Transcript Nonlinear Dynamics and Bifurcation

Bifurcation in Econometrics
Implications for Robustness of
Dynamical Inferences to
Measurement Error
Closing Words to Edmund
Malinvaud’s, Statistical Methods
of Econometrics
"Finally, we must never forget that our progress
in understanding economic laws depends
strictly on the quality and abundance of
statistical data. Nothing can take the place of
the painstaking work of observation of the
facts. All improvements in methodology would
be in vain if they had to be applied to mediocre
data."
Kenneth E. Boulding, “After Samuelson, Who
Needs Adam Smith?” History of Political
Economy, fall 1971, 3(2), pp. 225-237.
“We seem to be producing a generation of
economists now whose main preoccupation
consists of analyzing data which they have not
collected and who have no interest whatever in
what might be called a data-reality function,
that is, to what extent a set of data corresponds
to any significant reality in the world.“ (p. 233)
“The economic statistics that the
government issues every week should
come with a warning sticker: User
beware. In the midst of the greatest
information explosion in history, the
government is pumping out a stream of
statistics that are nothing but myths
and misinformation.”
=====================================
Michael M. Mandel, “The Real Truth About the Economy: Are
Government Statistics So Much Pulp Fiction? Take a Look,”
Business Week, cover story, November 7, 1994, pp. 110-118.
The Economist Magazine
“A long way from dismal,” January 10-16, 2015,
v. 414, n. 8920, p. 8.
“Established macroeconomists
would do well to pay attention.
They should start by being much
more careful about data.”
Systems Theory and Engineering
• Small changes in data or in
parameters can cause large changes
in dynamics.
• Auto industry lawsuits.
•
Rochester Products Division of GM
• Rocketdyne F-1 rocket engine.
• Expenditure on measurement.
Kraemer, Robert S. (2005), Rocketdyne: Powering
Humans into Space, American Institute of
Aeronautics and Astronautics, Reston, VA.
From the first American orbiting
satellite, to Neil Armstrong, and Buzz
Aldrin's historical walk on the Moon,
virtually every major achievement in
American Space history was made
possible by a Rocketdyne engine.
Pogo Effect
Harmonic oscillations.
Resonant frequencies.
Vehicle failure.
Econometrics: Errors in the
Variables
• Mapping from a Euclidian
space to another Euclidian
space.
• E.g., from data space to the
estimated elasticity of
substitution.
Dynamics: Errors in the Variables
• With Known Parameters: Mapping
from Euclidian data space to
solution path function space.
• With Estimated Parameters:
Mapping from data space to space
of stochastic processes.
Simpler Procedures in Practice
• Usual approach: Policy simulations
produced with parameters set at
their point estimates.
• Better approach: Conduct
simulations with parameters set at
various points within their
confidence region.
Robustness of Dynamical Inferences
• Without measurement errors.
• Stratified confidence regions.
• With measurement errors.
• Effect of measurement error on
confidence region.
• Effect of measurement error on
bifurcation boundaries.
Economic Theory
• J. M. Grandmont (1985).
• Rational expectations, Cobb Douglas
consumers and firms, continuous
market clearing, no rigidities or
market failures, perfect competition.
• Parameter space stratified into an
infinite number of bifurcation
subsets.
Market Mechanisms in Model
• Transversality conditions rule out
explosive instability.
• No mechanism in economy to
rule out nonexplosive instability.
• Distinction between damped
stability and instability can be
hard to detect.
Economic Theory
• Unstable cases.
• Period doubling bifurcation.
• Feigenbaum recursion.
• In most macro text books.
• Hopf bifurcation.
• Can be hard to distinguish from
stability.
Stable Bifurcations.
• Soft bifurcations.
• Infinite number of cases.
• Similar dynamics
attained only after near
steady state.
Papers with Yijun He on the UK
Continuous Time
Macroeconometric Model
(Bergstrom et al)
"Analysis and Control of Bifurcations in Continuous Time Macroeconomic
Systems," with Yijun He, Proceedings of the 37th IEEE Conference on Decision
and Control, December 1998, pp. 2455-2460.
