Dynamic Factor Models, Cointegration, and Error Correction Mechanisms

The paper studies Non-Stationary Dynamic Factor Models such that: (1) the factors $\mathbf F_t$ are $I(1)$ and singular, i.e. $\mathbf F_t$ has dimension $r$ and is driven by a $q$-dimensional white noise, the common shocks, with $q < r$, (2) the idiosyncratic components are $I(1)$. We show that $\mathbf F_t$ is driven by $r-c$ permanent shocks, where $c$ is the cointegration rank of $\mathbf F_t$, and $q-(r-c)<c$ transitory shocks, thus the same result as in the non-singular case for the permanent shocks but not for the transitory shocks. Our main result is obtained by combining the classic Granger Representation Theorem with recent results by Anderson and Deistler on singular stochastic vectors: if $(1-L)\mathbf F_t$ is singular and has rational spectral density then, for generic values of the parameters, $\mathbf F_t$ has an autoregressive representation with a finite-degree matrix polynomial fulfilling the restrictions of a Vector Error Correction Mechanism with $c$ error terms. This result is the basis for consistent estimation of Non-Stationary Dynamic Factor Models. The relationship between cointegration of the factors and cointegration of the observable variables is also discussed.