Financial crisis, Omori's law, and negative entropy flo · 2013-06-05 · Financial crisis,...

Financial crisis, Omori’s law, and negative entropy flow

Jianbo Gao

PMB InTelliGence, LLC, West Lafayette, IN 47906

Mechanical and Materials Engineering, Wright State University

[email protected]://www.gao.ece.ufl.edu/

Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori’s law, and negative entropy flow July 2013 1 / 36

Outline

Background and significance

Exposure network preceding the crisisI Omori-like lawI Information flow

Forewarning crisis through distribution analysis

Forewarning crisis using entropy

Nonlinear dynamics associated with crisisI Recurrence plot analysisI Stability of Okun’s law

Concluding remarks


Background

In the past three decades, many countries have experienced financialcrises with different degrees of severity

Especially costly is the 2008 global financial crisis, which has affectedessentially all the industrialized countries, as well as a large number ofdeveloping economies

The 2008 global financial crisis has again pushed early warning system(EWS) models into the spotlight for reducing the risks of future crises

EWS models aim to anticipate whether and when individual countriesmay be affected by a financial crisis

I Types of crises: currency crises, banking crises, sovereign debt crises,private sector debt crises, and equity market crises

I Most EWS models focuses primarily on currency crises


Background (Cont’)

Existing EWS models are not very effective in forewarning crisesI Rose and Spiegel (2009) examined Multiple-Indicator Multiple Cause

(MIMIC) model of Goldberger (1972)I They found that few of the characteristics suggested as potential

causes of the crisis actually help predict the intensity and severity ofthe crisis across countries

The best indicators for the 2008 crisis include asset price inflation,rising leverage, large sustained current account deficits, and a slowingtrajectory of economic growth (Reinhart and Rogoff 2008)

Overall, economists have not had a particularly good track record atpredicting the timing of crises (Rose and Spiegel 2009)


Background (Cont’)

In economics, an important assumption is the economic equilibriumI economic forces are balancedI in the absence of external influences, the equilibrium values of

economic variables will not change

The assumption clearly is violated during a crisis

Existing EWS models employ aggregated variables that cannotexamine the nonlinear dynamics of participating players on scalessmaller than a country in unstable, non-equilibrium economies

Most desirable approach: Understand the large scale emergenteconomic behavior by studying the detailed interactions among theparticipating players of an unstable economy

We propose an anatomical approach toI analyze the exposure networks associated with Fannie Mae/Freddie

Mac, Lehman Brothers, and American International GroupI help understand the mechanisms of financial crisesI identify new robust indicators for financial crises and economic

recessions


Data used in this study

Two types of data were used for this study

One is the amount of investments exposed to FNM, FRE, LEH, andAIG, obtained by exhaustively searching the relevant files on theInternet, and then extracting the amount of investments

The other is income data of thousands of companies in 9 sectors ofUS economy, from 1990 - present

I 9 sectors: Financial, Consumer Goods, Consumer Services, BasicMaterials, Health Care, Industrials, Oil/Gas, Tech-Telecommunications,and Utilities

I Data were obtained from COMPUSTAT data base, by inputting stocksymbol lists for those 9 sectors

I Pretax income data were partitioned into 2 clusters, positive andnegative


Exposure network and Omori law

Exposure network: network nodes were the companies which investedin FNM, FRE, LEH, and/or AIG, and the strength of the links wascharacterized by the amount of the investment

The exposure networks we constructed contain 34, 151, and 146companies worldwide, exposed to AIG, LEH, and FNM/FRE,respectively

The complementary cumulative distribution function (CCDF)

P(X ≥ x) = Probability that X ≥ x million

is well fitted by an Omori-law-like distribution for earthquakeaftershocks:

P(X ≥ x) =(

1 +x

β

)−α, α > 0, β > 0


Exposure network and Omori law (Cont’)P(X ≥ x) =

(1 + x

β

)−α, (α, β) are (2, 118), (1.6, 116), and (0.5, 2), for

AIG, LEH, and FNM/FRE

10−2

100

102

104

10−2

10−1

100

Data x

CC

DF

P(X

≥ x

)

AIGLEHFNM/FRE


Properties of the Omori-like law

When x � β, Omori-like law becomes a power-law, P(X ≥ x) ∼ x−α

I When α < 2, the distribution is heavy-tailed having infinite varianceI When α ≤ 1, the mean also becomes infinite

If we introduce a new random variable, Y = X + β, then Y followsthe Pareto distribution,

P(Y ≥ y) = P(X ≥ y − β) =(

1 + y−ββ

)−α=(

βy

)α, y ≥ β


General considerations about business operation

How is a business operated?