“Stability Analysis of Continuous Time Macroeconometric Systems”, Studies in
Nonlinear Dynamics and Econometrics, January 1999, vol 3, no. 4, pp. 169-188.
"Bifurcation Theory in Economic Dynamics," in Shri Bhagwan Dahiya (ed.), The
Current State of Economic Science, vol. 1, 1999, pp. 435-451.
"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, 2001, pp. 167-187.
"Unsolved Econometric Problems in Nonlinearity, Chaos, and
Bifurcation," with Yijun He, Central European Journal of
Operations Research, vol 9, July 2001, pp. 147-182.
“Stabilization Policy as Bifurcation Selection: Would Stabilizaton
Policy Work if the Economy were Unstable?” Macroeconomic
Dynamics, vol 6, no 5, Nov 2002, pp. 713-747.
“Bifurcation 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, 2004, pp. 95-112.
A. R. Bergstrom UK Model
Bergstrom, A. R., K. B. Nowman and C. R. Wymer
(1992), “Gaussian Estimation of a Second Order
Continuous Time Macroeconometric Model of the
UK,” Economic Modelling, vol 9, pp. 313-351.
Bergstrom, A. R. and K. B. Nowman (2006), A
Continuous Time Econometric Model of the United
Kingdom with Stochastic Trends, Cambridge U. Press,
2007
Model structure:
14 second order differential equations
Structural parameters:
63 structural parameters, including 27 longrun propensities and elasticities and 33
speed of adjustment parameters
Free parameters:
3 trend parameters
Hopf Bifurcation Example
• Dx = -y + x(θ - (x2 + y2))
• Dy = x + y(θ - (x2 + y2))
The equilibria are found by setting Dx = Dy = 0.
All equilibria satisfy x* = y* = 0, with the stable
equilibria occurring for θ < 0 and the unstable
equilibria occurring for θ > 0.
Phase Diagram for Hopf Bifurcation
y
q
x
Transcritical Bifurcation Example
•Dx = θx –
2
x
It is stable around the
*
equilibrium x = 0 for θ < 0, and
unstable for θ > 0. The
equilibrium x*= θ is stable for
θ > 0, and unstable for θ < 0.
The solid dark lines represents stable equilbria, while
the dashed lines display unstable ones
• Transcritical bifurcation diagram:
x
q
Bergstrom UK Model: 2-dimensional
sections of stable region
• 1 is the confidence region.
•  is the theoretically feasible region.
θ62 versus θ2
θ23 versus θ62
Bergstrom UK Model: 3-dimensional
sections of stable region
• 1 is the confidence region.
•  is the theoretically feasible region.
θ2, θ23, θ62
θ12, θ23, θ62
Bergstrom’s Fiscal Policy Design
• Instrument:
• Target:
total taxation variable
real net output
• Policy intent:
Use the instrument to
stabilize the target with the ultimate objective of
stabilizing the economy's dynamics. Closed loop
stabilization policy rule, feeding back observed
values of the target.
Symbols in Figures
• Fiscal Policy Rule Parameters:
  = strength of feedback
  = speed of policy adjustment
The adjustment lag is caused by delays in sampling
the target variable and delays in adjusting the
instrument to the observed target variable.
Private Sector Parameters in Figures
• θ2 = coefficient of the real interest rate in the
consumption function.
• θ5 = a measure of the importance of capital in
production.
• θ62 = rate of growth of expected labor supply
trend.
Parameter Settings
• Private sector parameters set at their
estimated values.
• Parameters of fiscal policy feedback rule set at
various settings.
θ62 versus θ2
θ5 versus θ2
Leeper and Sims Euler Equations Model of
the US Economy
• Deep parameters solve Lucas critique.
• Model first appeared in:
Eric Leeper and Christopher Sims, “Toward a
Modern Macro Model Usable for Policy
Analysis,” NBER Macroeconomics Annual,
1994, pp. 81-117.
Our Findings with the Leeper and Sims
Model
Consists of differential equations with
algebraic constraints. Singularity
bifurcation boundary found near the
model’s parameter point estimates. To
our knowledge, this kind of bifurcation
had not previously been observed in
macroeconomics.
