On time-varying factor models: Estimation and testing

Conventional factor models assume that factor loadings are fixed over a long horizon of time, which appears overly restrictive and unrealistic in applications. In this paper, we introduce a time-varying factor model where factor loadings are allowed to change smoothly over time. We propose a local version of the principal component method to estimate the latent factors and time-varying factor loadings simultaneously. We establish the limiting distributions and uniform convergence of the estimated factors and factor loadings in the standard large N and large T framework. We also propose a BIC-type information criterion to determine the number of factors, which can be used in models with either time-varying or time-invariant factor models. Based on the comparison between the estimates of the common components under the null hypothesis of no structural changes and those under the alternative, we propose a consistent test for structural changes in factor loadings. We establish the null distribution, the asymptotic local power property, and the consistency of our test. Simulations are conducted to evaluate both our nonparametric estimates and test statistic. We also apply our test to investigate Stock and Watson's (2009) U.S. macroeconomic data set and find strong evidence of structural changes in the factor loadings.