多元整数值GARCH模型的统计分析

Modeling integer-valued time series data is one of the hot topics in modern statistical research, current research focuses on two kinds of one-dimensional models: integer-valued autoregressive models and integer-valued GARCH models, and the latter have better performances in modeling volatility in data. Because of inherent characteristics of data (discreteness and correlation) and difficulties in dealing with multivariate data, research on multivariate integer-valued time series is still on the initial stage, and there is only a few research related to multivariate integer-valued autoregressive models. Based on success of one-dimensional integer-valued GARCH models and demand of multivariate models, this project constructs three classes multivariate integer-valued GARCH models based on multivariate discrete distributions, Copula functions and factor structures, respectively. For each class model, we will establish stability (stationarity, ergodicity or weak dependence), and consider parameter estimating methods (including conditional least square, maximum likelihood, empirical likelihood, Bayesian and other methods), and discuss large-sample properties of estimators (including consistency and asymptotic normality). We will also consider hypothesis test, model selection and forecasting and other issues, and generalize the results for the latter two classes to large-dimensional cases via introducing the sprase structures. We evaluate modeling performances between the newly proposed models and existing models via simulations and real-data examples. These new results will be applied to real-life practices, and provide modeling guides for practitioners.

建模整数值时间序列数据是当今统计学研究中的一个国际热点问题，已有的研究集中在两类一元模型：整数值自回归模型和整数值GARCH模型，并且后者能更好地建模数据的波动性。由于数据本身的特征(离散性和相关性)和处理多元数据的困难性,多元整数值时间序列的研究还处于起步阶段，只有很少的关于多元整数值自回归模型的研究。基于一元整数值GARCH模型的成功和多元模型的需求，本项目分别构造基于多元离散分布、Copula函数和因子结构的三类多元整数值GARCH模型。针对每类模型，首先建立稳定性质(平稳性、遍历性或弱相依性)，然后考虑多种参数估计方法，并讨论估计量的大样本性质(相合性和渐近正态性)。考虑模型的假设检验、模型选择和预测等问题，并通过引入稀疏结构将后两类模型推广到大维情形。利用模拟数据和实际数据，评判提出的模型和已有模型的优劣。将模型的统计推断结果应用于生产实践中，为实际工作者提供建模指南。

建模整数值时间序列数据是当今统计学研究中的一个国际热点问题，一元模型还有许多重要待解决的问题，而多元整数值时间序列的研究还处于起步阶段。本项目研究了一元整数值自回归模型和整数值GARCH模型的模型推广、新的估计方法和检验方法。基于一元整数值GARCH模型的成功和多元模型的需求，本项目构造基于二元离散分布的整数值GARCH模型，建立了稳定性质(平稳性和遍历性)，考虑了参数估计，并讨论估计量的大样本性质(相合性和渐近正态性)。利用模拟数据和实际数据，评判提出的模型和已有模型的优劣。将模型的统计推断结果应用于生产实践中，为实际工作者提供建模指南。本项目共正式或在线发表SCI论文24篇。