Absence of 1/f Spectra in DowJones Daily PriceWentian Li

SFI WORKING PAPER: 1990--009

SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent theviews of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our externalfaculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, orfunded by an SFI grant.©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensuretimely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rightstherein are maintained by the author(s). It is understood that all persons copying this information willadhere to the terms and constraints invoked by each author's copyright. These works may be reposted onlywith the explicit permission of the copyright holder.www.santafe.edu

SANTA FE INSTITUTE

Absence of 1/f Spectra in DoW" Jones Daily Price

Wentian LiSanta Fe Institute, 1120 Canyon Road, Santa Fe, NM 87501

April 9, 1990 *

Abstract. The power spectra for the daily Dow Jones average, boththe original series and the de-trended series, are calculated. The spec­tra are of the form 1/ fCi with a close to 2 instead of 1. Also studiedis the distribution of daily price changes which is in-between an expo­nential and a power law function. With certain criterion, this resultsuggests that stock market activities may not be qualified as "critical".

1. Introduction

Noise can·be classified according to its power spectrum - a decomposition ofthe time series into components with different frequencies by their contribu­tion to the variance. If there is no preferred frequency component, the powerspectrum is flat, and the time series is called white noise. Sometimes, due tothe large contributions from the slow-varying, non-periodic components inthe time series, the spectrum peaks at lower frequencies. The increase of thespectrum at lower frequencies can be a power-law function 1/ fa with a ~ 2,f being the frequency, then the time series is called 1/ f2 noise, or it is apower-law function 1/ fCi with a~ 1, then the series is called 1/f noise (see[3]).

1/f noise has been found in traffic flow [22], electric current throughsolids [31], fluctuation of the frequency of quartz oscillators [1], luminosityfluctuation of stars [24], the tune of classical music [30], as well as a largenumber of other systems and other signalse Unlike white noise and 1/ f2noise, which can be easily generated by a sequence of uncorrelated randomvariables and a sequence with uncorrelated random increment (random walkor Brownian motion), the theory of 1/f noise is wide open. The numberof the models for 1/f noise is as numerous as the number of systems whichshow 1/f noise.

Although 1/f noise is very common, there is little, or perhaps no, recordson whether stock market prices are one of theme Ref. [2] lists stork exchangeprice as one example of the 1/f noise, quoting from an earlier paper [19].Nevertheless, by checking the literature more carefully, one will find that

*The main result in this paper was presented at the Dynamics Days'90 workshop(Austin, Texas, January 3-6, 1990).

1

Dow Jones 2

the most solid result of frequency domain analysis on' economic time seriesis that the lower-frequency components of the power spectrum are typicallylarger than the higher-frequency components [5], but no specific forms of thespectra are given.

In this paper, straightforward power spectral analysis has been performedfor the most generic stock market index: the Dow Jones daily closing price.The result shows that this time series is very closer to a 1/ f2 noise, evenafter the trend is removed.1 Although it is not impossible for other economictime series to show 1// noise, by the existing evidence on the analysis ofeconomic data (most of them are restricted in time domain), it is temptingto believe that 1//2 noise is more generic for economic series.

2. Numerical Results of Power Spectra

Power spectra are only well defined for stationary series. One should pro­ceed with care to apply them to non-stationary series because spectra mightchange with time. For a finite series, there is no problem since it is difficultto judge whether a finite series is stationary or not. Even if the finite serieslooks non-stationary, we can always make it stationary by using the periodicboundary condition. In this sense, it is always possible to calculate the powerspectrum for a finite number of data points.

Nevertheless, if the finite sequence is indeed "non-stationary" in the infi­nite time limit, the power spectrum determined from N data points will bea function of N. Here we will only consider the situation where the shape ofthe spectrum is not sensitive to N, only its overall level may change.

Another way to make spectral analysis meaningful for non-stationary se­ries is to "remove" the non-stationary part of the series. For example, a seriesmay commonly have a trend:

X(t) - met) + Xt(t)

where the trend met) is a simple given function and Xt(t), the de-trendedseries, is stationary with zero mean. If the series can be decomposed thisway, one can calculate the power spectrum for the de-trended part of thetime series Xt(t). The power spectrum of the original series X(t) will have anoverestimation of the lower-frequency components because the slow-varyingfunction m(t) contributes to ,the lower-frequency parts of the spectrum.