I Suppose a company engages in a number of businessesI Different business areas make different profitsI The one with the highest profit will be privileged and expands rapidlyI While attracting large investments, it also requires larger liquidity and

costs to run itI In a profitable time, all parties will be happy, and investments will be

enhancedI In a troubled time, however, the dominant business area may bring

down the entire company — housing bubble


Stability of the Omori-like economy

From a mathematical modeling perspective, we may assume thatwhen a promising business area is just beginning, the situation isstable

The second law of thermodynamics states that entropy cannotdecrease

The most stable situation is the one with the highest entropyH(f ) = −

∫f (x) ln f (x)dx where f (x) is the PDF

When the mean investment x is given, exponential distributionF (X ≥ x) = e−λx , x ≥ 0maximizes entropy, where λ = 1/x

Entropies for exponential and Omori-like law:HExp(f ) = 1 + ln xHOmori(f ) = 1 + ln x + 1/α + ln[(α− 1)/α], when α > 1

Entropy difference: under the condition that mean is the same,x = β/(α− 1), entropy difference = 1/α + ln[(α− 1)/α] < 0— Less stable


Deriving the Omori-like law

Mathematically, treating λ as a random variable with a PDF f (λ) isequivalent to treating x = 1/λ as a random variable

Then the PDF for x becomes

f (x) =

∫ ∞0

λe−λx f (λ)dλ

and CCDF is

F (X ≥ x) =

∫ ∞x

f (x)dx =

∫ ∞x

∫ ∞0

λe−λx f (λ)dxdλ

Assuming uniform convergence and exchange order of integration, weobtain

F (X ≥ x) =

∫ ∞0

e−λx f (λ)dλ

— Laplace transform of f (λ)

Gamma distributed f (λ) yields Omori-like law !


Deriving the Omori-like law (Cont’)

Gamma distribution:f (λ) = 1

Γ(α)βαλα−1e−βλ, λ ≥ 0, α > 0, β > 0

I A special case: chi-square distribution of degree n,p(λ) = 1

2n/2Γ(n/2)λn/2−1e−λ/2I{λ≥0}

I chi-square distribution is the distribution for the summation of nindependent, standard normal random variables

Q =n∑

i=1

X 2i

I Except for a constant scaling coefficient, Q amounts to the totalenergy of a mechanical system

I chi-square distribution is the distribution that maximizes the entropy ofthe compound system (called variational principle by Chakraborti andPatriarca (2009))


100

102

10−2

10−1

100

P(X

≥ x

)

100

102

104

10−3

10−2

10−1

100

100

102

104

10−3

10−2

10−1

100

P(X

≥ x

)

100

102

104

10−3

10−2

10−1

100

100

102

104

10−3

10−2

10−1

100

Pretax income x

P(X

≥ x

)

100

102

104

10−3

10−2

10−1

100

Pretax income x

(b) Industrial sector 1992 Q1

(c) Technology sector 2001 Q2

(d) Technology sector 2001 Q4

(f) Financial sector 2008 Q2

(e) Financial sector 2007 Q4

α = 0.94 ± 0.13 α = 1.22 ± 0.16

α = 1.78 ± 0.27 α = 1.82 ± 0.40

α = 1.00 ± 0.05

(a) Industrial sector 1991 Q4

α = 0.91 ± 0.10

α = 0.97 ± 0.06

α = 0.91 ± 0.07

α = 1.23 ± 0.10

α = 0.63 ± 0.10α = 0.91 ± 0.08

α = 0.96 ± 0.04

Distribution of lossesaround recession times

CCDF (log-log scale) fornegative (red square) andpositive (black circle) pretaxincomes amongst U.S.companies

Omori-like law still applies !

