The Canonical Decomposition of Factor Models: Weak Factors are Everywhere
Philipp Gersing, Matteo Barigozzi, Christoph Rust, Manfred Deistler
TL;DR
The paper addresses how static and dynamic factor models relate in high-dimensional time series, introducing a canonical threefold decomposition of $y_{it}$ into a static common component $C_{it}$, a weak common component $e_{it}^ hi$, and a dynamic idiosyncratic term $ _{it}$, with $e_{it}^ hi= hi_{it}-C_{it}$. It reframes the problem in terms of dynamic and static aggregation spaces and proves that the static aggregation space is contained in the dynamic one, enabling a unified representation where weak factors live in the dynamically common space. The authors develop a practical estimation procedure combining dynamic PCA (or spectral-density-based dynamic PCA) to estimate $chi$, followed by static PCA on $chi$ to obtain $C$, with the residual forming the weak component; they establish consistency with explicit rates and illustrate the approach via simulations and a US macroeconomic data application (FRED-MD). Empirically, the weak component explains a meaningful share of variation across many series and improves forecasting relative to the conventional static diffusion-index approach, highlighting the importance of accounting for weak, non-pervasive factors in high-dimensional panels. Overall, the work extends the traditional factor-model toolkit by clarifying the complex link between static and dynamic representations and providing actionable estimation tools for the canonical decomposition and its components.
Abstract
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.