Unit 3 used stationarity for one series: a reasonably stable typical level, a reasonably stable amount of variation, and a dependence structure that does not keep changing in a systematic way. A multivariate stationary process extends that idea to several series at once. Not only each series, but also the relationships among the series (including lagged cross-links), stay reasonably stable over time.
Unit 3 used stationarity for one series: a reasonably stable typical level, a reasonably stable amount of variation, and a dependence structure that does not keep changing in a systematic way. A multivariate stationary process extends that idea to several series at once. Not only each series, but also the relationships among the series (including lagged cross-links), stay reasonably stable over time.
Two related economic series, such as a regional price index and a wage index, recorded monthly. If both wander upward without bound and the gap between them keeps changing regime, the pair is not behaving as a stationary multivariate process. If, after a simple transformation or differencing, the pair fluctuates around stable means and a stable co-movement, modelling is on safer ground.
Covariance at the same time is “do they move together this month?” Cross-covariance at a lag is “does today's A move with last month's B?” Both are association measures. They are not causation. Advanced matrix proofs of these quantities are not required at this level.
| Univariate stationarity | Multivariate stationarity | |
|---|---|---|
| Focus | One series: level, variation, own lags | The whole vector: own properties plus cross-series links |
| Can fail even if… | The single series wanders or changes volatility | Each series looks calmer but the relationship between them keeps shifting |
Joint models and simultaneous forecasts assume that the pattern you estimated will still be relevant next month. If the means, spreads or cross-links are still evolving, the fitted joint structure can be a poor guide to the future.
As in Unit 3, not every series needs differencing. Check plots and whether associations look stable. Do not difference automatically “because the data are multivariate.”