Mathematical Representations in Economics and Finance: Empirical Relevance And Philosophical Preference

Ping ChenCenter for New Political Economy, Fudan University in Shanghai

National School of Development, Peking University in Beijingpchen@nsd.pku.edu.cn

Workshop on Finance, Mathematics, and Philosophy

Sapienza University of Rome, June 12-13, 2014

Mathematical Representation of Quantitative Observations

• Why need mathematical representation in economic and financial analysis?

• Large number of time series > hard to judge by naked eyes• Conflicting images between non-stationary data and static concepts

> mathematical transformation changes perception of reality• Complex systems with many time-varying variables > hard to

understand by simple causality and linear statistics

• Choice of mathematical representation depends on philosophical preference in economics or empirical relevance in complexity science

Competing Thoughts in Economics and Finance

• Equilibrium school: self-stabilizing markets:• Smith “invisible hand”, Fama “efficient market”, Lucas

“rational expectations”.

• Disequilibrium school: endogenous instability: Keynes “animal spirits”, Minsky “Ponzi scheme”, Shiller “herd behavior”, Delong et al. “noise traders”

• Organism school: self-organization: Schumpeter “biological clock”, Hayek “spontaneous order”, Chen “color chaos”

Polarizing Images of Equilibrium vs. Disequilibrium School

• Equilibrium school:• Single equilibrium + white noise = linear stability > linear

DD and DD curves, Gaussian distribution, smooth trajectory, random walk, Brownian motion, Frisch model of noise-driven cycles

• Disequilibrium school:• Multi-equilibrium + fat tails = marginal stability >• non-Gaussian distribution (Levy distribution, fat tail,

power law, black swan), fractal, unit root, co-integration, sand pile, etc.

New Math Tools for Complex Economics

• Growth: log-linear > logistic (S-shaped)• Cycles: long wave, short noise > finite wavelet• Dynamics: linear algebra equation > nonlinear

differential equation• Solution: single stable equilibrium state > multiple

stable/unstable equilibrium states > deterministic chaos

• Probability distribution: Gaussian, static distribution > non-Gaussian, time-varying distribution

• Stochastic process: mean-variance analysis > high moments, birth-death process, transition probability

Neoclassical Economics > Math Foundation for Self-Stabilizing Market:

Linear Demand-Supply Curves > Unique Equilibrium+ White Noise

Complex Economics:Nonlinear Demand & Supply Curves>

Multiple Equilibriums >Cycles + Crisis

Why IS-LM Curve Breaks Down during the Great Recession?

Complex Causality in Open Economy• Lower interest rate i 》 Y

• Optimistic market > increasing investment 》 Y • Uncertain market > holding cash 》 Y • Pessimistic market > capital flight 》 Y

i IS i LM Y Y

Slim (equilibrium school) vs. Fat (disequilibrium school) distribution in statistics

Gaussian and Cauchy Distribution. The N(0, 1) is the tallest in solid line. The Cauchy(1, 0) distribution is in the middle in dashed line, and Cauchy(π, 0) is the lowest and fattest distribution in dotted line.

Auto-Correlations of Harmonic Waves & White Noise

New Economic Thinking:Where Went Wrong Economics?

• Hendry, David F. Econometrics: Alchemy or Science? Essays in Econometric Methodology, Oxford University Press, Oxford (2001).

• Foley, Duncan K. Adam's Fallacy: A Guide to Economic Theology, Harvard University Press (2008).

• Two cases of philosophical bias and mathematical illusion:

• Whitening filter in econometrics• Frisch model of noise-driven cycles in macro & finance

Equilibrium Illusion made by Economertric Filter 》FD（first difference of logarithmic series) ~

Percentage rate of changes: X(t) = FD[S(t)]=S(t+1)-S(t)Whitening filter > High frequency noise amplifier

Frequency Response Function for FD filter

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4FD Whitening Filter

f

Freq

Res

pons

e R(

f)

Normal Band-Pass Filter

•

Frisch Model of Noise-Driving Cycles = Perpetual Motion Machine (Chen 1999)

• The observed auto-correlation will be damped exponentially (Wang and G. E. Unlenbeck 1945);

• For the Frisch model of the U.S.business cycles, American business cycles would be ceased within 4~10 years!

2 2( ) exp( )[cos( ) sin( )]

2r

r r

T

T T T T

π π π

Frisch model (1933) = Never published > Proved wrong by (1930, 1945) > Shared the first Nobel Prize in economics (1969) > Dominating macro &

finance

• Frisch was not the FIRST: G.E.Uhlenbeck and L.S. Ornstein, Physical Review, 36(3), 823-841 (1930).

• Frisch’s Informal conference paper: R. Frisch, “Propagation Problems and Impulse Problems in Dynamic Economics”, in Economic Essays in Honour of Gustav Cassel, George Allen & Unwin, London (1933).

• Frisch's promised paper, "Changing harmonics studied from the point of view of linear operators and erratic shocks," was advertised three times under the category "papers to appear in early issues" in Econometrica, including Issue No. 2, 3, and 4 of Volume I (April, July, and October 1933) but never appeared in Econometrica since 1934.

