Understanding Economic Complexity and Coherence ... - CiteSeerX

Understanding Economic Complexity and Coherence: Market Crash, Excess Capacity, and Technology Wavelets Ping Chen China Center for Economic Research, Peking University Beijing 100871, China. [email protected] And I. Prigogine Center for Studies in Statistical Mechanics and Complex Systems University of Texas, C1609, Austin, Texas 78712-1081, USA [email protected]

Presented at Shanghai International Symposium on Complexity Science Aug.6, 2002 Abstract There are two conflicting views of market economies. Equilibrium school gives a stable image of efficient market without internal instabilities while econophysics reveals an unstable picture at the edge of chaos. The question is how to provide a consistent framework in understanding the Great depression, the market crash, and over investment in business cycles. It is found out that equilibrium models could not explain observed patterns of business cycles. Economic order and variability can be directly observed from time path of relative deviation and intrinsic period from macro indicators. Birth-death process and technology wavelets provide a new framework in understanding growth fluctuations and product cycles. Key Words: Birth-Death Process, Business Cycles, Color Chaos, Economic Coherence, Economic Complexity, Excess Capacity, Growth Fluctuations, Intrinsic Period, Market Crash, Relative Deviation, Technology Wavelets

Introduction There are two conflicting views of market economies. Equilibrium school gives a stable image of efficient market without internal instabilities, while econophysics reveals an unstable picture at the edge of chaos. The question is how to provide a consistent framework in understanding the Great depression, the market crash, and over investment in business cycles. We find that two new observations reveal the coherent nature of persistent business cycles: the stable pattern of relative deviations and continuious nature of time-frequency path from macro indicators. Based on these observations, we conclude that popular models of random walk, Brownian motion (noise-induced diffusion), Frish model of noise-driven oscillator, Lucas model of microfoundations, and fractal Brownian motion, fail to give a consistent picture of economic stability and variability. Persistent business cycles are endogenous in nature. Birth-death process, color chaos, and dynamic competition provide an integrating picture of economic complexity and coherence. The disordered image of white noise or edge of chaos should be re-examined by empirical evidence and theoretical analysis. The time scale plays a critical role in studying economic dynamics. We will show that the endogenous nature of business cycles, which is not a Brownian motion driven by white noise but a color chaos shaped by a stream of technology wavelets. I. References in Trend-Cycle Decomposition and the Role of Time Scale A distinctive feature of many macroeconomic time series is their uneven growth trends in addition to fluctuating movements. There are three types of detrending methods in econometric analysis. The first is to the percentage rate of changes in a specific time unit. In econometric analysis, it is called the first difference (FD) of the logarithmic time series, which is very erratic and looks like random noise. In macro econometrics, two methods are widely used in trend-cycle decomposition: the log-linear (LL) detrending

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and the Hodrick-Prescott (HP) filter. The images of business cycles under these observation references are quite different. Our question is if there is a preferred reference which provides a consistent picture of growth and cycles in macro movements. The stylized feature of macro indicators can be illustrated by the logarithmic GDPC1, the US real gross domestic product in billions of chained 1996 dollars (Figure 1). We have a quarterly time series from the first quarter in 1947 to the second quarter in 2000. The data source is the Bureau of Economic Analysis (BEA) of the U.S. Department of Commerce. LL & HP Growth Trends of GDPC1Ln 9.5

9

Ln S(t)

8.5

8 GDPC1Ln LLg HPg

7.5

7 1950

1960

1970

1980

1990

2000

(1a) Detrended Series of GDPC1Ln 0.1 LLc FDs HPc

cycles

0.05

0

-0.05

-0.1 1950

1960

1970

1980

1990

2000

(1b) Figure 1. Time Series Patterns under Three Observation References. (a) LL and HP trend of GDPC1Ln quarterly data (1947Q1 –2000Q2). (b) The FD series and the cyclic components of LLc and HPc. N=194. The HP growth trend (HPg ) is quite smooth. The FD s series is very erratic, since the FD is a highfrequency band-pass filter, which amplifies high-frequency noise. The time scale is critical in studies of business cycles. If we consider a normal range in Fourier spectra is about one eighth to one fourth of frequency, then the time interval of observed time series determines the time scale of economic movements (see Table I).

