Financial crisis, Omori s law, and negative entropy flow

Transcription

1 Financial crisis, Omori s law, and negative entropy flow Jianbo Gao PMB InTelliGence, LLC, West Lafayette, IN 479 Mechanical and Materials Engineering, Wright State University jbgao.pmb@gmail.com Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 1 /

2 Outline Background and significance Exposure network preceding the crisis Omori-like law Information flow Forewarning crisis through distribution analysis Forewarning crisis using entropy Nonlinear dynamics associated with crisis Recurrence plot analysis Stability of Okun s law Concluding remarks Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 2 /

3 Background In the past three decades, many countries have experienced financial crises with different degrees of severity Especially costly is the 28 global financial crisis, which has affected essentially all the industrialized countries, as well as a large number of developing economies The 28 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 countries may be affected by a financial crisis Types of crises: currency crises, banking crises, sovereign debt crises, private sector debt crises, and equity market crises Most EWS models focuses primarily on currency crises Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 /

4 Background (Cont ) Existing EWS models are not very effective in forewarning crises Rose and Spiegel (29) examined Multiple-Indicator Multiple Cause (MIMIC) model of Goldberger (1972) They found that few of the characteristics suggested as potential causes of the crisis actually help predict the intensity and severity of the crisis across countries The best indicators for the 28 crisis include asset price inflation, rising leverage, large sustained current account deficits, and a slowing trajectory of economic growth (Reinhart and Rogoff 28) Overall, economists have not had a particularly good track record at predicting the timing of crises (Rose and Spiegel 29) Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 4 /

5 Background (Cont ) In economics, an important assumption is the economic equilibrium economic forces are balanced 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 cannot examine the nonlinear dynamics of participating players on scales smaller than a country in unstable, non-equilibrium economies Most desirable approach: Understand the large scale emergent economic behavior by studying the detailed interactions among the participating players of an unstable economy We propose an anatomical approach to analyze the exposure networks associated with Fannie Mae/Freddie Mac, Lehman Brothers, and American International Group help understand the mechanisms of financial crises identify new robust indicators for financial crises and economic recessions Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 5 /

6 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, and AIG, obtained by exhaustively searching the relevant files on the Internet, and then extracting the amount of investments The other is income data of thousands of companies in 9 sectors of US economy, from present 9 sectors: Financial, Consumer Goods, Consumer Services, Basic Materials, Health Care, Industrials, Oil/Gas, Tech-Telecommunications, and Utilities Data were obtained from COMPUSTAT data base, by inputting stock symbol lists for those 9 sectors Pretax income data were partitioned into 2 clusters, positive and negative Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 /

7 Exposure network and Omori law Exposure network: network nodes were the companies which invested in FNM, FRE, LEH, and/or AIG, and the strength of the links was characterized by the amount of the investment The exposure networks we constructed contain 4, 151, and 14 companies 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 earthquake aftershocks: ( P(X x) = 1 + x ) α, α >, β > β Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 7 /

8 Exposure network and Omori law (Cont ) P(X x) = AIG, LEH, and FNM/FRE ( 1 + x β ) α, (α, β) are (2, 118), (1., 11), and (.5, 2), for 1 CCDF P(X x) AIG LEH FNM/FRE Data x Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 8 /

9 Properties of the Omori-like law When x β, Omori-like law becomes a power-law, P(X x) x α When α < 2, the distribution is heavy-tailed having infinite variance When α 1, the mean also becomes infinite If we introduce a new random variable, Y = X + β, then Y follows the Pareto distribution, ( ) α ( ) P(Y y) = P(X y β) = 1 + y β β = β α, y y β Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 9 /

10 General considerations about business operation How is a business operated? Suppose a company engages in a number of businesses Different business areas make different profits The one with the highest profit will be privileged and expands rapidly While attracting large investments, it also requires larger liquidity and costs to run it In a profitable time, all parties will be happy, and investments will be enhanced In a troubled time, however, the dominant business area may bring down the entire company housing bubble Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 1 /

