Empirical assessment of bifurcation regions within new Keynesian models

Barnett, William A. and Duzhak, Evgeniya A. (2008): Empirical assessment of bifurcation regions within new Keynesian models.

Abstract

As is well known in systems theory, the parameter space of most dynamic models is stratified into subsets, each of which supports a different kind of dynamic solution. Since we do not know the parameters with certainty, knowledge of the location of the bifurcation boundaries is of fundamental importance. Without knowledge of the location of such boundaries, there is no way to know whether the confidence region about the parameters’ point estimates might be crossed by one or more such boundaries. If there are intersections between bifurcation boundaries and a confidence region, the resulting stratification of the confidence region damages inference robustness about dynamics, when such dynamical inferences are produced by the usual simulations at the point estimates only.

Recently, interest in policy in some circles has moved to New Keynesian models, which have become common in monetary policy formulations. As a result, we explore bifurcations within the class of New Keynesian models. We study different specifications of monetary policy rules within the New Keynesian functional structure. In initial research in this area, Barnett and Duzhak (2008) found a New Keynesian Hopf bifurcation boundary, with the setting of the policy parameters influencing the existence and location of the bifurcation boundary. Hopf bifurcation is the most commonly encountered type of bifurcation boundary found among economic models, since the existence of a Hopf bifurcation boundary is accompanied by regular oscillations within a neighborhood of the bifurcation boundary. Now, following a more extensive and systematic search of the parameter space, we also find the existence of Period Doubling (flip) bifurcation boundaries in the class of models.

Central results in this research are our theorems on the existence and location of Hopf bifurcation boundaries in each of the considered cases. We also solve numerically for the location and properties of the Period Doubling bifurcation boundaries and their dependence upon policy-rule parameter settings.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bifurcation; dynamic general equilibrium; Hopf bifurcation; flip bifurcation; period doubling bifurcation; robustness; New Keynesian macroeconometrics; Taylor rule; inflation targeting
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C30 - General; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E37 - Forecasting and Simulation: Models and Applications; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook > E60 - General

References

Aiyagari, S. R. (1989), “Can there be short-period deterministic cycles when people are long lived?” Quarterly Journal of Economics 104, 163-185.

Andronov, A. A. (1929), “Les Cycles Limits de Poincaré et la Théorie des Oscillations Autoentretenues,” Comptes-rendus de l’Academie des Sciences 189, 559-561.

W. A. Barnett and W. A. Duzhak, "Non-robust dynamic inferences from macroeconometric models: Bifurcation stratification of confidence regions," Physica A, vol. 387, issue 15, June 15, 2008, pp. 3817-3825.

Barnett, William A. and Yijun He (1999), "Stability Analysis of Continuous-Time Macroeconometric Systems," Studies in Nonlinear Dynamics and Econometrics, January, vol 3, no. 4, pp. 169-188.

Barnett, William A. and Yijun He (2001), "Nonlinearity, Chaos, and Bifurcation: A Competition and an Experiment," with Yijun He, in Takashi Negishi, Rama Ramachandran, and Kazuo Mino (eds.), Economic Theory, Dynamics and Markets: Essays in Honor of Ryuzo Sato, Kluwer Academic Publishers, 167-187.

Barnett, William A. and Yijun He (2002), "Stabilization Policy as Bifurcation Selection: Would Stabilization Policy Work if the Economy Really Were Unstable?," Macroeconomic Dynamics, vol 6, no 5, November, 713-747.

Barnett, William A. and Yijun He (2004), "Bifurcations in Macroeconomic Models," in Steve Dowrick, Rohan Pitchford, and Steven Turnovsky (eds), Economic Growth and Macroeconomic Dynamics: Recent Developments in Economic Theory , Cambridge University Press, 95-112.

Barnett, William A. and Yijun He (2006), "Robustness of Inferences to Singularity Bifurcation," Proceedings of the Joint Statistical Meetings of the 2005 American Statistical Society, vol. 100, American Statistical Association, February.

Barnett, William A. and Susan He (2008), “Existence of Singularity Bifurcation in an Open-Economy Euler-Equations Model of the United States Economy,” Open Economies Review, submitted.

Benhabib, J., Day, R. H. (1982),”A characterization of erratic dynamics in the overlapping generations model.” Journal of Economic Dynamics and Control 4, 37-55.

Benhabib J., Nishimura K. (1979), The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth, Journal of Economic Theory, vol. 21, pp. 421-444.

Benhabib, J., Rustichini, A. (1991), “Vintage capital, investment and growth.” Journal of Economic Theory 55, 323-339.

Bergstrom, A. R. (1996), “Survey of Continuous Time Econometrics,” in W. A. Barnett, G. Gandolfo, and C. Hillinger (eds.), Dynamic Disequilibrium Modeling, Cambridge University Press, 3-26.