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Table I. The Time Interval and Basic Period in Time Series

∆t =

1min

10min

Day

Week

Mon

Qtr

Yrs

Pmin(Fmax)

2 min

20min

2D

2W

2M

2Qtr

2 Yrs

P(1/2Fmax)

4min

40min

4D

1Mon

1Qtr

1Yrs

4Yrs

P(1/4Fmax)

8min

1.2hrs

1.6W

2Mon

2Qtr

2Yrs

8Yrs

P(1/8Fmax)

0.3hrs

3hrs

3.2W

1.3Qtr

1Yrs

4Yrs

16Yrs

P(1/16Fmax)

0.5hrs

5.3hrs

1.4M

2.7Qtr

2Yrs

8Yrs

32Yrs

According to Table I, the high frequency data (1minute to 10 minutes used in econophysics) is useful for intra-day dynamics, say trading psychology, while daily and weekly data is useful for short-term speculators, monthly and quarterly are good for investors and policy makers in dealing with business cycles. II. Stable Relative Deviation and the Statistical Nature of Many Agents The Brownian motion model of efficient market in finance theory implies a Gaussian distribution with finite mean and variance (Osborne 1959, Fama 1967, Black and Scholes 1973). The fractal Brownian motion model by Mandelbrot suggests a Levy distribution with infinite variance, which is observed from the first differenced high frequency data from stock market (Mandelbrot 1963, Mantegna and Stanley 2000). The Lucas question of microfoundations of macro fluctuations stimulates our investigation of relative deviations from macro indicators (Lucas 1972, Chen 2002a). For non-stationary time series, the relative deviation is measured by the standard deviation of the cseries to the ratio of the g-series within a moving time window, which should be larger than a typical period of business cycles. The relative deviation of real GDP annual series measured by a moving time window of 20 years is shown in Figure 2.

relative deviation (%)

10

LL HP

5

0 1900

1920

1940

1960

1980

Figure 2. The relative deviation of GDP96 under the LL and HP reference. The moving time window is 20 years for annual data. The observed RD can be divided into three periods: (a) 1890 to 1920, RD ~ 1%; (b) 1920-1945, RD ~ 3%; (c) 1950-1990, RD~0.3-0.5%. The large RD was striking during the great depression. The reduction of RD since 1950 is a clear signal of effective control in macroeconomic policies. The different magnitude in three periods can be understood by different institutions and macroeconomic policies in each period. The stable pattern of RD is visible, since there is no explosive or damping trend in each period. There is a debate on the nature of the Great Depression. New classical school recently argue that the persistece of the Great Depression was caused by Keynesian policies during the New Deal. The effectiveness of government policies can be seen from the reduction of relative deviation of US real GDP after 1950s. Most of quaterly macro series are started from 1947. The stable pattern of US real GDP, real consumption, and real investment can be seen from Figure 3 .

3

8 gdpc1 gpdic1 pcecc96

rel dev (%)

6

4

2

0 1950

1960

1970

1980

1990

2000

(a) RD measured under the LL reference. 8 gdpc1 gpdic1 pcecc96

rel dev (%)

6

4

2

0 1950

1960

1970

1980

1990

2000

(b) RD measured under the HP reference. Figure 3. The RDs of gdpc1 (US real GDP), gpdic1 (real investment), and pcecc96 (real consumption) quarterly series (1947-2001). N=220. Moving time window is 10 years.(a) Measured under the LL observation refeence. (b) Measured under the HP reference. The control parameter of the HP filter is 1600 for quarterlyl data. The RD of investment is much larger while the RDs of real GDP and real consumption is quite close. Certainly, the observed RDs have wavelike behavior. However, their mean values could be considered as a constant under the first appxoximation. We will find a proper model to explain this remarkable feature in macroeconomic fluctuations. (2.1) Three Types of Linear Stochastic Models of Growth In stochastic process of growth, three linear models have analytical solutions: the random walk model, the diffusion or geometric Brownian motion, and the birth-death process [5-7]. Their mean and variance are known so that we can calculate their relative deviation. Their stylized behavior is given in Table I.

