Modelling Multivariate Time Series: Vector Autoregressions and Granger Causality

Abstract

So far our focus has just been on modelling individual time series but we now extend the analysis to multivariate models. To develop methods of modelling a vector of time series, consider again the AR(1) process, now written for the stationary series y _t and with a slightly different notation to that used before:

$$$$

(7.1)

The standard dynamic regression model adds exogenous variables, perhaps with lags, to the right-hand side of (7.1); to take the simplest example of a single exogenous variable x _t having a single lag, consider

$$$$

(7.2)

Again, note the change of notation as coefficients and innovations will not, in general, be the same across (7.1) and (7.2): c and a will differ from θ and ϕ, as will the variance of e _t , σ² _e , differ from that of a_t,σ²_a with, typically σ²_e < σ² _a if the additional coefficients b₀and b₁ are non-zero.