Bifurcation of macroeconometric models and robustness of dynamical inferences

William, Barnett and Guo, Chen (2015): Bifurcation of macroeconometric models and robustness of dynamical inferences.

Abstract

In systems theory, it is well known that the parameter spaces of dynamical systems are stratified into bifurcation regions, with each supporting a different dynamical solution regime. Some can be stable, with different characteristics, such as monotonic stability, periodic damped stability, or multiperiodic damped stability, and some can be unstable, with different characteristics, such as periodic, multiperiodic, or chaotic unstable dynamics. But in general the existence of bifurcation boundaries is normal and should be expected from most dynamical systems, whether linear or nonlinear. Bifurcation boundaries in parameter space are not evidence of model defect. While existence of such bifurcation boundaries is well known in economic theory, econometricians using macroeconometric models rarely take bifurcation into consideration, when producing policy simulations from macroeconometrics models. Such models are routinely simulated only at the point estimates of the models’ parameters. Barnett and He (1999) explored bifurcation stratification of Bergstrom and Wymer’s (1976) continuous time UK macroeconometric model. Bifurcation boundaries intersected the confidence region of the model’s parameter estimates. Since then, Barnett and his coauthors have been conducting similar studies of many other newer macroeconometric models spanning all basic categories of those models. So far, they have not found a single case in which the model’s parameter space was not subject to bifurcation stratification. In most cases, the confidence region of the parameter estimates were intersected by some of those bifurcation boundaries. The most fundamental implication of this research is that policy simulations with macroeconometric models should be conducted at multiple settings of the parameters within the confidence region. While this result would be as expected by systems theorists, the result contradicts the normal procedure in macroeconometrics of conducting policy simulations solely at the point estimates of the parameters. This survey provides an overview of the classes of macroeconometric models for which these experiments have so far been run and emphasizes the implications for lack of robustness of conventional dynamical inferences from macroeconometric policy simulations. By making this detailed survey of past bifurcation experiments available, we hope to encourage and facilitate further research on this problem with other models and to emphasize the need for simulations at various points within the confidence regions of macroeconometric models, rather than at only point estimates.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bifurcation, Hopf, robustness, dynamical inference, policy simulations
• Subjects: E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E37 - Forecasting and Simulation: Models and Applications; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E52 - Monetary Policy; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E58 - Central Banks and Their Policies

References

Andrews, Donald W. K. (1993), “Test for Parameter Instability and Structural Change with Unknown Change Point.” Econometrica, 1993, 61(4), pp. 821-856.

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

Arnold, L. G. (2000a), “Stability of the Market Equilibrium in Romer’s Model of Endogenous Technological Change: A Complete Characterization.” Journal of Macroeconomics 22, pp. 69-84.

Arnold, L. G. (2000b), “Endogenous Technological Change: A Note on Stability.” Economic Theory, 16, pp.219-226.

Arnold, L. G. (2006), “The Dynamics of Jones R&D Model.” Review of Economic Dynamics 9, pp. 143-152.

Bala, V. (1997), “A Pitchfork Bifurcation in the Tatonnement Process.” Economic Theory, Vol. 10, pp. 521-530.

Banerjee, S., W. A. Barnett, E. A. Duzhak, and R. Gopalan (2011), “Bifurcation Analysis of Zellner’s Marshallian Macroeconomic Model.” Journal of Economic Dynamics and Control 35, pp. 1577-1585.

Barnett, W. A. and J. Binner (2004), Functional Structure and Approximation in Econometrics, Amsterdam: Elsevier.

Barnett, W. A. and P. Chen (1988), “The Aggregation Theoretic Monetary Aggregates are Chaotic and Have Strange Attractors: an Econometric Application of Mathematical Chaos.” In W. A. Barnett, E. R. Berndt, and H. White, ed., Dynamic Econometric Modeling, Cambridge University Press, pp. 199-246.

Barnett, W. A. and E. A. Duzhak (2008), “Non-Robust Dynamic Inferences from Macroeconometric Models: Bifurcation of Confidence Region.” Physica A, Vol. 387, No. 15, June, pp. 3817-3825.

Barnett, W. A. and E. A. Duzhak (2010), “Empirical Assessment of Bifurcation Region within New Keynesian Models.” Economic Theory, Vol. 45, pp. 99-128.

Barnett, W. A. and E. A. Duzhak (2014), “Structural Stability of the Generalized Taylor Rule.” Macroeconomic Dynamics, forthcoming.

