Prediction of Singular VARs and an Application to Generalized Dynamic Factor Models

Vector autoregressive processes (VARs) with innovations having a singular covariance matrix (in short singular VARs) appear naturally in the context of dynamic factor models. Estimating such a VAR is problematic, because the solution of the corresponding equation systems is numerically unstable. For example, if we overestimate the order of the VAR, then the singularity of the innovations renders the Yule‐Walker equation system singular as well. We are going to show that this has a severe impact on accuracy of predictions. While the asymptotic rate of the mean square prediction error is not impacted by this problem, the finite sample behaviour is severely suffering. This effect will be reinforced, if the predictor variables are not coming from the stationary distribution of the process, but contain additional noise. Again, this happens to be the case in context of dynamic factor models. We will explain the reason for this phenomenon and show how to overcome the problem. Our numerical results underline that it is very important to adapt prediction algorithms accordingly.