Qualitative Matrices and Information

This paper proposes a method for assessing the information content and validity of a mathematical structural model for which only the directions of influence among its endogenous and exogenous variables are known, as expressed by the sign patterns of associated arrays. The traditional literature on this issue presents extremely restrictive conditions under which such a “qualitative analysis” can be conducted. As a result, there have been very few successful applications of the traditional method. We propose a means of vastly expanding the scope of such an analysis to virtually any applied model. Our method works with the restrictions found for the sign patterns of complete rows and columns, or even the entire sign pattern, of the reduced form, rather than only individual entries. The information provided by the model is measured by the Shannon entropy of the possible sign patterns of the reduced form and the frequency of occurrence of each possibility. An example of the method is provided for Klein’s Model I. Although this model has been used for over fifty years for a variety of purposes, we found that the sign pattern of the estimated, unrestricted reduced form from the original data set was not consistent with the proposed, structural directions of influence among the model’s variables.