Another simple situation of the non-stationarity is when the sequence ismade stationary by differencing it. For example, the difference series

X(t + 1) - X(t) == x2(i),

the log-difference series

logX(t + 1) -logX(t) = x2(i),

(2.2)

(2e3)

ITrends contribute heavily to the zero-frequency and lower-frequency components ofthe spectrum. Actually, linear trends can produce' a background 1//2 spectrum. See p.132of [4] as well as the Appendix in this paper.

Dow Jones

or the percentage change series

X(t + 1) - X(t) _ '()X(t) - X2 t

3

(2.4)

is stationary. In anyone of the above cases, one can calculate the powerspectrum for X2(t), the series after differencing.

It is well known that if X2(t) is a white noise with a flat power spectrum,the original series X (t), or log X (t), or dX (t) / dt, is Brownian noise witha 1/ f2 power spectrum (see, e.g., Ref. [18] as well as the Appendix in thispaper). So differencing a series can greatly change the shape of the spectrum.This is in contrast with the case of de-trending, where the power spectrumfor X1(t) is more or less the same as the original series X(t). In this section,I will mainly use the de-trending method to calculate power spectra of stockprices, and use differencing method only as an independent check.

The collected data are the daily closing price of Dow Jones Industrial,the average of the 30 stocks. There are 5346 data points available (fromOctober 14, 1974 to November 3, 1989 after filling the weekends and holidayswith Friday's prices), but I delete the first 1200 points, and use the next4096 (=212) points because the standard fast Fourier transformation (FFT)program requires the number of points to be a power of 2.

The definition of power spectrum P(f) is standard (see any textbook onFourier transformation):

P(f) = NIA(f)f2, (2.5)

I use the FFT subrouting.in C.language given by Ref. [27]. Figure 1 shows theDow Jones daily price. X(t), its trend approximated by a linear function, andthe de-trended series X1(t). The trend fitted by the straight line m(t) = a+bthas slope b~ 0.308 and y-intercept a ~ 445.94.

The power spectrum for both the original series X(t) and the de-trendedseries Xl (t) is shown in Figure 2 (a) and (b), respectively. The original2048 components of the power spectrum are shown by dotso I averaged theneighboring 8 components to make it smooth:

Pave(8i < ! < 8(i + 1)) = ~ t P(! = 8i + j) for 0 < i < 256. (2.6)j=l

Straight lines are used to fit the spectra Pave(f) in log-log plot from 1=8 tof=2000. In Figure 2(a), the slope of the fitting line is approximately -1879;in Figure 2(b), the slope is -1.76.2 It is clear that de-trending the series notonly reduces the total power (the area under the spectrum), but also slightlydecreases the exponent a in 1/ fO:.

2Since the slope of the fitting line depends on at which frequency range the spectrumis fitted, the last digit of the exponent may not be reliable.

Dow Jones 4

Although approximating the trend by a linear function is a common prac­tice in spectral analysis of non-stationary series [7], we want to check whetherchoosing other functional forms of m(t) will affect the spectra of the de­trended series. As an example, I use the quadratic trend: m(t) = a +bt +ct2

•

The fitting curve and the de-trended series for the same Dow Jones data areshown in Figure 3(a), with a ~ 971065, b ~ -0.28 and c ~ 0.00011. Thepower spectrum for the de-trended series is shown in Figure 3(b). After aver­aging the neighboring eight components, the spectrum can be approximatedby P(f) ~ 1//1.74 by a fitting from point 1=8 to 1=2000. No drastic changeof the exponent a is observed when we change from linear fitting to quadraticfitting. It is consistent with what was observed in Ref. [4]: as long as mostof the trend is removed, the spectrum estimation is reliable.