11 Stability of the Omori-like economy From a mathematical modeling perspective, we may assume that when a promising business area is just beginning, the situation is stable The second law of thermodynamics states that entropy cannot decrease The most stable situation is the one with the highest entropy H(f ) = f (x) ln f (x)dx where f (x) is the PDF When the mean investment x is given, exponential distribution F (X x) = e λx, x maximizes entropy, where λ = 1/x Entropies for exponential and Omori-like law: H Exp (f ) = 1 + ln x H Omori (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] < Less stable Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

12 Deriving the Omori-like law Mathematically, treating λ as a random variable with a PDF f (λ) is equivalent to treating x = 1/λ as a random variable Then the PDF for x becomes and CCDF is F (X x) = f (x) = x f (x)dx = λe λx f (λ)dλ x λe λx f (λ)dxdλ Assuming uniform convergence and exchange order of integration, we obtain F (X x) = Laplace transform of f (λ) e λx f (λ)dλ Gamma distributed f (λ) yields Omori-like law! Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

13 Deriving the Omori-like law (Cont ) Gamma distribution: f (λ) = 1 Γ(α) βα λ α 1 e βλ, λ, α >, β > A special case: chi-square distribution of degree n, 1 p(λ) = 2 n/2 Γ(n/2) λn/2 1 e λ/2 I {λ } chi-square distribution is the distribution for the summation of n independent, standard normal random variables Q = n i=1 Except for a constant scaling coefficient, Q amounts to the total energy of a mechanical system chi-square distribution is the distribution that maximizes the entropy of the compound system (called variational principle by Chakraborti and Patriarca (29)) X 2 i Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 1 /

14 P(X x) P(X x) P(X x) α =.9 ±.4 (c) Technology sector 21 Q α = 1. ±.5 (a) Industrial sector 1991 Q4 α =.91 ±.1 α =.91 ±.8 (e) Financial sector 27 Q Pretax income x (b) Industrial sector 1992 Q (d) Technology sector 21 Q α =.94 ±.1 α = 1.22 ± α =.97 ±. α = 1.2 ±.1 α = 1.78 ± α = 1.82 ±.4 α =.91 ±.7 α =. ±.1 (f) Financial sector 28 Q Pretax income x Distribution of losses around recession times CCDF (log-log scale) for negative (red square) and positive (black circle) pretax incomes amongst U.S. companies Omori-like law still applies! During crises or recessions, the distribution for the negative income cluster is heavier than that for the positive income cluster Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

15 Health of US financial industry: far from recovered Red: negative income; black: positive income 1 28 Q Q Q P(X x) Q Q Q2 P(X x) Pretax income x Pretax income x Pretax income x Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

16 Health of US financial industry: far from recovered 1 29 Q 1 29 Q Q1 P(X x) Q Q Q4 P(X x) Pretax income x Pretax income x Pretax income x Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 1 /

17 Health of US financial industry: far from recovered Q Q P(X x) 1 2 P(X x) Pretax income x Pretax income x Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

18 Entropy 4 2 (a) Financial 2_Q1 2_Q4 27_Q 28_Q2 Entropy for negative incomes: 28 crisis H = P i log P i Entropy Entropy Entropy Entropy 4 2 (b) Consumer goods 2_Q1 2_Q4 27_Q 28_Q2 4 2 (c) Consumer services 2_Q1 2_Q4 27_Q 28_Q2 4 2 (d) Technology 2_Q1 2_Q4 27_Q 28_Q2 4 2 (e) Healthcare 2_Q1 2_Q4 27_Q 28_Q2 Time Red: negative income black: positive income By the 2nd law of thermodynamics, large entropy is more stable When entropy for negative incomes exceeds that for positive incomes, negative income cluster is even stronger than positive income cluster very indication of onset of crises Crisis propagates Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