Order Mean

Table II. The Statistical Properties of Linear Stochastic Processes Random-Walk Diffusion Birth-Death ~t X exp( rt ) X exp( rt ) 0

RD ~

σ t 2

0

2

1

e

t

(1 − e − tσ ) 2

C BD

N0 Here, N 0 is the size of initial population of micro agents in the birth-death process. If N 0 is estimated from observed macro series, which is called the implied number N * from a positive time series. Both the first moment of the diffusion and the birth-death process described an exponential growth. In the sense of the first moment, they are equivalent to a deterministic process of exponential growth. However, they differ significantly in the second moment of variance.

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The Fokker-Planck equation of the diffusion process has two types of formulation in stochastic calculus. In Table I, the mean of diffusion process is calculated by the Ito formulation. For the Stratonovich formulation, the exponential growth rate is not r, but

(r +

σ2 ) . However, both Ito and Stratonovich 2

formulation gave the same result for RD in Table I. This fact indicates that RD is a better measure of system coherence than the traditional measure of variance [Chen and Li 2002]. Obviously, only the birth-death process is capable of explaining the stable pattern of RD from macro indicators. In the case of GDP96 annual data, the average annual growth rate is 3.28% and the variance of LL cycles is 0.0172 in the period of 1889 to 2001. We can calibrate the model by the observed time series under LL trend. The RD of random walk would decline 90% while the RD of diffusion process would multiply 22 times in 100 years! These finding has three implications in economic theory (2.2) Statistical nature of endogenous fluctuations The three stochastic models bear distinguishing features in economic theory. Both random walk and the noise-driven diffusion belong to the representative agent model in macroeconomic literature, which ignore the statistical nature of the many-body problems in macro dynamics. In contrast, the birth-death process is endogenous fluctuation with a large number of elements. The statistical nature of macroeconomic movements is often ignored by most microfoundations models in business cycle theory [Lucas 1972, Chen 2002a]. The fact that economic fluctuations are better described by the birth-death process is strong evidence to support the endogenous school in business cycle theory [Schumpeter 1939]. (2.3) Non-Brownian motion of stock prices Diffusion process is a noise-driven growth [Osborne 1959, Mantegna, and Stanley 2000]. The exogenous school in business cycle theory assumes the stable nature of market movements, so external shocks are the ultimate cause of economic fluctuations [Frisch 1933, Kydland and Prescott 1990]. In the widely used Black-Scholes model of option pricing, it is assumed that stock prices follow a geometric Brownian motion in diffusion process [Black & Scholes 1973]. It is known that the Black-Scholes theory works well for short time transactions, but theoretical error would explode for longer time horizon of more than three months. Most economists doubt the stationary assumption for the variance of stock prices. We also find that the RD of Standard & Poor 500 stock price index has similar stable pattern. Therefore, we have reason to suspect that the Brownian motion or diffusion process may not be a proper type model for stock prices. We will examine the issue elsewhere. (2.4) Weak microfoundations in labor market but strong evidence in financial intermediaries The new classical school led by Robert Lucas believes that fluctuations in labor market are capable of explaining the microfoundations in macro fluctuations [Lucas 1972]. This view is challenged by the findings of the implied numbers from macroeconomic indexes. Consider a macro system with N identical stochastic elements, which have positive values such as output and price. Their mean µ is always larger than zero , and their standard deviation is σ. According to the law of large numbers in probability theory, the mean of the macro system is N µ. Based on the central limit theorem, the variance of the macro system is N σ2 . Therefore, the relative deviation RD of a macro system with N elements is in the order of

1

. We call this rule the Principle of Large Numbers. This

N pattern is still valid for the linear birth-death process [Reichl 1998]. It is easy to figure out that ht e fluctuations generated in financial markets would be much larger than that in labor and producer markets, Empirical observations of implied numbers and actual numbers of economic agents in the US economy are shown in Table III & IV.