• Frisch never mentioned a word about his prize-winning model in his Nobel speech in 1970 (Frisch 1981) > Many followers: Friedman, Solow, Lucas, Sargent, Fama, Granger, Scholes, Merton, Kydland, Prescott, (Bernanke), etc.

From New Complexity Science >Complex Evolutionary Economics

• Complexity in living & social systems:• Nonlinearity > bifurcation, chaos, instability• Nonequilibrium > two-way evolution, non-stationary time

series analysis > wavelets• Many-body problem > non-integrable system, master

equation in statistical mechanics

• E. Schrondinger: quantum biology > Principle of Large Numbers

• I. Prigogine: non-equilibrium themodynamics > chemical reaction, multi-humped distribution

What Can We Learn from Physics?

Copernicus: observation reference> geocentric to heliocentric Prescott: FD to HP filterKepler: base function > from circle to ellipse Chen (1996,2005): from white noise to color chaos(biological clock)/wavelet

Einstein: from special (Newton) to general theory Chen (2005): from Brownian motion (BS option pricing, representative agent) to birth-death process (population dynamics)Tang & Chen (2014): calm & turbulent market

White Noise or Color Chaos in Financial Market?Phase Portraits of S&P 500 Index:Noisy Image vs. Strange Attractors

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

X(t

+T

)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3X(t)

FS PCOM Raw HP Cycles

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3

X(t

+T

)

-0.2 -0.1 0 0.1 0.2 0.3X(t)

FS PCOM Filt ered HP Cycles

Why Economics Are Afraid of Economic Chaos?

• Existence of economic chaos implies:• Nonlinear interactions > reject linear causality

in micro and macro theory • Non-integrable system > little room for

regression analysis in econometrics• Continuous-time evolutionary dynamics >

reject static statistics > pave foundation for market regulation > shake the economic matrix of self-stabilizing market and “invisible hand”

Separating Trend from Cycles & Noise

• Copernicus problem in economics, Why ?• Identify macro trend > define market expectations in

economic growth

• Strategy:• FD (of logarithmic series) = percentage change of

growth = short-term observation > assume zero-trend

• HP = define macro smooth trend by cycle length > adjust λ so that average cycle = NBER business cycle (~4-5 years)

S&P500 Stock Price Index FSPCOM with log-linear (LL) and

HP growth trends

2. 5

3

3. 5

4

4. 5

5

log

S(t)

5. 5

1945 1955 1965 1975 1985 1995

l og S(t)H Ps

t

Trends o f F SPCOM ln Index

l og- l i near trend

Detrended cycles with Varying auto-correlation lengths

-0. 6

-0. 4

-0. 2

0

0. 2

0. 4

X(t)

0. 6

1945 1955 1965 1975 1985 1995t

Cyc les of F SPCOM ln Index

H PcFDLLD c

Stationary (Discrete Fourier Transform) vs.Non-Stationary (Joint-Time-Frequency Analysis)

Time Series Analysis (Qian & Chen 1994)

Gaussian Wavelet in Quantum Mechanics

• Minimum uncertainty in time-frequency domain exists only for Gaussian wavelet

)exp()2

exp()1

()(2

24

1

2ti

tth

π

Gabor Space (1946) in discrete time-frequency space

• Minimum uncertainty in time-frequency space

• Un-orthogonal base function

Separating signals and noise in Gabor (time-frequency) space

Un filtered & F il tered Gab o r Distrib u tio n

Strange Attractor from S&P 500 Monthly Correlation dimension = 2.5

Variance of color chaos = 69 %

-4

-2

0

2

4

1945 1955 1965

S(t)

1975 1985 1995t

F SPCOM Original & F iltered Cyc les (H= 0.5)

SoSg

Economic diagnosis

1965 1970 1975 1980 1985 19900

5

10

15

20Pc History of FSPCOM HP Cycles

Pc

(yrs

)

Pc OilShock StockCrash

Phase Transition in Equilibrium & Non-Equilibrium Statistical Mechanics

• Equilibrium statistical mechanics in physics:• Under high & low equilibrium temperature• Gaussian (Maxwell) & polarized distribution

• Non-equilibrium statistical mechanics in economics

• Intensity of mutual interaction• Single-hump & U-shaped distribution

Ising Model of Social Opinion (Weidlich 1972)

The steady state of probability distribution function in Ising Model with h=0. (a) Centered distribution with k=0. High social temperature and weak interaction. (b) Marginal distribution at the phase transition with k=2. Phase transition.(c) Polarized distribution with k=2.5. Low social temperature and strong group interaction.

Logistic Interaction without Temperature (Chen 1991)

The steady state of probability distribution function in socio-psychological model of collective choice.(a) Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. (b) Horizontal flat distribution with b = a (denoted by long dashed line.) Marginal case when individualistic orientation balances the social pressure. (c) Polarized distribution with b > a (denoted by solid line).It occurs when social pressure through mutual communication is stronger than independent judgment.

