The Canonical Decomposition of Factor Models: Weak Factors are Everywhere

There are two approaches to time series approximate factor models: the static factor model, where the factors are loaded contemporaneously by the common component, and the Generalised Dynamic Factor Model, where the factors are loaded with lags. In this paper we derive a canonical decomposition which nests both models by introducing the weak common component which is the difference between the dynamic- and the static common component. Such component is driven by potentially infinitely many non-pervasive weak factors which live in the dynamically common space (not to be confused with rate-weak factors, being pervasive but associated with a slower rate). Our result shows that the relation between the two approaches is far more rich and complex than what usually assumed. We exemplify why the weak common component shall not be neglected by means of theoretical and empirical examples. Furthermore, we propose a simple estimation procedure for the canonical decomposition. Our empirical estimates on US macroeconomic data reveal that the weak common component can account for a large part of the variation of individual variables. Furthermore in a pseudo real-time forecasting evaluation for industrial production and inflation, we show that gains can be obtained from considering the dynamic approach over the static approach.