It is common in finance community to study the log-price series logX(t)instead of the original series X(t). Figure 4(a) shows the log~price log10X(t)and the linear fitting m(i) = 208 + O.000097t. The power spectrum of thede-trended series log10X(t) - m(t) is shown in Figure 4(b). It is a powerlaw function: 1//1077, with the exponent determined by fitting the averagedspectrum from point 8 to point 2000.

It is clear from Figure 2(a), 2(b), 3(b) and 4(b) that the Dow Jones dailyprice is not 1// noise, but 1//2 noise. If the 1//2 noise is resulted from arandom walk, the difference series should be white noise. As an independentcheck, the power spectrum of the difference series X2(t) = X(t + 1) - XCi),shown in Figure 5(a), is calculated and shown in Figure 5(b). This powerspectrum is almost white (also shown is a best-fit straight line in the log-logplot with the slope 0.045). Similarly, the difference series X2(t) = loglo X(T+1) -log10X(T) is plotted in Figure 6(a) and its power spectrum is shown inFigure 6(b). The best-fit line in the log-log plot has a slope 0.0046.

The small deviation from the zero slope in the fitting of the spectrummight either be a statistical fluctuation, or an indication of the real weakcorrelation. In the latter case, the small exponent 0.045 and 0.0046 indicatesthat even if we have a correlation, the magnitude is very small. Noticefrom Figure 5(a), that the variance of X2(t) seems to increase - anotherphenomenon under active investigation (see, e.g., [10]).

The next question we would like to ask is whether the power spectradetermined from the daily Dow Jones industrial price is representative ofother stock prices or other economic data. Figure 7(a) shows the time series ofthe average price of20 bonds X(i) (total 4919 points), its best-fit linear trendm(t) = 77.87 +O.Oa08t, as well as the de-trended signal X1(t) = X(t) - m(t).Figure 7(b) shows the power spectrum of Xl(t), for the 212 = 4096 pointsstarting from the SOOth point. A particular fitting (from f=8 to 1=600)estimates the spectrum to be P(f) ~ 1//2.2. Since there is almost no trendin the bond series, the spectrum for Xl (t) should be almost the same as theoriginal series X (t).

In Figure 7(b), the higher frequency components are considerably flatterthan the lower frequency components. Similar shape of the spectrum isalso observed in US-Canada dollar exchange rate [28]0 In this case, the

Dow Jones 5

exponent of the power law can be greatly underestimated if the end pointof the fitting line is moved to higher frequency values. The stock tradingvolume series are known to be less "persistent" than the price series [11]0 Inother words, their power spectra are "whiter" than those of price series, atleast at higher frequencies. The spectral as well as other statistical analysisof trading volume series is currently carried out [28, 11]. The results will bediscussed elsewhere.

In summary, the Dow Jones daily price, both the de-trended and the orig­inal series, show 1/JOt spectra with a closer to 2 instead of 1. The differenceseries has a flat spectrum. This result is not surprising considering that therandom walk model is to some extent a reasonable good model for real stockreturn series [23]. Many recent studies on stock price are.re-examinations ofthe random walk assumption [17], but the deviation from the random walkmodel is not large enough to change the spectrum from 1/ f2 to 1/f. Thepreliminary spectral analysis on other stock prices, such as S&P 500, drawsthe similar conclusion [28].

3. Are Stock Markets "Critical" Systems?

In statistical physics it is known that there is a critical temperature separat­ing ordered and disordered phases such as solid and liquid. At the criticalpoint or transition point, the system does not belong to either the orderedor the disordered state, and it typically has fluctuations at many time andlength scales [32].

This concept of phase-transition and criticality can be generalized to com­plex systems with other tuning parameters rather than temperature. Onesuch study was carried out for cellular automata (discrete dynamical sys­tems with local coupling) rule spaces [15]. Indeed, it has been discoveredthat there are "complex" rules residing in regions of the rule space whichseparate periodic dynamics and the chaotic dynamics [8, 9, 13, 15].