During crises or recessions,the distribution for thenegative income cluster isheavier than that for thepositive income cluster


Health of US financial industry: far from recoveredRed: negative income; black: positive income

100

105

10−3

10−2

10−1

100

2008 Q1P

(X ≥

x)

100

105

10−3

10−2

10−1

100

2008 Q2

100

105

10−3

10−2

10−1

100

2008 Q3

100

105

10−3

10−2

10−1

100

2008 Q4

Pretax income x

P(X

≥ x

)

100

105

10−3

10−2

10−1

100

2009 Q1

Pretax income x10

010

510

−3

10−2

10−1

100

2009 Q2

Pretax income x


Health of US financial industry: far from recovered

100

105

10−3

10−2

10−1

100

2009 Q3

P(X

≥ x

)

100

105

10−3

10−2

10−1

100

2009 Q4

100

105

10−3

10−2

10−1

100

2010 Q1

100

105

10−3

10−2

10−1

100

2010 Q2

Pretax income x

P(X

≥ x

)

100

105

10−3

10−2

10−1

100

2010 Q3

Pretax income x10

010

510

−3

10−2

10−1

100

2010 Q4

Pretax income x


Health of US financial industry: far from recovered

100

105

10−3

10−2

10−1

100

2011 Q1

Pretax income x

P(X

≥ x

)

100

105

10−3

10−2

10−1

100

2011 Q2

Pretax income x

P(X

≥ x

)


2006_Q1 2006_Q4 2007_Q3 2008_Q20

2

4E

ntro

py

2006_Q1 2006_Q4 2007_Q3 2008_Q20

2

4

6

Ent

ropy

2006_Q1 2006_Q4 2007_Q3 2008_Q20

2

4

Ent

ropy

2006_Q1 2006_Q4 2007_Q3 2008_Q20

2

4

Ent

ropy

2006_Q1 2006_Q4 2007_Q3 2008_Q20

2

4

Ent

ropy

Time

(a) Financial

(b) Consumer goods

(c) Consumer services

(d) Technology

(e) Healthcare

Entropy for negative incomes:2008 crisis

H = −∑

Pi log Pi

Red: negative incomeblack: positive income

By the 2nd law ofthermodynamics, large entropy ismore stable

When entropy for negativeincomes exceeds that for positiveincomes, negative income clusteris even stronger than positiveincome cluster — very indicationof onset of crises

Crisis propagates


1999_Q1 2000_Q1 2001_Q1 2002_Q1 2003_Q10

2

4E

ntro

py

1999_Q1 2000_Q1 2001_Q1 2002_Q1 2003_Q10

2

4

Ent

ropy

1999_Q1 2000_Q1 2001_Q1 2002_Q1 2003_Q10

2

4

Ent

ropy

1999_Q1 2000_Q1 2001_Q1 2002_Q1 2003_Q11

2

3

4

Ent

ropy

1999_Q1 2000_Q1 2001_Q1 2002_Q1 2003_Q10

2

4

Ent

ropy

Time

(a) Financial

(b) Consumer goods


(d) Technology

(e) Industrial

Entropy for negative incomes:2001 recession




Crisis propagates


1990_Q1 1991_Q1 1992_Q1 1993_Q10

1

2

3E

ntro

py

1990_Q1 1991_Q1 1992_Q1 1993_Q10

1

2

3

Ent

ropy

1990_Q1 1991_Q1 1992_Q1 1993_Q10

1

2

3

Ent

ropy

1990_Q1 1991_Q1 1992_Q1 1993_Q10

1

2

3

Ent

ropy

1990_Q1 1991_Q1 1992_Q1 1993_Q10

1

2

3

Ent

ropy

Time

(b) Consumer goods


(d) Technology

(e) Industrial

(a) Financial

Entropy for negative incomes:1991 recession




Crisis propagates


Identification of crises/recessions using various metrics

1990 1993 1996 1999 2002 2005 2008

−2

0

2

Time

Ent

ropy

diff

eren

ce

Consumer ServicesIndustrialTechnologyFinancialTotal

1990 1993 1996 1999 2002 2005 2008−200

−100

0

100

Time

Tot

al in

com

e


1990 1993 1996 1999 2002 2005 2008−5

0

5

Time

I(i)/

|I(i−

1)|


(a)

(b)

(c)

Comparison of metrics: entropy difference,total income, and total income ratio

(a) variation of difference between entropy ofnegative and positive incomes with time

(b) variation in time of total income (wherethe income of the 1st quarter of 1990 is takenas 1 unit)

(c) variation in time of total income ratio(= IQj (i)/|IQj (i − 1)|

), where i denotes year

and j = 1, · · · , 4 denotes quarter, so IQ1 (1990)means 1st quarter income in 1990 (this ratiocrudely measures GDP contraction/expansion)

The grey and orange vertical dashed lines

indicate, respectively, the downturn onset

times determined by NBER and the dates the

NBER announced their onset identifications


When has the US financial crisis ended?