Index RD(%)

N*

Table III. The Relative Deviation and Implied Numbers of Macro Quarterly Indexes (1947-2001) under the LL reference GDPC1 PCECC96 GDPIC1 0.34 0.31 1.6 80,000 100,000 4,000

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Table IV. Real Numbers of the US households and Firms in 1980 Micro-Agents Households Corporations Public Companies 80.7(million) 2.9(million) 20,000 N 0

RD(%) 0.01 0.05 0.7 Here, we count only those corporations with more than $100,000 in assets. Obviously, households and small companies contribute little to observed business cycles because of their large numbers. Large business organizations, such as investment banks, multi-national companies, and labor unions have much larger weights in generating macro fluctuations. Correlations among business activities may also amplify internal fluctuations. III. White Noise and Color Chaos in Time -Frequency Space The efficient market theory in finance simply assets that stock price changes can be described by Brownian motion or white noise with normal d istribution (Osborne 1959, Fama 1967). The Frisch model of noise-driven harmonic cycles is inherently unstable because of its linearity (Frisch 1933, Chen 1999). The Schumpeter theory of business cycles is a model of biological clock, which can be represented by a color chaos model in continuous-time (Schumpeter 1939, Chen 1988). Time-frequency representation in Gabor space is a constructed based on Gabor wavelets. It is a powerful tool in separating white noise and persistent cycles. For S&P 500 stock price index, only about 30% of variance of HP cycles is white noise but 70% can be explained by color chaos. The global stability and local instability of color chaos is a key to understanding complexity and coherence of economic movements (Chen 2002b). (3.1) An Exponential Damping of Noise-Driven Cycles In an informal conference paper in 1933, Frisch claimed without proof that damped harmonic cycles could be maintained by persistent shocks (Frisch 1933). In physics literature, the Frisch model is called the Brownian motion of a harmonic oscillator, which can be described by the Langevin equation. (Uhlenbeck and Ornstein 1930). The Langevin equation can be transformed into a Fokker-Planck equation. Its analytic conclusion is contrary to the Frisch claim: the harmonic oscillation will be dampened in an exponential way. Persistent cycles cannot be maintained by random shocks. For a representative model of noise-driven damped harmonic oscillator, the observed autocorrelation is (Wang and Unlenbeck 1945, Chen 1999):

ρ (τ ) = exp( −

β 2π βT 2π τ )[cos( τ ) + 1 sin( τ )] 2 T1 4π T1

(3.1)

We can apply the harmonic Brownian model to the logarithmic US real GDP data. The GDP cycles will cease within 20 years for the HP cycles and in less than 7 years for the FD fluctuations. Here is another problem with the FD reference: it needs such a strong driven-noise, which is larger than the US economy! If the Frisch claim could be realized, it implies a perpetual motion machine. This scenario violates the second law of thermodynamics. (3.2) The Uncertainty Principle and Gabor Wavelet The uncertainty principle in time and frequency is the very foundation of signal processing (Qian and Chen 1996):

∆f ∆t ≥

1 4π

(3.2)

Here, f is the frequency and t is time. Minimum uncertainty occurs for a harmonic　wave modulated by a Gaussian envelope, which is called the Gabor wavelet in signal processing or a coherent state in quantum mechanics. This is the very foundation of time -frequency analysis in the two-dimensional time-frequency Gabor space (Figure 4).

6

. Figure 4.A two-dimensional Gabor wavelet with minimum uncertainty: WDb (t, f) = 2 exp{- [ (t/σ)2 + (2πfσ)2 ] }* exp[i (2πft)] (3.3) Separating Noise and Cycles in Time-Frequency Space For analyzing a time-dependent series, we introduce a new analytic tool, the joint time-frequency analysis (Qian and Chen 1996, Chen 1996a,b). A time-varying filter in a two-dimensional time-frequency lattice space can be applied for separating cycles and noise. Its localized bases are the Gabor wavelets. The Gabor distribution and time series of filtered and unfiltered HP cycles are shown in Figure 5. The deterministic pattern of filtered HP cycles can be clearly seen from the phase portrait in Figure 6.

Unfiltered & Filtered Gabor Distribution

(a). The Gabor distribution of the unfiltered (upper) and filtered (lower) time series.

4

Filtered & Original HP Cycles Xo Xg

X(t)

2

0

-2

-4 1945

1955

1965

1975

1985

1995

t

(b) The filtered Xg and the original time series X o .

7

Figure 5. The filtered FSPCOM (S&P Price Index) HP cyclic series Xg closely resembles the original time series Xo . The correlation coefficient between Xg and Xo is 0.85. The ratio of their variance is 69 %. The correlation dimension of Xg is 2.5.