Since stock markets are not deterministic dynamical systems with knowndynamical rules, it is subtle to apply the concept of criticality here. Althoughwe can roughly speak about complex or critical as being neither regular norchaotic, more specific definitions are needed.. In Ref. [15], the poor conver­gence of the statistical properties is used as a criterion of complex dynamics,but again the term "poor" needs clarification. Another criterion of criticalityspecifically requires the "poor" convergence as being a power law distribu­tion of the responses: h(~) I'J 1/~a with small exponent 0:, preferably a ~ 1[2].3 In this section, I will adopt this criterion, and systems with irregularresponses but their distributions are not of the above mentioned form willbe rejected as "critical" systems.

Note that the distribution of responses can be directly related to thepower spectrum of the "response series" when we assume that each response

3If the exponent of the power law is too large, it can be indistinguishable from anexponential function.

Dow Jones 6

contributes to the power spectrum in a particular form. The standard as­sumption is that each response relaxes exponentially, which contributes aLorentzian spectrum [29]. Whether the same assumption can be made tostock market is questionable. Also we may not look at the "response se­ries" but some other time series. Therefore, I will treat the distribution of"responses" (daily price change) as a calculation independent of the powerspectrum of the price series.

How can we measure the responses for a stock market? The difficultycomes from the fact that we basically cannot design experiments in realstock markets. We have to rely on the existing data instead. Here, I definethe "perturbation" to be the beginning of a day, and the "response" to bethe stock price change on that day. The difference series for the Dow Jonesaverage is our response series, which has already been plotted in Figure 5(a).The distribution (histogram) for this daily price change is shown in semi-logscale in Figure 8(a), and log-log scale in Figure 8(b).

In Figure 8, I plot the distribution of the rise and fall of daily pricesseparately - the two seem to be identical. Each bin in the histogram has asize 10. In the semi-log plot, an exponential function

h(~) I"V e-O.08~

is plotted in comparison. In the log-log plot, a power law function

h(Ll) I"V 1/Ll3

(3.1)

(3.2)

is plotted. There are 5 data points left out from the plots: 508 down ("BlackMonday" in 1987), 186 up, 156 down, 140 down, and 190 down. These largefluctuations are the most recent (all after the Black Monday), and exceptionalevents, perhaps reflecting the fact that the underlying dynamics is changing.

Note that both the exponential function with slow decaying rate (a = 0.08in h(t:..) = e-(8

) and the power-law function with large exponent (0: =3 in 1/ !:lOt) can fit the distribution quite well. One interpretation of thisambiguity is perhaps that the distribution is neither a typical power-law nora typical exponential function. Another example of this type of function isthe "stretched exponential" function: h(fj,,) = e-Ot8f3 with 0 < f3 < 1, whichis frequently used in studying slow-varying phenomena such as the relaxationof spin-glass to its minimum energy state [26].

Another factor causing the ambiguity is that most of the data are withinone decade. Larger daily price changes ~ are extremely rare. The increase ofthe variance in recent time makes it easier to gather more countings for ~ >100 in the future. Nevertheless, the very fact that there will be more large~'s may mean that the stock market activity is changing its characteristics,and we are obtaining statistics from a changed dynamics system [10].

If we define the "response" to be the daily change of the log-price, we willget a different histogram. This histogram is shown in Figure 9 in semi-logscale. The straight line is plotted for comparison:

h(Ll) I"V e-4006.. (3.3)

Dow Jones 7

Again, thirteen large events are left out (the maximum change of log-price is-0.11 on "Black Monday"). We can also choose the daily trading volume asthe "response". Typically, it is not expected to have small trading volumesso it is unlikely that they obey a power law distributione

In summary, by choosing one criterion of the criticality (power-law dis­tribution of the responses 1/~a with a ~ 1), as well as one particular choiceof perturbation (the starting of a day) and the response (the change of theprice), the stock market does not seem to be qualified as "critical". Ofcourse, there are many other choice of "responses" , so the definition of criti­cality seems to be arbitrary to certain degree. The fast-decay functional formin the daily price change distribution implies that the stock market (charac­terized by the price change) is more stable than critical systems, which canbe called subcritical, or the "left edge of the chaos".4 What we have donehere is essentially to study a complex system with many interacting degreesof freedom by only one variable. Whether a single variable can capture thecharacteristics of the system is still an open question, which is similar tothe situation of using electro-encephalograms (EEG) to measure the brainactivities.

