Bistability, Bifurcations and Regime Changes in Economics and Finance Gerald Silverberg
UNU-MERIT
Maastricht University
and
International Institute for Applied Systems Analysis*
Bistability, Bifurcations and Regime Changes in Economics and Finance Gerald Silverberg
UNU-MERIT
Maastricht University
and
International Institute for Applied Systems Analysis
*
Economic Systems Occasionally Seem to be Characterized by Rapid and Large Change without Apparent External Cause
Recent common descriptions in the business press:
‘financial meltdown’
‘the economy is in free fall’
‘house of cards’
The system may then remain in the new state for an indefinite length of time
The USA did not exit from the great depression until rearmament for WW2 began in earnest around 1939
Asymmetry between rapid onset of recession and more gradual expansion phases
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Time series of Industrial Capacity Utilization, USA 1967-2010 (Federal Reserve, monthly data seasonally detrended)
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Skewness and Multimodality in Cap-Util Data
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Difficulties for Economic Theory
The standard model of a shocked, linear stable system (impulse-propagation model, Frisch 1933) would not generate fluctuations of sufficiently large amplitude without very large exogenous shocks
Internal stochastic noise should decline markedly in relative size as the numbers of actors increases
Asymmetry and persistence are inexplicable in a linear model *
Pedigree of Bistability Perspective in Macroeconomics
N. Kaldor, 1940, “A Model of the Trade Cycle”, Economic Journal, 50: 78-92: nonlinear multiplier and accelerator *
Pedigree of Bistability Analysis II
J. T. Schwartz, 1965, Theory of Money: the essence of Keynesianism is the assertion that there are coordination full and underemployment Nash equilibrium
Cooper, R. and John, A., 1988, “Coordinating Coordination Failures in Keynesian Models”, Quarterly Journal of Economics, 103: 441-461
Hamilton, JD, 1989, “A new approach to the economic analysis of nonstationary time series and the business cycle”, Econometrica 57:357–384: Markov-switching model
A. Manning, 1990, “Imperfect Competition, Multiple Equilibria and Unemployment Policy”, Economic Journal, 100: 151-162
Durlauf, Steven N., 1991, “Multiple Equilibria and Persistence in Aggregate Fluctuations”, American Economic Review. Papers and Proceedings, 81: 70-74: social interaction models
Duncan Foley, 2009, “The Anatomy of Financial and Economic Crisis”, Gildersleeve Lecture: role of externalities
W. A. Brock, S.R. Carpenter, M. Scheffer, 2010, “Regime Shifts, Environmental Signals, Uncertainty, and Policy Choice”, in J. Norberg, G.S. Cumming (eds), Complexity Theory for a Sustainable Future
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Attempts to Confirm Multistability Empirically
John Dagsvik, Boyan Jovanovic, 1991, “Was The Great Depression A Low-level Equilibrium?”, NBER Working Paper No. 3726 (rejects)
Alan Manning, 1992, “Multiple Equilibria in the British Labour Market: Some Empirical Evidence”, European Economic Review, 36: 1333-1365 (accepts based on evidence for increasing returns)
Extensive results of Markov-switching literature, which often finds evidence for three states (reviewed in e.g. Hamilton and Raj, 2002, “New directions in business cycle research and financial analysis”, Empirical Economics, 27:149–162)
Fuzzy-logic cluster analysis and Markov-switching model also finds three states relating unemployment and inflation (P. Ormerod, B. Rosewell, P. Phelps, 2009, “Inflation/Unemployment Regimes and the Instability of the Phillips Curve”) *
Results of Ormerod, Rosewell, Phelps, 2009
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Is a More Thoroughgoingly Dynamic and Structural Approach Needed?
In Markov-switching models, in general the number and location of regimes is fixed, as are the transition probabilities
One can conjecture, however, that there may be a ‘meta-regime shift’ corresponding to the transition from a unique equilibrium, self-restorative economy (normal business cycle, ‘efficient financial markets’) to a bistable regime (prosperity/depression, bubble/crash), as a function of (slowly) varying structural parameters such as
the extent of highly leveraged debt
trade imbalances
government stimulus investment *
Nonlinear Dynamic Approach
* From bifurcation theory (cf. e.g., Kuznetsov 1998) we know that the one-dimension deterministic system can locally bifurcate generically in one of only two ways:
a fold bifurcation, from no stable equilibrium to one stable and one unstable one, with parameter space codimension one:
a cusp bifurcation, from one stable equilibrium to two stable ones separated by one unstable equilibrium, with parameter space codimension two Start with nonlinear dynamic system in one dimension y with parameter vector p, subject to small-amplitude stochastic noise ε:
Canonical form of cusp bifurcation
PV:
Equilibrium condition:
Bifurcation set in parameter space:
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Relation of Canonical Form to Empirical System (x, p)
The state space variable and parameters will be related to the canonical ones by a smooth invertible transformation:
To a first approximation we can take the functions g and h to be linear.
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Bistability/Bifurcation Models
* Scheffer et al, Nature 2001
Cusp Catastrophe Derived from a Potential Function
* Slow changes in parameters can push system between one and two-state regimes
perturbations can push system over barrier between regimes
hysteresis Scheffer et al, Nature 2001
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The catastrophe-theoretic approach of Haag, G., Weidlich, W. Mensch, G.O., 1985, “A Macroeconomic Potential Describing Structural Change of the Economy”, Theory and Decision, 19: 279-299
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Constructing a (time-dependent) PV from a time series (Haag, Weidlich & Mensch 1985)
Filtering structural from high-frequency, low-amplitude fluctuations: First calculate deviation from trend:
The potential determines the dynamics as follows:
In a window [t-T,t+T], calculate p(t) and q(t) by regressing
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Estimated parameters for different window sizes
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6 Year Window: Trajectory Moves in and out of Bistability Zone
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