Factor and Idiosyncratic Empirical Processes

The distributions of the common and idiosyncratic components for an individual variable are important in forecasting and applications. However, they are not identified with low-dimensional observations. Using the recently developed theory for large dimensional approximate factor model for large panel data, the common and idiosyncratic components can be estimated consistently. Based on the estimated common and idiosyncratic components, we construct the empirical processes for estimation of the distribution functions of the common and idiosyncratic components. We prove that the two empirical processes are oracle efficient when T = o(p) where p and T are the dimension and sample size, respectively. This demonstrates that the factor and idiosyncratic empirical processes behave as well as the empirical processes pretending that the common and idiosyncratic components for an individual variable are directly observable. Based on this oracle property, we construct simultaneous confidence bands (SCBs) for the distributions of the common and idiosyncratic components. For the first-order consistency of the estimated distribution functions, T=o(p)$\sqrt{T} =o(p)$ suffices. Extensive simulation studies check that the estimated bands have good coverage frequencies. Our real data analysis shows that the common-component distribution has a structural change during the crisis in 2008, while the idiosyncratic-component distribution does not change much. Supplementary materials for this article are available online.