2008_Q1 2009_Q1 2010_Q1 2011_Q12

2.5

3

3.5

4E

ntro

py

Time

Financials

Positive incomeNegative income


On the meaning of negative entropy flow

Consider entropy change associated with the transition from anexponential to an Omori-like distribution

Denote the mean investment at t = 0 by x0.

At t = i , after weighing profits and losses, the mean investmentbecomes xi

The profit is given by ri − 1, where xi = rix0

Using the entropy formula for exponential and Omori distribution,∆Hi = ln ri + 1/α + ln[(α− 1)/α]

I ln ri is directly related to profits or lossesI 1/α + ln[(α− 1)/α < 0 is due to distributional changes of the

investments and may be termed entropy change due to structuralchanges in a business

Since 1/α + ln[(α− 1)/α < 0, to make total entropy changenon-negative, ln ri has to be large enough

I Take AIG and LEH for examples, where α = 2 and 1.6I Then 1/α + ln[(α− 1)/α = −0.19315 and −0.35583I When ∆Hi = 0, ri = 1.2131 and 1.4274, respectively — impossible

during crises


Summary of basic results

Losses in exposure networks can be modeled by a two-parameterOmori-law-like distribution for earthquake aftershocks

I Such a distribution suggests that losses will be widespread aroundcrises or recessions

Indeed during crises or recessions, the heavy-tailed distributions forthe negative income cluster are even heavier than those for thepositive income cluster

Consequently, the entropies associated with the distribution of thenegative income cluster exceed that of the positive income cluster

Distribution and entropy based indicators for crises are very accurate,and can monitor general economic recessions, besides financial crises

Moreover, instability propagates from the crisis initiating sector toother sectors, just as cancer spreads from one part of a body to otherparts


Stability of Okun’s law and economic recessions

Okun’s law: rising unemployment typically coincides with growthslowdowns

1950 1960 1970 1980 1990 2000 2010

−2

0

2

4

6

Time (year)

Diff

eenc

e si

gnal

∆ (unemployment)∆ (GNP)


Model specification of Okun’s law

There are two basic forms of Okun’s law. One is the first-differenceform, described byyt − yt−1 = α + β(ut − ut−1) + εtwhere yt is the natural log of observed real output, ut is the observed

unemployment rate, α is the intercept, β, which is negative, is Okun’s

coefficient measuring how much changes in the unemployment rate,

ut − ut−1, can cause changes in output, yt − yt−1, and εt is the disturbance

term

yt − yt−1 may be written as yt − yt−1 = ln Pt − ln Pt−1 =

ln Pt−1

(1 + Pt−Pt−1

Pt−1

)− ln Pt−1 = ln

(1 + Pt−Pt−1

Pt−1

)≈ Pt−Pt−1

Pt−1

Gap form of Okun’s lawyt − y∗t = α + β(ut − u∗t ) + εtwhere y∗t represents the log of potential output, u∗t is the natural rate of

unemployment, and y∗t and u∗t are complicated functions of time


Complicating factors of Okun’s law

Okun’s coefficient, originally thought to be close to 3, has been foundto be well below 3, and to vary substantially with time and withspatial samples under consideration

Okun’s coefficient depends on the model specification and themethod employed to estimate it

Using regional data, Okun’s coefficient has been found to vary fromregion to region

There appears to be an asymmetry in Okun’s law, i.e., cyclicalunemployment is more sensitive to negative than to positive cyclicaloutput

There is a time varying aspect of Okun’s law, as can be revealed byrolling regressions, or by explicitly allowing for time-varyingcoefficients

Violation of Okun’s law has also been observed


Cross recurrence plot (CRP) of ∆(GNP) and ∆(unemployment)

Construct vectors Xi = (xi , xi+L, · · · , xi+(m−1)L) and Yj

CRP: a dot is at (i , j) whenever ε2 ≤ ||Xi − Yj || ≤ ε1

where ε1 and ε2 are pre-specified scale parameters

1940 1960 1980 2000 20201940

1950

1960

1970

1980

1990

2000

2010

2020

∆ (Unemployment)

