VAR modeling
The AR(p), MA(q), ARMA(p,q), ARIMA(p,d,q)m, and SARIMA(p,d,q) models we looked at in the last chapter form the basis of multivariate VAR modeling. In this chapter, we have discussed ARIMA with exogenous variables (ARIMAX). We will now begin discussion on the VAR model. First, it is important to understand that while ARIMAX requires leading (future) values of the exogenous variables, no future values of these variables are required for the VAR model as they are all autoregressive to each other – hence the name vector autoregressive – and by definition not exogenous. To start, let us consider the two-variable, or bivariate, case. Consider a process y t that is the output of two different input variables, y t1 and y t2. Note that in matrix form, we are discussing the case of an nxm matrix (y n,m) where n corresponds to the point in time and m corresponds to the variables involved (variables 1,2, … , m). We exclude the comma from notation...