19 Entropy Entropy Entropy Entropy Entropy 4 2 (a) Financial 1999_Q1 2_Q1 21_Q1 22_Q1 2_Q1 4 2 (b) Consumer goods 1999_Q1 2_Q1 21_Q1 22_Q1 2_Q1 4 2 (c) Consumer services 1999_Q1 2_Q1 21_Q1 22_Q1 2_Q1 4 2 (d) Technology _Q1 2_Q1 21_Q1 22_Q1 2_Q1 4 2 (e) Industrial 1999_Q1 2_Q1 21_Q1 22_Q1 2_Q1 Time Entropy for negative incomes: 21 recession Red: negative income black: positive income By the 2nd law of thermodynamics, large entropy is more stable When entropy for negative incomes exceeds that for positive incomes, negative income cluster is even stronger than positive income cluster very indication of onset of crises Crisis propagates Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

20 Entropy Entropy Entropy Entropy Entropy _Q1 1991_Q1 1992_Q1 199_Q1 2 1 (b) Consumer goods 199_Q1 1991_Q1 1992_Q1 199_Q1 2 1 (c) Consumer services 199_Q1 1991_Q1 1992_Q1 199_Q1 2 1 (d) Technology 199_Q1 1991_Q1 1992_Q1 199_Q1 2 (a) Financial 1 (e) Industrial 199_Q1 1991_Q1 1992_Q1 199_Q1 Time Entropy for negative incomes: 1991 recession Red: negative income black: positive income By the 2nd law of thermodynamics, large entropy is more stable When entropy for negative incomes exceeds that for positive incomes, negative income cluster is even stronger than positive income cluster very indication of onset of crises Crisis propagates Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 2 /

21 Identification of crises/recessions using various metrics Total income Entropy difference 2 (a) Consumer Services Industrial Technology Financial Total Time 1 (b) 1 Consumer Services Industrial Technology Financial Total Time 5 (c) I(i)/ I(i 1) Consumer Services Industrial Technology Financial Total Time Comparison of metrics: entropy difference, total income, and total income ratio (a) variation of difference between entropy of negative and positive incomes with time (b) variation in time of total income (where the income of the 1st quarter of 199 is taken as 1 unit) (c) variation in time of total income ratio ( = IQj (i)/ I Qj (i 1) ), where i denotes year and j = 1,, 4 denotes quarter, so I Q1 (199) means 1st quarter income in 199 (this ratio crudely 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 Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

22 When has the US financial crisis ended? Financials 4 Positive income Negative income.5 Entropy _Q1 29_Q1 21_Q1 211_Q1 Time Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

23 On the meaning of negative entropy flow Consider entropy change associated with the transition from an exponential to an Omori-like distribution Denote the mean investment at t = by x. At t = i, after weighing profits and losses, the mean investment becomes x i The profit is given by r i 1, where x i = r i x Using the entropy formula for exponential and Omori distribution, H i = ln r i + 1/α + ln[(α 1)/α] ln ri is directly related to profits or losses 1/α + ln[(α 1)/α < is due to distributional changes of the investments and may be termed entropy change due to structural changes in a business Since 1/α + ln[(α 1)/α <, to make total entropy change non-negative, ln r i has to be large enough Take AIG and LEH for examples, where α = 2 and 1. Then 1/α + ln[(α 1)/α =.1915 and.558 When Hi =, r i = and , respectively impossible during crises Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 2 /

24 Summary of basic results Losses in exposure networks can be modeled by a two-parameter Omori-law-like distribution for earthquake aftershocks Such a distribution suggests that losses will be widespread around crises or recessions Indeed during crises or recessions, the heavy-tailed distributions for the negative income cluster are even heavier than those for the positive income cluster Consequently, the entropies associated with the distribution of the negative 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 to other sectors, just as cancer spreads from one part of a body to other parts Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

25 Stability of Okun s law and economic recessions Okun s law: rising unemployment typically coincides with growth slowdowns (unemployment) (GNP) Diffeence signal Time (year) Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

