Polynomial Factor Models

Polynomial factor models (henceforth, PFM) represent a new class of factor models for high-dimensional panel data. We develop several econometric theories for factor models of latent factor interactions. Unlike approximate factor models (AFM), which are based on linear combinations of observed variables, PFM is based on polynomial combinations of latent factors. We show that PFM can capture nonlinear relationships between latent factors and observed variables and can be used to estimate latent factor interactions. We propose a general estimation approach for the PFM, labeled Polynomial Factor Analysis (PFA), to consistently estimate all the factor interaction-dependent factors and loadings. Their asymptotic distributions are established, and the proposed estimator is shown to be consistent and asymptotically regular. Simulation results demonstrate the effectiveness of the proposed method, and the empirical results in macroeconomic aggregates and asset pricing show that PFA can provide more accurate estimations than the existing methods