Latent ancestral graph of structure vector autoregressive models

Abstract

Abstract: A class of latent ancestral graph for modelling thedependence structure of structural vector autoregressive (VAR)model affected by latent variables is proposed. The graphs aremixed graphs with possibly two kind of edges, namely directed andbidirected edges. The vertex set denotes random variables at dif-ferent times. In Gaussian case, the latent ancestral graph leads toa simple parameterization model. A modified iterative conditionalfitting algorithm is presented to obtain maximum likelihood esti-mation of the parameters. Furthermore, a log-likelihood criterionis used to select the most appropriate models. Simulations areperformed using illustrative examples and results are provided todemonstrate the validity of the methods. Keywords: multivariate time series, latent ancestral graph, itera-tive conditional fitting. DOI: 10.3969/j.issn.1004-4132.2010.02.010 1. Introduction Graphical models have become an important tool for an-alyzing multivariate data and recently have been appliedalso to stationary multivariate time series [1–4]. One ma-jor problem in the application of graphical models is thepossible presence of latent variables that affect the ob-served variables and thus lead to false dependence struc-ture when we establish a model for observed variables.Richardson, et al [5] proposed an ancestral graph to rep-resent the dependence among the observed variables. Fortime series, Eichler [6] first proposed latent a graphi-cal modelling approach for the discussion of causal re-lationships in systems that are affected by latent vari-ables. Recently, Chu, et al [7] described a feasible pro-cedureforlearningaclass ofadditivenonlineartimeseries