Reconciling the Theory of Factor Sequences

Factor Sequences are stochastic double sequences $(y_{it}: i \in \mathbb N, t \in \mathbb Z)$ indexed in time and cross-section which have a so called factor structure. The name was coined by Forni et al. 2001, who introduced dynamic factor sequences. We show the difference between dynamic factor sequences and static factor sequences which are the most common workhorse model of econometric factor analysis building on Chamberlain and Rothschild (1983), Stock and Watson (2002) and Bai and Ng (2002). The difference consists in what we call the weak common component which is spanned by a potentially infinite number of weak factors. Ignoring the weak common component can have substantial consequences for applications of factor models in structural analysis and forecasting. We also show that the dynamic common component of a dynamic factor sequence is causally subordinated to the output under general conditions. As a consequence only the dynamic common component can be interpreted as the projection on the common structural shocks of the economy whereas the static common component models the contemporaneous co-movement.