multivariate time series analysis with interactions among multiple dependent variables

Question

I have been playing with time series data and using models like facebook prophet which uses time series decomposition and models the signal using trend, seasonalities and holidays.

The stuff I found about these time series methods use time as the independent variable and a scalar value as the dependent variable. What about time series models with multiple y-values.

As an example, in my case I am tracking people and I track the counts of people but also the average speed at which they move through a particular section of a shop. Also, there may also be a correlation between these quantities. Are there established time series modelling techniques which can use such multi variate and non independent data sources?

Answer 1

I am not sure I fully got your question. But if I did, and you are asking about a multivariate time series, well you have many solutions. Take the following as an introduction more than a fully comprehensive answer.

The models that, among the others, take into account linear interactions between variables that are relevant to their conditional means are VAR models. Then if you assume that residuals are not independent and there may be some dependencies in the conditional second moments of the series, then you can combine a VAR model for conditional means with a multivariate GARCH allowing you to model dependencies in the conditional second moment here for an introduction and then here. Then, if you want a very broad and comprehensive specification of a multivariate model, that may include external regressors, constraints, time-varying coefficients etc, in addition to the above mentioned specifications for conditional means and covariances, then see MARSS