Boundary with Stable Subset not
Yet Found
•Unstable on both sides of singularity
bifurcation boundary.
•Location of bifurcation boundary with
stable region difficult to locate, even if
near point estimates, since the
parameter space is high dimensional
Singularity Bifurcation
As parameters approach a singularity boundary,
one eigenvalue of the linearized part of the
model rapidly moves to infinity, while others
remain bounded. This implies nearly
instantaneous response of some variables to
changes of other variables.
On the singularity boundary, the number of
differential equations will decrease, while the
number of algebraic constraints will increase.
Singularity bifurcations thereby cause a change in
the order of the dynamics.
Consider a continuous time model in the
following form:
A(x(t),θ)Dx(t) = F(x(t), θ),
in which x(t) is the state vector, D is the
differentiation operator, t is time, and
A(x(t),θ) is a matrix valued function of
time. The matrix A can be singular.
Singularity-induced bifurcation
occurs when the rank of A(x(t),θ)
changes, such as from an invertible
matrix to a singular one. In such
cases, the dimension of the
dynamical part of the system
changes.
Example of Singularity Bifurcation
• Dx = ax - x2,
• θDy = x - y2,
In which a > 0. The equations consist of two
differential equations with no algebraic
equations for nonzero θ. But when θ = 0, the
system has one differential equation and one
algebraic equation.
By setting Dx = Dy = 0, we can find that for
every θ, the equilibria are at (x,y) = (0,0)
and (x,y) = (a,a1/2). In this case, the
system is unstable around the equilibrium
(x*,y*) = (0,0) for any value of θ. The
equilibrium (x*,y*) = (a,+a1/2 ) is unstable
for θ < 0 and stable for θ > 0. The third
equilibrium (x*,y*) = (a,-a1/2 ) is unstable
for θ > 0 and stable for θ < 0.
Normalized Figure
The figure below is normalized at a = 1
with positive θ. When θ is negative, the figure
remains valid, but with the diagram flipped
over about the x axis, so that (1,1) becomes
unstable and (1,-1) becomes stable.
The equilibrium (0,0) remains unstable for
either positive or negative θ.
Singularity Bifurcation Phase Portrait with θ > 0
On the Singularity Boundary
• Recall that Dx = ax - x2 and θDy = x - y2.
• When θ = 0, the system’s behavior degenerates
into movement along the one dimensional curve
x – y2 = 0, as shown in the figure below, with (0,0)
being an unstable equilibrium and both (1,1) and
(1,-1) being stable equilibria. Note that the
second equation changes from a differential
equation to an algebraic equation
Phase Portrait with θ = 0 on Singularity
Bifurcation Boundary
Bifurcation of New Keynesian Models
Research joint with Evgeniya A. Duzhak.
Three economic agents:
Households
Firms
Central Banks
Linearize around zero inflation steady state.
Linearized Model
Three Equations:
Phillips curve relating inflation to output gap. The
output gap is the gap between the actual sticky
prices output and the flexible-price equilibrium
output.
An IS (investment-savings) curve determining the
output gap.
A monetary policy rule.
Monetary Policy Rules
Taylor rules:
Feed back inflation rate and output gap to set
interest rate.
Inflation targeting:
Feed back only the inflation rate as a final target,
in setting the interest rate.
Taylor Rule Types
•
•
•
•
Current looking: it = a1πt + a2xt
Backward looking: it = a1πt-1 + a2xt-1
Forward looking: it = a1πt+1 + a2xt+1
Hybrid: it = a1πt+1 + a2xt
where it = interest rate, πt = inflation rate, and xt =
output gap.
Taylor Rules with Interest Rate
Smoothing
• Current looking:
it = a1πt + a2xt + a3it-1
• Backward looking:
it = a1πt-1 + a2xt-1 + a3it-1
• Forward looking:
it = a1πt+1 + a2xt+1 + a3it-1
• Hybrid:
it = a1πt+1 + a2xt + a3it-1
Inflation Targeting Types
• Current looking: it = a1πt
• Backward looking: it = a1πt-1
• Forward looking: it = a1πt+1
Active Versus Passive Policy Rules
• A Taylor rule or an inflation targeting rule is
called “active,” if the coefficient of the
inflation rate, a1, exceeds one.
