Circularly Projected Common Factors for Grouped Data

To extract the common factors from grouped data, multilevel factor models have been put forward in the literature, and methods based on iterative principal component analysis (PCA) and canonical correlation analysis (CCA) have been proposed for estimation purpose. While iterative PCA requires iteration and is hence time-consuming, CCA can only deal with two groups of data. Herein, we develop two new methods to address these problems. We first extract the factors within groups and then project the estimated group factors into the space spanned by them in a circular manner. We propose two projection processes to estimate the common factors and determine the number of them. The new methods do not require iteration and are thus computationally efficient. They can estimate the common factors for multiple groups of data in a uniform way, regardless of whether the number of groups is large or small. They not only overcome the drawbacks of CCA but also nest the CCA method as a special case. Finally, we theoretically and numerically study the consistency properties of these new methods and apply them to studying international business cycles and the comovements of retail prices.