Center Manifold, Stability, and Bifurcations in Continuous Time Macroeconometric Systems

In a recent paper, we studied bifurcation phenomena in continuous time macroeconometric models. The objective was to explore the relevancy of Grandmont's (1985) ndings to models permitting more reasonable elasticities than were possible in Grandmont's Cobb Douglas overlapping generations model. Another objective was to explore the relevancy of his ndings to a model in which some solution paths are not Pareto optimal, so that policy rules can serve a clearly positive purpose. We used the Bergstrom, Nowman, and Wymer (1992) UK continuous time second order di erential equations macroeconometric model that permits closer connection with economic theory than is possible with most discrete time structural macroeconometric models. We do not yet have the ability to explore these phenomena in a comparably general Euler equations model having deep parameters, rather than structural parameters. It was discovered that the UK model displays a rich set of bifurcations including transcritical bifurcations, Hopf bifurcations, and codimension two bifurcations. The point estimates of the parameters are in the unstable region. But we did not test the null hypothesis that the parameters are actually in the stable region. In addition, we did not investigate the dynamical properties on the bifurcation boundaries; and we did not investigate the relevancy of stabilization policy rules. In this paper, we further examine the stability properties and bifurcation boundaries of the UK continuous time macroeconometric models by analyzing the stability of the model along center manifolds. The results of this paper show that the model is unstable on bifurcation boundaries for those cases we consider. Hence calibration of the model to operate on those bifurcation boundaries would produce no increase in the model's ability to explain observed data. However, we have not yet determined the dynamic properties of the model on the Hopf bifurcation boundaries, which sometimes do produce useful dynamical properties for some models. Of more immediate interest, it is also shown that bifurcations exist within the Cartesian product of 95% con dence intervals for the estimators of the individual parameters. This seems to suggest that we cannot reject the null hypothesis of stability, despite the fact that the point estimates are in the unstable region. However, when we decreased the con dence level to 90%, the intersection of the stable region and the Cartesian product of the con dence intervals became empty, thereby suggesting rejection of stability. But a formal sampling theoretic hypothesis test of that null would be very di cult to conduct, since some of the sampling distributions are truncated by boundaries, and since there are some corner solutions. A Bayesian approach might be possible, but would be very di cult to implement. A new formula is also given for nding the closed forms of transcritical bifurcation boundaries. Finally, e ects of scal policy on stability are considered. It is found that change in scal policy may a ect the stability of the continuous time macroeconometric models. But we nd that the selection of an advantageous stabilization policy is more di cult than expected. Augmentation of the model by feedback policy rules chosen from plausible economic reasoning can contract the stable region and thereby be counterproductive, even if the policy is time consistent and has insigni cant e ect on structural parameter values.