Vector Autoregressive Moving Average Processes

Abstract

In this chapter, we extend our standard finite order VAR model,

$$y_t = \nu + A_1 y_{t - 1} + \ldots + A_p y_{t - p} + \varepsilon _t , $$

by allowing the error terms, here ε_t, to be autocorrelated rather than white noise. The autocorrelation structure is assumed to be of a relatively simple type so that ε_t has a finite order moving average (MA) representation,

$$\varepsilon _t = u_t + M_1 u_{t - 1} + \ldots + M_q u_{t - q} ,$$

where, as usual, u _t is zero mean white noise with nonsingular covariance matrix Σ_u. A finite order VAR process with finite order MA error term is called a VARMA (vector autoregressive moving average) process.