On the existence of macro variables and of macro relationships

A common practice is to restrict the implied definition of a macro variable to an aggregate of micro components. Using such a definition, the aggregation problem is to determine when one can interpret macro coefficient estimates in terms of micro economic theory and when one can recover micro coefficients from macro estimates. In this paper it is argued that a more useful abstract definition of a macro variable is in terms of properties of dynamical systems as a whole; such a definition may include aggregates of micro components as relevant macro variables, but they are not necessarily included and system-wide variables are included in this definition. The idea of macro variables and their relationships is claimed to be more complex than a mere aggregation of micro components. An operational definition of a macro variable is given that defines macro relationships as well. The operational definition is based on examining the time variation of the transitional probabilities of system variables. One practical conclusion is that in many cases an analysis of a system variable, such as mean income or consumption, may require examining the variances of those variables simultaneously. More precisely, even to a first order of approximation the time varying variances are of equal importance to the time varying means and one must estimate the interactive effect of both together. Further, the results in this paper suggest a deep relationship between the ubiquitous observance of ARCH processes and the nonlinearity in the Fokker-Planck equation that approximates the time variation in the probability densities of macro variables.