4. Conclusion

Power spectral analysis can only provide very crude statistical characteriza­tion of time series. Typically, spectra analysis alone cannot decide the dy­namical rules which model the time series. For example, both deterministic(e.g., logistic map with the maximum parameter value [21]) and stochas­tic systems can generate white noise, so a flat spectrum cannot tell whichmodel is correct. Similarly, both short-ranged deterministic rules (e.g., in­termittency mapping [20]) and long-ranged stochastic rules (e.g., fractionaldifferencing of white noise [19]) are capable of generating 1/f spectra. As aresult, the study in this paper is not directly related to the modelings of thestock activities or the prediction of stock prices.

On the other hand, since power spectra measure the magnitude of fluctu­ations in frequency domain, it is an appropriate measure not only for systemswith characteristic time scales, but also for systems with all frequency COffi­

ponents..A certain form of the spectra gives us general information aboutwhether the slow or the fast fluctuation is the dominant component in atime series. The absence of 1/f spectrum in the Dow Jones daily price asshown in this paper indicates an absence of a certain scaling relation betweenfluctuations at different time scales.

Our result should not be surprising by examining the current literatureon analysis of stock price series. Actually, many spectra of the economictime series were calculated more than twenty years ago (see Ref. [4]). Byredrawing them in a log-log instead of a semi-log plot, one will find thatthey are not 1/f spectra either. Recently, there are other studies on the

4The name "edge of chaos" is coined in Ref. [25] to represent systems showing complex,critical dynamics.

Dow Jones 8

long-range effect of the stock return series using other statistical quantities.5

A deviation from the white noise in stock return implies a deviation fromthe 1/ f2 spectra for the original stock price series. Such long-term memoryeffects seem to be very small, if they exist at all [16], and the stock priceseries should have a 1/ f2 instead of 1/f spectrum.

Acknowledgement

I would like to thank Stephen Eubank for providing the data, Doyne Farmer,Martin Casdagli and John Scalo for comments. Prof. William Brock wasmost helpful in pointing out the relevant papers on stock market. I appreciateAndy Lo for sending me his papers, and Ronda K. Butler-Villa for proofread­ing the draft. Kurt Thearling has checked independently some of the spectrapresented here, and Blake LeBaron made many valuable suggestions to thedraft. I thank them for their contributions. The work at SFI was supportedby NSF Grant PHY-87-14918 and DOE Grant DE-FG05-88ER25054.

Appendix

A. Simple Time Series with 1/ f2 Spectra

1/ f2 spectrum is typically not discussed in standard textbooks on spectralanalysis, because the lower frequency divergence is an indication of the non­stationarity while most of the textbooks discuss only stationary series .. Somenumerically obtained '''1/ f2 spectra" do not actually diverge though, eitherbecause they are not 1/ f2 at all frequencies (such as the Lorentzian spectruma2/(f2 + a2). see, e.g., [12]), or because the series does not have infinitenumber of points.

Power spectrum P(f) is the Fourier transformation of the autoCQVarialltfunction r(d) by"Wiener-Khinchin theorem" (the {=::} represents the Fourierand inverse Fourier transformation in either direction):

P(f) {=::} f(d) (A.I)

where the autocovariant function is defined for both continuous (and in­finitely long) and discrete (and finite). series

JOO 1 N-d

r(d)= _oo(X(t)-X)(X(t+d)-X)dt, r(d) = N E(Xi-X)(Xi+d-X)

(A.2)where X is the mean of XCi). Notice that the upper limit in the summationis N - d instead of N for the case of finite series. It is equivalent to the zeropadding in the end of the series (Xi = 0, if i > N), rather than the periodicboundary condition (Xi = Xi- N , if i > N).

5For example, the RIS test [6, 16].

