There has been an enormous interest in analysing and modelling periodic time series. The research on periodically integrated autoregressive (PIAR) models which capture the periodic structure and the presence of unit roots is widely applied in environmental, financial and energy areas. In this paper, we propose a multi-companion method which uses the eigen information of the multi-companion matrix in the multi-companion representation of PIAR models. The method enables the estimation and forecasting of PIAR models with a single, two and multiple unit roots. We show that the parameters of PIAR models can be represented in terms of the eigen information of the multi-companion matrix. Consequently, the estimation can be conducted using the eigen information, rather than directly estimating the parameters of PIAR models. A Monte Carlo experiment and an application are provided to illustrate the robustness and effectiveness of the multi-companion method.

翻译：分析和建模周期时间序列一直备受关注。捕获周期结构及单位根存在的周期积分自回归（PIAR）模型在环境、金融和能源领域具有广泛应用。本文提出一种多伴随方法，利用PIAR模型多伴随表示中多伴随矩阵的特征信息。该方法能够对含有单个、两个及多个单位根的PIAR模型进行估计与预测。研究表明，PIAR模型参数可通过多伴随矩阵的特征信息表示。因此，估计可基于特征信息进行，而无需直接估计PIAR模型参数。通过蒙特卡洛实验与应用实例，验证了多伴随方法的稳健性与有效性。