Multivariate Count Autoregressive Models and their Assessment

统计时间序列数据出现在经济和社会统计领域的各种应用中。例如，国家统计局(ONS)发布了关于商业活动和人口、劳动力市场状况、人口和人口等的时间序列数据集。这些数据大部分是由计数组成的多变量整数值时间序列，并且相当缺乏适当分析它们的方法。此外，英国国家统计局和更广泛的政府统计局正在探索与传统调查整合或取代传统调查的新数据来源，以增进对英国经济、社会和人口的了解，时间序列分析发挥着重要作用。对于整数值时间序列的分析，通常很难强加一个完整的参数模型来进行推断，因为文献中有许多可用的多变量泊松分布的版本。即使确定了一个合适的多变量模型，所涉及的计算通常也很繁琐。此外，上次更新于2018年8月的参数模型可能会遗漏数据中可能存在的真实相关形式(例如，考虑车祸数量或按邻近地区分列的失业人数)，并且不能边缘地描述在计数数据中通常发现的过度分散现象。这些点和其他点在Fokianos等人(2018)最近的工作中得到了彻底的讨论，他提供了一个框架，用于为多变量计数时间序列建立观测驱动的自回归线性和对数线性模型。这些作者的观点是基于麦卡拉和奈尔德(1989)所倡导的广义线性模型的方法论。Fokianos等人(2018)提出了一种数据生成过程，该过程不一定施加边缘的泊松假设，但所提出的模型的结构保持简单。未知矩阵参数的估计是由拟极大似然估计(QMLE，见Heyde(1997))实现的，并且在较温和的条件下，这些估计量具有良好的性质。这项工作将为进一步开发基于真实国家统计局数据的多变量计数自回归方法奠定基础。似然推理和广义线性模型的结合为定量和定性时间序列数据的分析提供了一个系统的框架。事实上，评估、诊断、模型评估和预测都是以一种简单的方式实现的，计算可以很容易地开发出来。

Count time series data are found in diverse applications arising in the field of Economic and Social Statistics. For example, the Office of National Statistics (ONS), publish time series data sets on business activity and demographics, labour market status, people and population, and so on. Much of these data are multivariate integer-valued time series that consist of counts and there is considerable lack of methodology for their proper analysis. Moreover, new data sources to integrate with or replace traditional surveys are being explored by ONS and the wider Government Statistical Service to improve understanding of the UK's economy, society and population, with time series analysis playing an important role. For analysing integer-valued time series, it is usually hard to impose a full parametric model for developing inference, as there are many available versions of the multivariate Poisson distribution in the literature. Even if one identifies a suitable multivariate model, the computations involved are usually cumbersome. In addition, a parametric model mightLast Updated August 2018miss the true correlation form that might exist in the data (consider, for example, number of car accidents or number of unemployed people by neighbourhing regions) and not be able to describe marginally the phenomenon of overdispersion that is usually found in count data. These points, and others, have been thoroughly discussed in the recent work by Fokianos et al (2018) who provides a framework for building observation-driven autoregressive linear and log-linear models for multivariate count time series. The point of view of these authors is based on generalized linear models' methodology as advocated by McCullagh and Nelder (1989). Fokianos et al (2018) have suggested a data generating process that does not necessarily impose marginally a Poisson assumption, yet the structure of the proposed models is kept simple. Estimation of unknown matrix parameters is implemented by Quasi Maximum Likelihood Estimation (QMLE, see Heyde (1997)) and, under mild conditions, it is shown that these estimators possess good properties. This work will be the basis for developing further methodology for multivariate count autoregressions motivated by real ONS data. The combination of likelihood inference and generalized linear models provide a systematic framework for the analysis of quantitative as well as qualitative time series data. Indeed, estimation, diagnostics, model assessment, and forecasting are implemented in a straightforward manner where computations can be easily developed.