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) findings 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 findings 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 differential 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% confidence 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 confidence level to 90%, the intersection of the stable region and the Cartesian product of the confidence intervals became empty, thereby suggesting rejection of stability. But a formal sampling theoretic hypothesis test of that null would be very difficult 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 difficult to implement. A new formula is also given for finding the closed forms of transcritical bifurcation boundaries. Finally, effects of fiscal policy on stability are considered. It is found that change in fiscal policy may affect the stability of the continuous time macroeconometric models. But we find that the selection of an advantageous stabilization policy is more difficult 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 insignificant effect on structural parameter values.
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