Spanning latent and observable factors

Factor analysis is a widely used tool to summarize high dimensional panel data via a small dimensional set of latent factors. Many applications in finance and macroeconomics, are often focused on observable factors with an economic interpretation. The objective of this paper is to provide a test to answer a question which naturally comes up in discussions regarding latent versus observable factors: do latent and observable factors span the same space? We derive asymptotic properties of a formal test and propose a bootstrap version with improved small sample properties. We find empirical evidence for a small number of factors common between a small number of traditional Fama–French risk factors – or returns on a few stocks (i.e. “magnificent” 5 or 7) – and large panels of US, North American and international portfolio returns.