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.

中文翻译：

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奇异VAR的预测及其在广义动态因子模型中的应用

在具有动态因子模型的情况下，具有奇异协方差矩阵（简称奇异VAR）的创新的向量自回归过程（VAR）自然出现。由于相应方程组的解在数值上是不稳定的，因此估计这种VAR是有问题的。例如，如果我们高估了VAR的阶数，则创新的奇异性也会使Yule-Walker方程组也变得奇异。我们将证明这对预测的准确性有严重的影响。尽管均方根预测误差的渐近率不受此问题的影响，但有限样本行为却受到严重影响。如果预测变量不是来自过程的平稳分布，而是包含额外的噪声，则将增强这种效果。再次，在动态因素模型的情况下恰好是这种情况。我们将解释这种现象的原因，并说明如何解决该问题。我们的数值结果表明，相应地调整预测算法非常重要。