Barnett, W. A. and U. Eryilmaz (2013), “Hopf Bifurcation in the Clarida, Gali, and Gertler Model.” Economic Modelling 31, pp. 401-404.

Barnett,W. A. and U. Eryilmaz (2014), ”An Analytical and Numerical Search for Bifurcations in Open Economy New Keynesian Models.” Macroeconomic Dynamics, forthcoming.

Barnett,W. A. and T. Ghosh (2013),“Bifurcation Analysis of an Endogenous Growth Model.” Journal of Economic Asymmetries, Elsevier, Vol. 10, No. 1, June 2013, pp. 53-64.

Barnett,W. A. and T. Ghosh (2014),“Stability Analysis of Uzawa-Lucas Endogenous Growth Model.” Economic Theory Bulletin (2014) 2: pp. 33-44.

Barnett, W. A. and Y. He (1999), “Stability Analysis of Continuous-Time Macroeconometric Systems.” Studies in Nonlinear Dynamics and Econometrics, January, Vol. 3, No. 4, pp. 169-188.

Barnett, W. A. and Y. He (2001a), “Nonlinearity, Chaos, and Bifurcation: A Competition and Experiment.” In Takashi Negishi, Rama Ramachandran, and Kazuo Mino (eds.), Economic Theory, Dynamics and Markets: Essays in Honor of Ryuzo Sato, Kluwer Academic Publishers, pp. 167-187.

Barnett, W. A. and Y. He (2001b), “Unsolved Econometric Problems in Nonlinearity, Chaos, and Bifurcation.” Central European Journal of Operations Research, Vol. 9, July 2001, pp. 147-182.

Barnett, W. A. and Y.He (2002), “Stability Policy as Bifurcation Selection: Would Stabilization Policy Work if the Economy Really Were Unstable?” Macroeconomic Dynamics 6, pp. 713-747.

Barnett, W. A. and Y. He (2004), “Bifurcations in Macroeconomic Models.” In Steve Dowrick, Rohan Pitchford, and Steven Turnovsky (eds.), Economic Growth and Macroeconomic Dynamics: Recent Development in Economic Theory, Cambridge University Press, pp. 95-112.

Barnett, W. A. and Y. He (2006a), “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, W. A. and Y. He (2006b), “Singularity Bifurcations.” Journal of Macroeconomics, invited special issue, Vol. 28, 2006, pp. 5-22.

Barnett, W. A. and Y. He (2008), “Existence of Singularity Bifurcation in an Euler-Equations Model of the United States Economy: Grandmont Was Right.” Physica A 387, pp. 3817-3815.

Barro, R. J. and X. Sala-i-Martín (2003), Economic Growth, Second Edition, The MIT Press.

Benhabib, J. and K. Nishimura (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. and R. Perli (1994), “Uniqueness and Indeterminacy on the Dynamics of Endogenous Growth.” Journal of Economic Theory 63, pp. 113-142.

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

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

Bergstrom, A. R., K. B. Nowman, and S. Wandasiewicz (1994), “Monetary and Fiscal Policy in a Second-Order Continuous Time Macroeconometric Model of the United Kingdom.” J. Economic Dynamics and Control, 18, pp. 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, pp. 313-352.

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

Bernanke, B. S., T. Laubach, F. S. Mishkin, and A.S. Posen (1999), Inflation Targeting: Lessons from the International Experience, Princeton, NJ: Princeton University Press. Binder M, Pesaran M. H. (1999). “Stochastic Growth Models and Their Econometric Implications.” Journal of Economic Growth, 4, pp.139-183.

Blanchard, O. J. and C. M. Kahn (1980), “The Solution of Linear Difference Models under Rational Expectations.” Econometrica, Vol. 48, No. 5, pp. 1305-1312.

Boldrin, M., and M. Woodford (1990), “Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey.” Journal of Monetary Economics, Vol. 25, pp. 189-222.

Bucci, A. (2008), “Population Growth in a Model of Economic Growth with Human Capital Accumulation and Horizontal R&D.” Journal of Macroeconomics 30, pp. 1124-1147.

Bullard, J. and K. Mitra (2002), “Learning about Monetary Policy Rules.” Journal of Monetary Economics, Vol. 49, Issue 6, September, pp. 1105-1129.

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

Carlstrom, C. T. and T. S. Fuerst (2000), “Forward-Looking Versus Backward-Looking Taylor Rules.” Working Paper 2009, Federal Reserve Bank of Cleveland.

