Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with $q<r$ . The present paper studies cointegration and error correction representations for an $I\left(1\right)$ singular stochastic vector ${\mathbf{y}}_t$ . It is easily seen that ${\mathbf{y}}_t$ is necessarily cointegrated with cointegrating rank $c\ge r-q$ . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to $I\left(1\right)$ singular vectors under the assumption that ${\mathbf{y}}_t$ has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of ${\mathbf{y}}_t$ has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

A popular validation procedure for Dynamic Stochastic General Equilibrium (DSGE) models consists in comparing the structural shocks and impulse-response functions obtained by estimation-calibration of the DSGE with those obtained in an Structural Vector Autoregressions (SVAR) identified by means of some of the DSGE restrictions. I show that this practice can be seriously misleading when the variables used in the SVAR contain measurement errors. If this is the case, for generic values of the parameters of the DSGE, the shocks estimated in the SVAR are not “made of” the corresponding structural shocks plus measurement error. Rather, each of the SVAR shocks is contaminated by noncorresponding structural shocks. We argue that High-Dimensional Dynamic Factor Models are free from this drawback and are the natural model to use in validation procedures for DSGEs.

A survey is provided dealing with the formulation of modelling problems for dynamic factor models, and the various algorithm possibilities for solving these modelling problems. Emphasis is placed on understanding requirements for the handling of errors, noting the relevance of the proposed application of the model, be it for example prediction or business cycle determination. Mixed frequency problems are also considered, in which certain entries of an underlying vector process are only available for measurement at a submultiple frequency of the original process. Certain classes of processes are shown to be generically identifiable, and others not to have this property.