Master Equation for Time-Varying Probability Distribution

• Master equation (partial differential equation)

is the probability distribution function

is is the transition probability

High Moment Risk and Financial Turmoil

High Moment Representation & Crisis Warning (One Quarter Ahead)

• High moments rising thousand times before crisis!

Multiple Regimes in Financial MarketTransition Probability & Birth-Death Process

Calm market ~ linear regime Turbulent market ~ nonlinear regime

From Natural Experiments to New Theoretical Thinking

• New Invariant for Macro & Finance:Stable RD (Relative Deviation) > constant without scale > anchor in macro & finance

• New Selection Rule for Math Tools:• Population model of birth-death process• Better than representative agent models of• Random walk & Brownian motion

Stable RD in US Macro IndexesObserved through HP filter

1950 1960 1970 1980 1990 20000

2

4

6

8

rel

de

v (

%)

gdpc1

gpdic 1

pcecc96

RD Behavior for Stochastic Models

• Random walk is damping over time• Brownian motion is exploding over time (Chen 2005) > BS

model is explosive! > warning before 2008 Financial Crisis

• Only the Birth-death process is stable in time, which is a statistical model of endogenous fluctuations

Order Brownian motion Birth-Death Random-Walk Mean )exp(~rt )exp(~rt t Variance }1){2exp(~

2

tert )1(~rtrtee t

RD )1(~2

2

2 t

tee

0

1~N

t

1

Schrödinger: What Is Life (1948):Why Life Is Stable & Variable ? >

The Principle of Large Numbersfor Positive Variables

Meso Foundation of Macro Behavior (Chen 2002)

• SN=X1+X2+ . . . . . . +XN

• RD(Relative Deviation) =

Fundamental Debate in Macroeconomics & Crisis Policy

• What is the cause of the Great Depression and large unemployment?

• Lucas (1972): microfoundations and rational expectations > household choice between work and leisure > no need for government policy > support Bernanke – Summers policy of saving financial oligarchs during 2008 Crisis

• Chen (2002): meso foundations at financial sector and industrial organization > excess concentration generates large macro fluctuations > competition policy > breaking-up financial oligarchs > support Paul Volcker proposal in dealing with 2008 Crisis

Observed RD(%) and Effective Number N From US macro indexes

Real Personal consumption ： 0.15% (800,000)• Real GDP: 0.2 %

(500,000)

• Real Private Investment: 1.2% (10,000)• Dow Jones Industrial (1928-09) ： 1.4% (9,000)• S&P 500 Index (1947-2009) ： 1.6% (5,000)• NASDAQ (1971-2009): 2.0 %

(3,000)

• Japan-US Exchange Rate (1971-09) 6.1% (300)• US-Euro Exchange Rate (1999-09) 4.9% (400)• Texas Crude Oil Price (1978-2008): 5.3 % (400)

Growth Wavelet of Resource Expansion: Capital Accumulation (Arrow 1962) vs. Technology Metabolism (Chen 2014)

0 100 200 300 400 5000

0.5

1

1.5

2

2.5Logistic Competition

time

popu

latio

n

Changing Paradigm in Physics: Impact to Biology & Economics

• Equilibrium physics > classical mechanics + equilibrium statistical mechanics with Maxwell (Gaussian) distribution

• Newton > Hamiltonian mechanics in closed system• Neoclassical economics simply imitates Hamiltonian mechanics >

ignoring energy dissipation in economic system (wrong model for economics)

• Non-equilibrium physics (1970s)> nonlinear dynamics + non-equilibrium thermodynamics & statistical mechanics > apply to biology & economics

• Deterministic chaos > sensitive to initial condition > limits of trajectory prediction (Poincare 1900, Lorenz 1963)

• Thermodynamics of evolution > no more potential function > no more optimization > pluralistic evolution > history matters (Prigogine 1972)

• Non-equilibrium statistical mechanics > time-dependent distribution > phase transition between calm & turbulent market (Chen 1991, 2014)

Conclusion• Complexity science provides general framework >

integrating conflicting economic schools as special cases (equilibrium, disequilibrium)

• More advanced mathematical representation is needed for complex systems > biological + social systems

• Proper choice of mathematical representation is essential for economics & finance > selection rule should based on empirical relevance, not economic doctrines

• New complex economics stimulate paradigm change > Economic philosophy evolves from Newton atomic world to Schumpeter organic world

References

• Chen, Ping. Economic Complexity and Equilibrium Illusion: Essays on Market Instability and Macro Vitality, London: Routledge (2010).

• Chen, Ping. “Metabolic Growth Theory: Market-Share Competition, Learning Uncertainty, and Technology Wavelets” Journal of Evolutionary Economics, 24(2), 239-262 (2014).

• Tang, Yinan, Ping Chen. “Time Varying Moments, Regime Switch, and Crisis Warning: The Birth-Death Process with Changing Transition Probability,” Physica A (2014).

• Complex_Economics: http://www.complexeconomics.org/