Vector autoregressions (VARs) are a widely used tool for modelling multivariate time-series. It is common to assume a VAR is stationary; this can be enforced by imposing the stationarity condition which restricts the parameter space of the autoregressive coefficients to the stationary region. However, implementing this constraint is difficult due to the complex geometry of the stationary region. Fortunately, recent work has provided a solution for autoregressions of fixed order $p$ based on a reparameterization in terms of a set of interpretable and unconstrained transformed partial autocorrelation matrices. In this work, focus is placed on the difficult problem of allowing $p$ to be unknown, developing a prior and computational inference that takes full account of order uncertainty. Specifically, the multiplicative gamma process is used to build a prior which encourages increasing shrinkage of the partial autocorrelations with increasing lag. Identifying the lag beyond which the partial autocorrelations become equal to zero then determines $p$. Based on classic time-series theory, a principled choice of truncation criterion identifies whether a partial autocorrelation matrix is effectively zero. Posterior inference utilizes Hamiltonian Monte Carlo via Stan. The work is illustrated in a substantive application to neural activity data to investigate ultradian brain rhythms.

翻译：向量自回归模型是多元时间序列建模中广泛使用的工具。通常假设VAR模型是平稳的；这可以通过施加平稳性条件来实现，该条件将自回归系数的参数空间限制在平稳区域内。然而，由于平稳区域几何结构的复杂性，实施这种约束十分困难。幸运的是，近期研究基于一组可解释且无约束的变换偏自相关矩阵重新参数化，为固定阶数$p$的自回归模型提供了解决方案。本研究聚焦于允许$p$未知这一难题，开发了充分考虑阶数不确定性的先验分布与计算推断方法。具体而言，采用乘性伽马过程构建先验分布，该先验会随着滞后阶数增加而增强对偏自相关系数的收缩效应。通过识别偏自相关系数等于零的临界滞后阶数即可确定$p$值。基于经典时间序列理论，通过严谨的截断准则可判定偏自相关矩阵是否有效为零。后验推断采用基于Stan的哈密顿蒙特卡洛方法实现。本研究通过神经活动数据的实质性应用进行演示，以探究超昼夜脑节律现象。