Explicit solutions to a vector time series model and its induced model for business cycles

This article gives the explicit solution to a general vector time series model that describes interacting, heterogeneous agents that operate under uncertainties but according to Keynesian principles, from which a model for business cycle is induced by a weighted average of the growth rates of the agents in the model. The explicit solution enables a direct simulation of the time series defined by the model and better understanding of the joint behavior of the growth rates. In addition, the induced model for business cycles and its solutions are explicitly given and analyzed. The explicit solutions provide a better understanding of the mathematics of these models and the econometric properties they try to incorporate.