Univariate time series models like the ARMA approach, we just discussed are limited to statistical relationships between a target variable and its lagged values or lagged disturbances and exogenous series in the ARMAX case. In contrast, multivariate time series models also allow for lagged values of other time series to affect the target. This effect applies to all series, resulting in complex interactions, as illustrated in the following diagram:

In addition to potentially better forecasting, multivariate time series are also used to gain insights into cross-series dependencies. For example, in economics, multivariate time series are used to understand how policy changes to one variable, for example, an interest rate, may affect other variables over different horizons. The impulse-response function produced by the multivariate model we will look at serves this purpose and allows us to simulate how one variable responds to a sudden change in other variables. The concept of Granger causality analyzes whether one variable is useful in forecasting another (in the least squares sense). Furthermore, multivariate time series models allow for a decomposition of the prediction error variance to analyze how other series contribute.