∆ (G

NP

)


Insights from the ratio time series

CRP suggests existence of an invariant aspect of the dynamics ofunemployment and production during recessions

Local extrema of the ratio time series coincides with economicrecessions

1950 1960 1970 1980 1990 2000 2010−10

−5

0

5

10

Time (year)

∆ (u

nem

ploy

)/ ∆

(G

NP

)


Partition data into cumulative normal and recessions timesRi =

∑11j=i rj , Ni =

∑12j=i nj

Name Dates Duration NotationRecession of 1949 Nov 1948 - Oct 1949 11 months r1

Recession of 1953 July 1953 - May 1954 10 months r2

Recession of 1958 Aug 1957 - April 1958 8 months r3

Recession of 1960 - 61 Apr 1960 - Feb 1961 10 months r4

Recession of 1969 - 70 Dec 1969 - Nov 1970 11 months r5

1973 - 75 recession Nov 1973 - Mar 1975 1 year 4 months r6

1980 recession Jan 1980 - July 1980 6 months r7

Early 1980s recession July 1981 - Nov 1982 1 year 4 months r8

Early 1990s recession July 1990 - Mar 1991 8 months r9

Early 2000s recession March 2001 - Nov 2001 8 months r10

Late-2000s recession Dec 2007-June 2009 1 year 6 months r11

Table: List of recessions from 1948 - present. Data from National Bureau ofEconomic Research, http://www.nber.org/cycles/cyclesmain.html


Okun’s law during 12 cumulative normal periods

−2 0 2−3

0

3

6 1N

−2 0 2−3

0

3

6 2N

−2 0 2−3

0

3

6 3N

−2 0 2−3

0

3

6 4N

∆ (G

NP

)

−2 0 2−3

0

3

6 5N

−2 0 2−3

0

3

6 6N

−2 0 2−3

0

3

6 7N

−2 0 2−3

0

3

6 8N

−2 0 2−3

0

3

6 9N

−2 0 2−3

0

3

6 10N

−2 0 2−3

0

3

6 11N

∆ (unemployment rate)−2 0 2

−3

0

3

6 12N


Okun’s law during 11 cumulative recession periods

−1 0 1 2−3

0

3

61R

−1 0 1 2−3

0

3

62R

−1 0 1 2−3

0

3

63R

−1 0 1 2−3

0

3

64R

∆ (G

NP

)

−1 0 1 2−3

0

3

65R

−1 0 1 2−3

0

3

66R

−1 0 1 2−3

0

3

67R

−1 0 1 2−3

0

3

68R

−1 0 1 2−3

0

3

69R

−1 0 1 2−3

0

3

610R

−1 0 1 2−3

0

3

611R

∆ (unemployment rate)


Variation of Okun’s coefficient with time

0 2 4 6 8 10 12−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Cumulative period

Oku

n co

effic

ient

Recession

Normal period


Variation of coefficient of determination with time

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

Cumulative period

R2 (

coef

f of d

eter

min

atio

n)

Recession

Normal period


Partial summary

CRP allows for detecting general, non-local correlations betweenunemployment and production and suggests that Okun’s law duringeconomic recessions appears to have captured an invariant aspect ofthe dynamics of unemployment and production

Regression analysis based on data during recessions and normaleconomic times separately shows that Okun’s coefficient is remarkablystable during recessions

However, Okun’s law is continuously weakening during normaleconomic conditions, with the Okun’s coefficient continuouslydecreasing, and approaching 0 in the recent few years, suggestingalmost a total breakdown of Okun’s law after the recent giganticrecession


Concluding remarks

Distribution and entropy based indicators for crises are very accurate,and can monitor general economic recessions, besides financial crises

Moreover, instability propagates from the crisis initiating sector toother sectors, just as cancer spreads from one part of a body to otherparts

Economic crises/recessions are inevitable since Okun’s law duringeconomic recessions appears to have captured an invariant aspect ofthe dynamics of unemployment and production

Entropy flow is close to zero in Marxian economics; is not notconsidered in major economic growth models, including Nobel prizewinning neo-classical growth model, the Solow-Swan model

It is time to seriously consider entropy flow when designing economicpolicies


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