FSPCOM Filtered HP Cycles 0.3

0.2

0.2

0.1

0.1

X(t+T)

X(t+T)

FSPCOM Raw HP Cycles 0.3

0

0

-0.1

-0.1

-0.2

-0.2 -0.3

-0.3 -0.3

-0.2

-0.1

0

0.1

0.2

-0.3

0.3

-0.2

-0.1

0

0.1

0.2

0.3

X(t)

X(t)

Figure 6. The phase portraits of the unfiltered (the left plot) and the filtered (the right plot) FSPCOM HP cycles. The time delay T is 60 months. The phase portrait of filtered FSPCOM (S&P 500 price index) HP cycles shows a clear pattern of deterministic spirals, a typical feature of color chaos. Color chaos here refers to the nonlinear oscillator in continuous time. Color means a strong peak in spectrum in addtion to a noisy background (Chen 1996b). (3.4) Natural Experiments of Economic Clock: Intrinsic Instabilities and External Shocks in Evolving Economies According to new classical economists, business cycles are all alike (Lucas 1981). From new observations in time-frequency analysis, we find that business cycles are not all alike. The time-frequency patterns of macroeconomic indicators resemble biological organisms with multiple rhythms. The frequency path can reveal valuable information in economic diagnostics and policy studies. Our picture of an economic clock is a dramatic contrast with those of a random walk in equilibrium economics. Can we conduct some out-of-sample tests to distinguish these two approaches? Perhaps not, because nonstationarity is the main obstacle to the application of statistics. However, the natural experiments of the oil price shock and the stock market crash demonstrate that time-frequency representation reveals more information than white-noise representation. The basic frequency of the S&P 500 Index was very stable compared to most macro indicators. We observe that the basic period Pb of FSPCOM HP cycles shifted after the oil price shock in October 1973, which signals an external shock. However, the frequency changes occurred before and after the stock market crash in October 1987. This fact suggests an internal instability in the stock market during the crash (Figure 7). Pb History of FSPCOM HPc 20

Pb OilShock StkCrash

Pb (yrs)

15

10

5

0 1965

1970

1975

1980

8

1985

1990

Figure 7. The time path of the intrinsic period Pi of FSPCOMln (the S&P 500 Price Index) HP cycles stock market indicators. We should note that the stock market crash in October 1987 led to a 23.1 percent drop in the level of the S&P 500 index in two months, but only a 6-percentage shift in its intrinsic period Pi . This is a typical feature of economic coherence, which could not be explained by the Brownian motion model of stock market. That is why the stock market crash did not trigger a Great Depression such as the event in 1929. Persistent cycle can be described by the color chaos model in delay feedback control (Chen 1988). IV. Product Cycles, Technology Wavelets, and Excess Capacity Technology advancement is the driving force of industrial economies. The birth and death of technologies and waves of product cycles are common features of a modern economy. Industry competition for increasing market share is largely technology driven, rather than price driven. The Arrow-Debreu model has only fixed number of commodities. Therefore, the competition model in ecological dynamics is a better model for technology advancement in modern economy. (4.1) Two-Species Competition Model When there are two competing technologies, their market shares are characterized by their resource ceilings N1 and N 2 . The Lotka -Volterra competition equation in population dynamics can be applied to market-share competition under conditions of limited resources (Pianka 1983).

d n1 = k1n1 (C1 − n1 − β n2 ) dt d n2 = k 2 n2 ( C 2 − n2 − β n1 ) dt

(4.1)

Where n1, n2 are population (or output) of species (technology or product) 1 and species 2; C1 and C2 is their carrying capacity (or resource limit) k1 and k2 their growth (or learning) rate;

β

is the overlapping (or

competition) coefficient in market or resource competition ( 0 ≤ β ≤ 1 ).

When β = 0, there is no competition between the two species. Both of them can grow the market share within the constraints of available resources, which ecologists refer to as effective carrying capacities. A firm without competition could realize its full capacity to occupy the full market share C. Species 2 will replace species 1 under the following condition:

β ( N2 −

R2 R ) = β C 2 > C1 = ( N 1 − 1 ) k2 k1

(4.2)

The winner may have a higher resource capacity, a faster learning rate, or a smaller death rate. When 0 < β < 1 , the two species can co-exist. However, the realized market share would decline to a level depending on the competition parameter β.

C2 1 < C1 β C − β C2 n1* = 1 < C1 1− β 2 C − β C1 n *2 = 2 < C2 1− β 2 β