26 Model specification of Okun s law There are two basic forms of Okun s law. One is the first-difference form, described by y t y t 1 = α + β(u t u t 1 ) + ɛ t where y t is the natural log of observed real output, u t 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, u t u t 1, can cause changes in output, y t y t 1, and ɛ t is the disturbance term y t y t 1 may be written ) as y t y t 1 ( = ln P t ln) P t 1 = ln P t 1 (1 + Pt P t 1 P t 1 ln P t 1 = ln 1 + Pt P t 1 P t 1 Pt P t 1 P t 1 Gap form of Okun s law y t yt = α + β(u t ut ) + ɛ t where yt represents the log of potential output, ut is the natural rate of unemployment, and yt and ut are complicated functions of time Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 2 /

27 Complicating factors of Okun s law Okun s coefficient, originally thought to be close to, has been found to be well below, and to vary substantially with time and with spatial samples under consideration Okun s coefficient depends on the model specification and the method employed to estimate it Using regional data, Okun s coefficient has been found to vary from region to region There appears to be an asymmetry in Okun s law, i.e., cyclical unemployment is more sensitive to negative than to positive cyclical output There is a time varying aspect of Okun s law, as can be revealed by rolling regressions, or by explicitly allowing for time-varying coefficients Violation of Okun s law has also been observed Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

28 Cross recurrence plot (CRP) of (GNP) and (unemployment) Construct vectors X i = (x i, x i+l,, x i+(m 1)L ) and Y j CRP: a dot is at (i, j) whenever ɛ 2 X i Y j ɛ 1 where ɛ 1 and ɛ 2 are pre-specified scale parameters (GNP) (Unemployment) Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

29 Insights from the ratio time series CRP suggests existence of an invariant aspect of the dynamics of unemployment and production during recessions Local extrema of the ratio time series coincides with economic recessions 1 (unemploy)/ (GNP) Time (year) Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July /

30 Partition data into cumulative normal and recessions times R i = 11 j=i r j, N i = 12 j=i n j Name Dates Duration Notation Recession of 1949 Nov Oct months r 1 Recession of 195 July May months r 2 Recession of 1958 Aug April months r Recession of 19-1 Apr 19 - Feb months r 4 Recession of Dec Nov months r recession Nov Mar year 4 months r 198 recession Jan July 198 months r 7 Early 198s recession July Nov year 4 months r 8 Early 199s recession July Mar months r 9 Early 2s recession March 21 - Nov 21 8 months r 1 Late-2s recession Dec 27-June 29 1 year months r 11 Table: List of recessions from present. Data from National Bureau of Economic Research, Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 /

31 Okun s law during 12 cumulative normal periods (GNP) N N4 N7 2 2 N2 2 2 N5 2 2 N8 2 2 N 2 2 N 2 2 N9 2 2 N1 2 2 N (unemployment rate) 2 2 N12 Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 1 /

32 Okun s law during 11 cumulative recession periods R 1 R 2 R (GNP) R R R R 1 R R R R (unemployment rate) Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 2 /

33 Variation of Okun s coefficient with time.5 Okun coefficient Recession Normal period Cumulative period Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 /

34 Variation of coefficient of determination with time.5 R 2 (coeff of determination) Recession Normal period Cumulative period Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 4 /

35 Partial summary CRP allows for detecting general, non-local correlations between unemployment and production and suggests that Okun s law during economic recessions appears to have captured an invariant aspect of the dynamics of unemployment and production Regression analysis based on data during recessions and normal economic times separately shows that Okun s coefficient is remarkably stable during recessions However, Okun s law is continuously weakening during normal economic conditions, with the Okun s coefficient continuously decreasing, and approaching in the recent few years, suggesting almost a total breakdown of Okun s law after the recent gigantic recession Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 5 /

36 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 to other sectors, just as cancer spreads from one part of a body to other parts Economic crises/recessions are inevitable since Okun s law during economic recessions appears to have captured an invariant aspect of the dynamics of unemployment and production Entropy flow is close to zero in Marxian economics; is not not considered in major economic growth models, including Nobel prize winning neo-classical growth model, the Solow-Swan model It is time to seriously consider entropy flow when designing economic policies Gao, Jianbo (PMB InTelliGence) Financial crisis, Omori s law, and negative entropy flow July 21 /