A cross-section average-based principal components approach for fixed-T panels

© 2020 The Authors. Journal of Applied Econometrics published by John Wiley & Sons Ltd. Because of the increased availability of large panel data sets, common factor models have become very popular. The workhorse of the literature is the principal components (PC) method, which is based on an eigen-analysis of the sample covariance matrix of the data. Some of its uses are to estimate the factors and their loadings, to determine the number of factors, and to conduct inference when estimated factors are used in panel regression models. The bulk of the underlying theory that justifies these uses is based on the assumption that both the number of time periods, T, and the number of cross-section units, N, tend to infinity. This is a drawback, because in practice T and N are always finite, which means that the asymptotic approximation can be poor, and there are plenty of simulation results that confirm this. In the current paper, we focus on the typical micro panel where only N is large and T is finite and potentially very small—a scenario that has not received much attention in the PC literature. A version of PC is proposed, henceforth referred to as cross-section average-based PC (CPC), whereby the eigen-analysis is performed on the covariance matrix of the cross-section averaged data as opposed to on the covariance matrix of the raw data as in original PC. The averaging attenuates the idiosyncratic noise, and this is the reason why in CPC T can be fixed. Mirroring the development in the PC literature, the new method is used to estimate the factors and their average loadings, to determine the number of factors, and to estimate factor-augmented regressions, leading to a complete CPC-based toolbox. The relevant theory is established, and is evaluated using Monte Carlo simulations.