Misspecification in dynamic models

Abstract

Within the class of ARMAX models we consider the effects that omitted explanatory processes have on the dynamic shape and the exogeneity properties of economic relations. We show that, under suitable assumptions, a well defined "misspecified" ARMAX model exists and is unique. The parameters of this model depend on the joint covariance structure of the omitted and the included variables. We completely characterize the properties of the misspecified model when no restrictions are imposed on this covariance structure. Moreover, we give necessary and sufficient conditions in order that the misspecified model may preserve (I) the same prediction error as the "true" model, (II) unidirectional causation, (III) the same dynamic multipliers and dynamic shape, (IV) cointegration. Our results apply to various misspecification problems, such as measurement errors, unobserved components, aggregation over agents, systematic sampling and temporal aggregation.