Carr, J. (1981), Applications of Center Manifold Theory, New York: Springer-Verlag.

Chen, G., D. Hill, and X. Yu (2003), Bifurcation Control: Theory and Applications, Springer-Verlag Berlin Heidelberg 2003.

Chow, S. and J. Hale (1982), Methods of Bifurcation Theory, Springer-Verlag New York Inc.

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

Clarida, R., J. Gali, and M. Gertler(1999), “The Science of Monetary Policy: A New Keynesian Perspective.” Journal of Economic Literature, 1999, 37(Dec).

Clarida, R., J. Gali, and M. Gertler (2000), “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.” The Quarterly Journal of Economics, Vol. 115, No.1, pp. 147-180.

Clarida, R., J. Gali, and M. Gertler (2001), “Optimal Monetary Policy in Open Versus Closed Economies.” American Economic Review, Vol. 91(2), May, pp. 248-252.

Clarida, R., J. Gali, and M. Gertler (2002), “A Simple Framework for International Monetary Policy Analysis.” Journal of Monetary Economics, Elsevier, Vol. 49(5), July, pp. 879-904.

Davig, Troy and E. M. Leeper (2006), “Generalizing the Taylor Principle.” American Economic Review, 97(3), pp. 607-635.

Demirel, U. D. (2010), “Macroeconomic Stabilization in Developing Economies: Are Optimal Policies Procyclical?”European Economic Review, Vol. 54, Issue 3, April, pp. 409-428.

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

Dockner, E. J., and G. Feichtinger (1991),“On the Optimality of Limit Cycles in Dynamic Economic Systems.” Journal of Economics, Vol. 51, pp. 31-50.

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

Evans, G. (1985), “Expectational Stability and the Multiple Equilibria Problem in Linear Rational Expectation Models.” Quarterly Journal of Economics, 1985, 100(4), pp. 1217-1233.

Farmer, Roger E. A., D. F. Waggoner, and Tao Zha (2007), “Understanding the New Keynesian Model When Monetary Policy Switches Regimes.” Working paper 2007-12, NBER Working Paper, No. 12965.

Gali, J. and T. Monacelli (1999), “Optimal Monetary Policy and Exchange Rate Volatility in a Small Open Economy.” Boston College Working Papers in Economics, 438, Boston College Department of Economics.

Gali, J. and T. Monacelli (2005), “Monetary Policy and Exchange Rate Volatility in a Small Open Economy.” Review of Economic Studies, Vol. 72, No. 3, July.

Gandolfo, G. (1996), Economic Dynamics, Springer-Verlag, New York.

Gandolfo, G. (2010), Economic Dynamics, 4th Edition, Springer.

Gantmacher F. R. (1996). The Theory of Matrices, Chelsa: New York.

Glendinning, P. (1994), Stability, Instability, and Chaos, Cambridge University Press.

Grandmont, J. M. (1985), “On Endogenous Competitive Business.” Econometrica, Vol. 53, pp. 995-1045.

Grandmont, J. M. (1998), “Expectations Formation and Stability of Large Socioeconomic System.s” Econometrica, Vol. 66, pp. 741-782.

Groen, Jan J. J and Haroon Mumtaz (2008), “Investigating the Structural Stability of the Phillips Curve Relationship.” Bank of England Working Paper, No. 350.

Grossman, G. and E. Helpman (1991), “Endogenous Product Cycles.” Economic Journal, vol. 101, pp. 1229-1241.

Guckenheimer, J. and P. Holmes (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.

Guckenheimer, J., M. Myers, and B. Sturmfels (1997), “Computing Hopf Bifurcations 1.” SIAM J. Numer. Anal., 34, pp. 1-21.

Hale, J. and H. Kocak (1991), Dynamics and Bifurcations, Springer-Verlag New York Inc.

Hamilton, James D. (1989), “A New Approach to the Economic Analysis of Non-stationary Time Series and the Business Cycle.” Econometrica, 1989, Vol. 57, No. 2, pp. 357-384.

Hansen, Bruce E. (1992), “Testing for Parameter Instability in Linear Models.” Journal of Policy Modeling, 14 (4), pp. 517- 533.

Henry, D. (1981), Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York.

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

Jones, Charles I. (1995), “R&D-Based Models Economic Growth in a World of Ideas.” Journal of Political Economy, Vol. 103, Issue 4, pp. 759-784.

