基于Bayes推断的ARIMAX模型的建模分析

Modeling Analysis of ARIMAX Model Based on Bayes Inference

摘要: ARIMAX模型在多个领域有着重要应用。但是针对ARIMAX模型的参数估计均是经典统计方法，利用贝叶斯估计是一个值得探究的问题。考虑到共轭先验分布的性质特点，对于ARIMAX模型中的系数的先验分布为正态分布，噪声项的方差先验分布为逆伽马分布假定，本文给出了参数的后验分布，并使用Gibbs采样的方式，给出各个参数的一个估计。模拟试验的结果表明，本文的估计方法具有很好的功效，借助参数的迭代图表明文中使用的方法具有稳健性。使用创业板数据与上证数据做出实证分析，并发现文中给出的方法不仅具有很好的解释性，同时能够提取出完整的数据信息。

Abstract: The ARIMAX model has important applications in various fields. However, parameter estimation for the ARIMAX model has traditionally relied on classical statistical methods, and exploring Bayesian estimation presents a worthwhile direction. Considering the properties of conjugate prior distributions, this study assumes a normal prior distribution for the coefficients in the ARIMAX model and an inverse gamma prior for the variance of the noise term. The posterior distributions of the parameters are derived, and estimates for each parameter are obtained using Gibbs sampling. Simulation results demonstrate the effectiveness of the proposed estimation method. Iterative plots of the parameters indicate the robustness of the method. An empirical analysis using data from the ChiNext Index and the Shanghai Composite Index further reveals that the proposed method not only offers strong interpretability but also effectively captures comprehensive information from the data.