Vector autoregressive moving average models

Abstract Vector autoregressive moving average (VARMA) processes constitute a flexible class of linearly regular processes with a wide range of applications. In many cases VARMA models allow for a more parsimonious parametrization than vector autoregressive (VAR) models. However, compared to VAR processes the relation between internal parameters and external characteristics (e.g., the autocovariance function) is more involved and estimation is harder since in general the maximum likelihood method here needs numerical optimization. In this contribution we want to give a broad overview of VARMA modeling with an emphasis on structure theory, estimation and practical implementation with the free software environment R and specialized R packages. First we present basic definitions and the interrelation between VAR, VARMA models and state space models. We will show how to compute important characteristics like autocovariance function, spectral density and impulse response functions and how to compute predictions. Then we discuss parametrization issues, including the question how to implement structural information. As mentioned above, estimation of VARMA models is quite involved. Consequently a substantial part of the paper will deal with maximum likelihood estimation and with alternative estimators which are cheaper to compute, but in general not asymptotically efficient. In addition to parameter estimation, model selection, in particular, choosing the best model order is treated.