk factors over the time horizon using a novel cross-sectional cross-validation method. We establish the consistency of our estimators under mild conditions as both the dimension $N$ and the sample size $T$ grow. Monte Carlo simulations demonstrate that the proposed method performs well in finite samples. Empirically, we apply our method to daily S&P 500 stock returns (2004--2016) and identify nine risk factors influencing the stock market.