• A Taylor rule or an inflation targeting rule is
called “passive,” if the coefficient of the
inflation rate, a1, is less than one.
Flip Bifurcation
• Also called period doubling bifurcation.
• The number of frequencies in the power
spectrum doubles, when a flip bifurcation
boundary is crossed.
• Possible only in discrete time.
• Made famous by the Feigenbaum recursion.
Results with New Keynesian Models:
Types of Bifurcation Found with Each
Version of Rule
Taylor Rule
Taylor Rule with
Interest Smoothing
Inflation Targeting
Current looking
Hopf bifurcation
Hopf bifurcation
Flip bifurcation
Hopf bifurcation
Backward looking
Hopf bifurcation
Hopf bifurcation
Flip bifurcation
Hopf bifurcation
Forward looking
Hopf bifurcation
Flip bifurcation
No bifurcation
boundaries found
within theoretically
feasible region.
Hopf bifurcation
Hybrid rule
Hopf bifurcation
Hopf bifurcation
Flip bifurcation
Does not apply
More Recently Published Research
•
"Non-Robust Dynamic Inferences from Macroeconometric Models: Bifurcation
Stratification of Confidence Regions," with Evgeniya A. Duzhak, Physica A, June
2008.
•
"Empirical Assessment of Bifurcation Regions within New Keynesian Models,"
with Evgeniya A. Duzhak, Economic Theory, vol 45, nos 1-2, 2010.
•
"Existence of Singularity Bifurcation in an Euler-Equations Model of the United
States Economy: Grandmont was Right," with Yijun He, Economic Modelling,
Nov. 2010.
•
"Bifurcation Analysis of Zellner's Marshallian Macro Model," with Sanjibani
Banerjee, Evgeniya Duzhak, and Ramu Gopalan, Journal of Economic Dynamics
and Control, Sept. 2011.
•
"Hopf Bifurcation in the Clarida, Gali, and Gertler Model," with Unal Eryilmaz,
Economic Modelling, vol 31, 2013.
•
"Bifurcation Analysis of an Endogenous Growth Model," with Taniya Ghosh,
Journal of Economic Asymmetries, Elsevier, June 2013.
•
"Stability Analysis of Uzawa-Lucas Endogenous Growth Model," with Taniya
Ghosh, Economic Theory Bulletin, April 2014.
Forthcoming Publications
• "Nonlinear and Complex Dynamics in Economics," with
Apostolos Serletis and Demitre Serletis, Macroeconomic
Dynamics, forthcoming.
• “Structural Stability of the Generalized Taylor Rule,” with
Evgeniya Duzhak, Macroeconomic Dynamics, forthcoming.
• "An Analytical and Numerical Search for Bifurcations in Open
Economy New Keynesian Models," with Unal Eryilmaz,
Macroeconomic Dynamics, forthcoming.
• “Bifurcation of Macroeconomic Models and Robustness of
Dynamical Inferences,” with Guo Chen, survey paper to
appear as an entire issue of Trends in Econometrics.
Most recent model used with Unal Eryilmaz: Galí, J.
and Monacelli T. (2005), “Monetary policy and
exchange rate volatility in a small open economy.”
Review of Economic Studies, Vol. 72, No 3, July.
• Open economy structure affects values of
bifurcation parameters and changes location of
bifurcation boundaries.
• More complex dynamics, a wider variety of
qualitative behaviors, and policy responses.
• Stratification of the confidence region remains
an important risk in open economy New
Keynesian functional structures.
The Bad News
Dynamical inferences need to be
qualified by the risk of bifurcation
boundaries crossing the confidence
regions.
As argued in the recent article in the
Economist magazine, macroeconomic
investment in measurement is
inadequate.
Positive Steps
Center for
Barnett, William A. (2012), Getting
It Wrong: How Faulty Monetary
Statistics Undermine the Fed, the
Financial System, and the Economy,
MIT Press.
• American Publishers Award for
Professional and Scholarly Excellence
(the PROSE Award) for the best book
published in economics during 2012.
The Good News:
This new society’s membership
includes many applied
macroeconomics, both in government
and academia.