Abstract

Stationarity is a very common assumption in time series analysis. A vector autoregressive process is stable if and only if the roots of its characteristic equation lie outside the unit circle, constraining the autoregressive coefficient matrices to lie in the stationary region. However, the stationary region has a highly complex geometry which impedes specification of a prior distribution. In this work, an unconstrained reparameterization of a stationary vector autoregression is presented. The new parameters are partial autocorrelation matrices, which are interpretable, and can be transformed bijectively to the space of unconstrained square matrices through a simple mapping of their singular values. This transformation preserves various structural forms of the partial autocorrelation matrices and readily facilitates specification of a prior. Properties of this prior are described along with an important special case which is exchangeable with respect to the order of the elements in the observation vector. Posterior inference and computation are described and implemented using Hamiltonian Monte Carlo via Stan. The prior and inferential procedures are illustrated with an application to a macroeconomic time series which highlights the benefits of enforcing stationarity and encouraging shrinkage toward a sensible parametric structure. Supplementary materials for this article are available online.

摘要

平稳性是时间序列分析中非常常见的假设。向量自回归过程是稳定的当且仅当其特征方程的根位于单位圆之外，约束自回归系数矩阵位于平稳区域。然而，静止区域具有高度复杂的几何形状，这阻碍了先验分布的规范。在这项工作中，提出了静止向量自回归的无约束重新参数化。新参数是可解释的偏自相关矩阵，并且可以通过奇异值的简单映射将其双射转换为无约束方矩阵空间。这种转换保留了偏自相关矩阵的各种结构形式，并且很容易促进先验的规范。此先验的属性与一个重要的特殊情况一起描述，该特殊情况可根据观察向量中元素的顺序进行交换。通过 Stan 使用 Hamiltonian Monte Carlo 描述和实现后验推理和计算。先验程序和推理程序通过应用于宏观经济时间序列来说明，该时间序列突出了强制平稳性和鼓励向合理的参数结构收缩的好处。本文的补充材料可在线获取。先验程序和推理程序通过应用于宏观经济时间序列来说明，该时间序列突出了强制平稳性和鼓励向合理的参数结构收缩的好处。本文的补充材料可在线获取。先验程序和推理程序通过应用于宏观经济时间序列来说明，该时间序列突出了强制平稳性和鼓励向合理的参数结构收缩的好处。本文的补充材料可在线获取。