Model selection with factors and variables

This paper provides consistent information criteria for the selection of forecasting models which use a subset of both the idiosyncratic and common factor components of a big dataset. This hybrid model approach has been explored by recent empirical studies to relax the strictness of pure factor-augmented model approximations, but no formal model selection procedures have been developed. The main difference to previous factor-augmented model selection procedures is that we must account for estimation error in the idiosyncratic component as well as the factors. Our first contribution shows that this combined estimation error vanishes at a slower rate than in the case of pure factor-augmented models in circumstances in which N is of larger order than sqrt(T), where N and T are the cross-section and time series dimensions respectively. Under these circumstances we show that existing factor-augmented model selection criteria are inconsistent, and the standard BIC is inconsistent regardless of the relationship between N and T. Our main contribution solves this issue by proposing new information criteria which account for the additional source of estimation error, whose properties are explored through a Monte Carlo simulation study. We conclude with an empirical application to long-horizon exchange rate forecasting using a recently proposed model with country-specific idiosyncratic components from a panel of global exchange rates.