I would like to stay away from finance and economic examples since they are too abstract for me to understand. Are there any "real world" examples for example healthcare, exam marks, environmental science and so on? From a managerial perspective I would like a basic understanding of how VAR could potentially be used to model real world phenomenon.
Do not get hung up by the Finance or Economics tags.
VAR is a time-series technique. This bears repeating: time-series, and nothing but. Its key innovation, back in the day, was that it removed all underlying domain-specific information encoded in a model and reduced it to pure time series interaction: in the simplest case, does the past of series A and B affect either one or both of A and B? If one series is shocked, how long does a change persist etc. Economics, particularly macroeconomics, simply became a big application area as there are often time series that are interrelated, and a model-free way (in the VAR sense) can not only be useful for model fitting but also for forecasts from these models.