Constrained Factor Models

This article considers estimation and applications of constrained and partially constrained factor models when the dimension of explanatory variables is high. Both the classical and approximate factor models are investigated. For estimation, we employ both the maximum likelihood and least squares methods. We show that the least squares estimation is based on constrained principal component analysis and provides consistent estimates for the model under certain conditions. The normality condition is not used in the derivation. We then propose likelihood ratio statistics to test the adequacy of factor constraints. The test statistic is developed under the normality assumption, but simulation results show that it continues to perform well even if the underlying distribution is Student-t. The constraints are useful tools to incorporate prior information or substantive theory in applications of factor models. In addition, the constraints also serve as a statistical tool to obtain parsimonious econometric models for forecasting, to simplify the interpretations of common factors, and to reduce the dimension. We use simulation and real examples to investigate the performance of constrained estimation in finite samples and to highlight the importance of noise-to-signal ratio in factor analysis. Finally, we compare the constrained model with its unconstrained counterpart both in estimation and in forecasting. This article has supplementary material online.