Sieve estimation of time-varying factor loadings: Estimating the conditional CAPM

This paper introduces a high-dimensional factor model with time-varying factor loadings. We show that both the factors and the time-varying loadings can be consistently estimated without rotations. We also propose a model-selection approach to determine the constancy of each factor loading for each cross-section. Theoretical results are supported by a simulation study. Applying the proposed methodology to the portfolio returns over the past 50 years, we find that there is only one factor for the most part of the sample period, and the number of factors tends to increase before or during crisis time. We also find evidence of time variations in most portfolio betas. The estimated betas of the portfolios formed with small firms resemble the log consumption-aggregate wealth ratio, which has been used as the conditioning variable in the literature of condition CAPM.