Vector Generalized Linear Time Series Models with an Implementation in R

Abstract

Since the introduction of the ARMA-class in the early 1970s many time series (TS) modelling extensions have been proposed involving linear and non-linear structures as part of a huge literature, for instance, the vector-ARMA class for multivariate TS and the ARCH-GARCH-type models for heteroskedasticity. The result has been an explosion of TS models and inference schemes having pockets of substructure but limited overriding framework. In this work, the class of vector generalized linear models (VGLMs) is shown to confer advantages towards data-types with trend and patterns that evolve over time. Specifically, we propose a new class of VGLMs, called vector generalized linear time series model (VGLTSMs), which are endowed with capabilites to handle autocorrelated data and can be thought of as multivariate generalized linear models directed towards time series. This work follows previous successful endeavours of developing VGLMs for other data types, such as categorical data, extremes, and quantile regression. The crucial VGLM ideas are constraint matrices, vector responses and covariate-specific linear predictors, and estimation by iteratively reweighted least squares and Fisher scoring. The only modification to the VGLM framework is to constrain its log-likelihood to depend on deterministic time-dependent data. We show how several popular sub-classes of TS models are accommodated as special cases of VGLTSMs, as well as new work that broadens TS modelling even more. A prominent example is cointegrated time series that is to be shown amenable to VGLTSMs by means of its ability to handle multiple responses. This work is accompanied with a new software implementation in R, called the VGAMextra package, which is available on CRAN. Its performance is compared to other software for TS analysis. Algorithmic details of its implementation, as well as many VGLTSM modelling features allowed by VGAMextra, are described here.