Bergstrom, A. R., K. B. Nowmann, and S. Wandasiewicz (1994), “Monetary and Fiscal Policy in a Second-order Continuous Time Macroeconometric Model of the United Kingdom,” Journal of Economic Dynamics and Control, 18, 731-761.

Bergstrom, A. R., K. B. Nowman, and C. R. Wymer (1992), “Gaussian Estimation of a Second Order Continuous Time Macroeconometric Model of the United Kingdom,” Economic Modelling, 9, 313-352.

Bergstrom, A.R., and C.R. Wymer (1976), “A Model of Disequilibrium Neoclassical Growth and its Application to the United Kingdom,” in A.R. Bergstrom, ed., Statistical Inference in Continuous Time Economic Models, Amsterdam: North Holland, 267-327.

Bergstrom, A. R. and K. B. Nowman (2006), A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends, Cambridge University Press, forthcoming.

Bernanke, Ben S., Thomas Laubach, Frederic S. Mishkin, and Adam S. Posen . (1999), Inflation Targeting: Lessons from the International Experience. Princeton, NJ: Princeton University Press.

Calvo, G. (1983), “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12, 383-398.

Carlstrom, Charles T. and Timothy S. Fuerst, 2000. "Forward-looking versus backward-looking Taylor rules," Working Paper 0009, Federal Reserve Bank of Cleveland.

Clarida, R., J.Gali and M.Gertler (1998), "Monetary Policy Rules in Practice: Some International Evidence", European Economics Review, June, 1033-1068

Clarida, Richard, Jordi Galí, and Mark Gertler (1999), “The Science of Monetary Policy: A New Keynesian Perspective,” Journal of Economic Literature 37, December, 1661–1707.

Dixit, Avinash and Joseph E. Stiglitz (1977), “Monopolistic Competition and Optimum Product Diversity,” American Economic Review 67, 297-308.

Eusepi, Stefano (2005). "Comparing forecast-based and backward-looking Taylor rules: a "global" analysis," Staff Reports 198, Federal Reserve Bank of New York.

Gali, J., and M. Gertler (1999), “Inflation dynamics: a structural econometric analysis,” Journal of Monetary Economics. 44, October, 195-222.

Gale, D. (1973), “Pure exchange equilibrium of dynamic economic models.” Journal of Economic Theory 6, 12-36.

Gandolfo, Giancarlo 1996, Economic Dynamics, Third edition, New York and Heidelburg, Springer.

Gavin, William T (2003), “Inflation Targeting: Why It Works and How To Make It Work Better,” Federal Reserve Bank of Saint Louis Working Paper 2003-027B.

Grandmont, J. M. (1985), "On Endogenous Competitive Business Cycles," Econometrica, 53, 995-1045.

Hopf, E. (1942), “Abzweigung Einer Periodischen Lösung von Einer Stationaren Lösung Eines Differetialsystems,” Sachsische Akademie der Wissenschaften Mathematische-Physikalische, Leipzig 94, 1-22.

Iooss, G. (1979), Bifurcation of Maps and Applications, Mathematical Studies Vol. 36, North-Holland, Amsterdam.

Kuznetsov,Yu.A (1998). Elements of Applied Bifurcation Theory.Springer - Verlag, New York.

Leeper, E. and C. Sims (1994), “Toward a Modern Macro Model Usable for Policy Analysis,” NBER Macroeconomics Annual, pp. 81-117.

McCallum, B.T. (1999), “Issues in the design of monetary policy rules,” Handbook of Macroeconomics, eds. J.B. Taylor and M. Woodford. North-Holland Pub. Co.

Poincaré, H. (1892), Les Methodes Nouvelles de la Mechanique Celeste, Gauthier-Villars, Paris.

Roberts, J. M. (1995), “New Keynesian Economics and the Phillips Curve”, Journal of Money, Credit and Banking, 27(4), November, 975-984.

Seydel, R. (1994), Practical bifurcation and stability analysis, Springer-Verlag, New York.

Shapiro, A. H. (2006), “Estimating the New Keynesian Phillips Curve: A Vertical Production Chain Approach” Federal Reserve Bank of Boston Working Paper No. 06-11.

Svensson, Lars E. O. (1999), “Inflation Targeting as a Monetary Policy Rule.” Journal of Monetary Economics 43, 607-54.

Taylor, John B. (1999), “A Historical Analysis of Monetary Policy Rules,” in John B. Taylor, ed., Monetary Policy Rules, Chicago: University of Chicago Press for NBER, 319-40.

Torre, V. (1977). Existence of Limit Cycles and Control in Complete Keynesian System by Theory of Bifurcations” Econometrica, Vol. 45, No. 6. (Sept.), pp. 1457-1466.

Walsh, Carl E. (2003), Monetary Theory and Policy, 2nd edition MIT Press: Cambridge Mass.

Wen, Guilin, Daolin Xu, and Xu Han (2002), “On creation of Hopf bifurcations in discrete-time nonlinear systems,” Chaos, Vol. 12, Issue 2, pp. 350-355.

Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton University Press.