Jones, Charles I. (2002), “Source of U.S. Economic Growth in a World of Ideas.” The American Economic Review, Vol. 92, No. 1, pp. 220-239.

Khalil, H. K. (1992), Nonlinear Systems, Macmillan Pub. Co, New York.

Kim, J. (2000), “Constructing and Estimating a Realistic Optimizing Model of Monetary Policy.” Journal of Monetary Economics, 45, pp. 329-359.

Kuznetsov, Yu. A. (1998), Elements of Applied Bifurcation Theory, 2nd Edition. Springer-Verlag, New York.

Kuznetsov, Yu. A. (1998), “Saddle-node bifurcation.” Scholarpedia, 1(10): 1859.

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

Leith, C., I. Moldovan, and R. Rossi (2009), “Optimal Monetary Policy in a New Keynesian Model with Habits in Consumption.” European Central Bank Working Paper Series, No 1076, July.

Lucas, R. E. (1976), “Econometric Policy Evaluation: A Critique.” In Brunner K, Meltzer AH (Eds.), “The Phillips Curve and Labor Markets.” Journal of Monetary Economics, supplement; pp. 19-46.

Lucas, R. E. (1988), “On the Mechanics of Economic Development.” Journal of Monetary Economics 22, pp. 3-42.

Luenberger, D. G. and A. Arbel (1977), “Singular Dynamic Leontief Systems.” Econometrica, 1977, Vol. 45, pp. 991-996.

Mattana, P. (2004), The Uzawa-Lucas Endogenous Growth Model, Ashgate Publishing, Ltd.

Medio, A. (1992), Chaotic Dynamics: Theory and Applications to Economics, Cambridge University Press.

Mondal, D. (2008), “Stability Analysis of the Grossman-Helpman Model of Endogenous Product Cycles.” Journal of Macroeconomics 30, pp.1302-1322.

Nieuwenhuis, H. J. and L. Schoonbeek (1997), “Stability and the Structure of Continuous-Time Economic Models.” Economic Modelling, 14, pp. 311-340.

Nishimura, K. and H. Takahashi (1992), “Factor Intensity and Hopf Bifurcations.” In G. Feichtinger, ed., Dynamic Economic Models and Optimal Control, pp. 135-149.

Nyblom, Jukka. (1989), “Testing for the Constancy of Parameters over Time.” Journal of the American Statistical Associaition, 84(405).

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

Powell, A. A. and C. W. Murphy (1997), Inside a Modern Macroeconometric Model: A Guide to the Murphy Model, Heidelberg, Springer.

Romer, P. M. (1990), “Endogenous Technological Change.” Journal of Political Economy, Vol. 98, pp. S71-S102.

Scarf, H. (1960), “Some Examples of Global Instability of Competitive Equilibrium.” International Economic Review, Vol. 1.

Schettkat, R. and R. Sun (2009), “Monetary Policy and European Unemployment.” Oxford Review of Economic Policy, Vol. 25, Issue 1, pp. 94-108.

Sims, Christopher A. and Tao Zha (2006), “Were There Regimes Switches in U.S. Monetary Policy?” American Economic Review, pp. 54-81.

Sotomayor, J. (1973),”Generic Bifurcations of Dynamic Systems”. M.M. Peixoto, ed., Dynamical Systems, pp. 561-582, New York: Academic Press.

Taylor, John B. (1993), “Discretion Versus Policy Rules in Practice.” Carnegie-Rochester Conferences Series on Public Policy, 39, December, pp. 195-214.

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, 1999, pp. 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.

Uzawa, H. (1965), “Optimum Technical Change in an Aggregate Model of Economic Growth.” International Economic Review 6, pp. 18-31.

Veloce, W. and A. Zellner (1985), “Entry and Empirical Demand and Supply Analysis for Competitive Industries.” Journal of Econometrics 30(1-2), pp. 459-471.

Walsh, E. C. (2003), Monetary Theory and Policy, Cambridge, MA, MIT Press, Second Edition.

Warne, A. (2000), “Causality and Regime Inference in a Markov Switching VAR.” Sveriges Riksbank Working Paper No. 118.

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

Wymer, C. R. (1997), “Structural Nonlinear Continuous Time Models in Econometrics.” Macroeconomic Dynamics, Vol. 1, pp. 518-548.

Zellner, A. and G. Israilevich (2005), “Marshallian Macroeconomic Model: A Progress Report.” Macroeconomic Dynamics 9(02), pp. 220-243.