  This paper proposes a novel method for sparse latent factor modeling using a
new sparse asymptotic Principal Component Analysis (APCA). This approach
analyzes the co-movements of large-dimensional panel data systems over time
horizons within a general approximate factor model framework. Unlike existing
sparse factor modeling approaches based on sparse PCA, which assume sparse
loading matrices, our sparse APCA assumes that factor processes are sparse over
the time horizon, while the corresponding loading matrices are not necessarily
sparse. This development is motivated by the observation that the assumption of
sparse loadings may not be appropriate for financial returns, where exposure to
market factors is generally universal and non-sparse. We propose a truncated
power method to estimate the first sparse factor process and a sequential
deflation method for multi-factor cases. Additionally, 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. Theoretically, we establish that
our estimators are consistent under mild conditions. Monte Carlo simulations
demonstrate that the proposed method performs well in finite samples.
Empirically, we analyze daily stock returns for a balanced panel of S&P 500
stocks from January 2004 to December 2016. Through textual analysis, we examine
specific events associated with the identified sparse factors that
systematically influence the stock market. Our approach offers a new pathway
for economists to study and understand the systematic risks of economic and
financial systems over time.
