The vector autoregression (VAR) has been widely used in system identification, econometrics, natural science, and many other areas. However, when the state dimension becomes large the parameter dimension explodes. So rank reduced modelling is attractive and is well developed. But a fundamental requirement in almost all applications is stability of the fitted model. And this has not been addressed in the rank reduced case. Here, we develop, for the first time, a closed-form formula for an estimator of a rank reduced transition matrix which is guaranteed to be stable. We show that our estimator is consistent and asymptotically statistically efficient and illustrate it in comparative simulations.

翻译：向量自回归(VAR)模型已广泛应用于系统辨识、计量经济学、自然科学等诸多领域。然而，当状态维度增大时，参数维度会呈指数级增长。因此，降秩建模方法因其吸引力而得到充分发展。但几乎所有应用场景中都有一个基本要求：拟合模型必须保持稳定性。这一问题在降秩情形下尚未得到解决。本文首次推导出保证稳定性的降秩转移矩阵估计量的闭式解公式。我们证明了该估计量具有一致性和渐近统计有效性，并通过对比仿真实验验证了其性能。