Dow Jones 9

It is known that by multiply the distance by a factor of k, we have thefollowing relation:

Using this equation, we can easily see that if the autocovariant function islinear, i.e., f(kd) = kf(d), the corresponding power spectrum will be 1/ f2,because

In other words, the 1/ f2 spectra are related to series with linear autocovariantfunction.6

There are many series with linear autocovariant function, one examplebeing the random arrangement of blocks of size n with either all O's or alII's[14]. In this appendix, only examples relevant to the discussions in this paperwill be given. They are: (1) random walk and (2) linear trend. For bothcases, there are equally easy ways to prove that these spectra are 1//2: (1)by doing the integral by parts, then using the fact that the difference seriesis a white noise (see [18]), and (2) by Fourier series of the linear function (see[4]). Nevertheless, our approach provides a better perspectives and a moreconsistent description.

In the following, I will prove that the 'autocovariant function for bothseries are linear (plus a constant), thus the spectra are 1/ f2 0 In the derivationof the autocovariant function, I assume the zero padding boundary condition.

1. Random walk: defined by

i

Xi = L: €jj=l

with ti random variables with zero mean and variance 0'2. The mean of {Xi}is also approximately zero

(A.6)

because random €i'S are cancel to each other. The autocovariant function is

1 N -d i i+d 1 N -d i i+d

r(d) = - L: (L: tj - X)(L: tk - X) ~ - L: L: L: tjtkN . 1 . k N. .

2= J=l =1 t=l J=l k=l

2. ~d. 2 '" (N - d + 1)(N - d) 2~ N ~ 'to' "-I 2N a

2=1

N(N + 1) - (2N + l)d + d2(J'2 ~ (N _ d)a2

2N 2 (A.7)

60r a linear function plus a constant term - the latter will only contribute to thezero-frequency component of the spectrum.

Dow Jones 10

where the only fj €k '8 that escape from being canceled to each other are theterms with j = k. In the last approximation, I assume N ~ d > 1.

2. Linear trend: defined by

The mean is

·X=.!..I>=N+lN i=l 2

and the autocovariant function is

(A.9)

f(d) = ~ I:d(i_ N+l)(i+d_ N+l)N i=l 2 2

(N - d)(N - d + 1)(2N + d + 1)6N

N(N2 - 1) - (3N2- l)d + 2d3

12NN2 N

~ ---d.12 4

Again I assume that N ~ d > 1 in the last line.Two comments: (1) Although we assume the zero padding boundary

condition (Xi = 0, if i > N), readers can check that the main conclusionis unchanged for periodic boundary conditions (Xi = X i - N , if i > N). Inthe latter case, there will be an extra term in the autocovariant function:

N - -1/N ~i=N-d+l (Xi - X)(Xi+d- N - X). (2) Both Eq. (A.7) and (A.IO) arefunction ofN, the number of data points, which confirms the statement thatautocovariant functions as well as power spectra for non-stationary seriesdiverge in the limit of N ~ 00 . Nevertheless, the fact that r (d) 's are linearfunctions of d is unaffected.

Dow Jones

References

11

[1] W.R. Attkinson, L. Fey, J. Newman, "Spectrum analysis of extremely lowfrequency variation of quartz oscillators," Proc. IEEE, 51,379 (1963) ..

[2] P. Bak, C .. Tang, and K. Wiesenfeld, "Self-organized criticality," PhysicalReview A, 38(1), 364-374 (1988).

[3] E ..g., Martin Gardner, "Mathematical games: white and brown music, fractalcurves and l/f fluctuations," Scientific American 238(4), 16-32 (1978).

[4] C.W.J. Granger and M. Hatanaka, Spectral Analysis of Economic TimeSeries (Princeton University Press, 1964).

[5] C.W.J. Granger, "The typical spectral shape of an economic variable,"Econometrica, 34, 150-161 (1966).

[6] M. Green and B.. Fielitz, "Long-term dependence in common stock returns,"Journal of Financial Economics, 4, 339-349 (1977).

[7] E.g.. , Cythia Kuo, Craig Lindberg and David J. Thomson, "Coherence es­tablished betweened atmospheric carbon dioxide and global temperature,"Nature, 343, 709-714 (1990).

[8] C. Langton, "Computation at the edge of chaos: phase transitions and emer­gent computation," to be published in Physica D, 1990.

[9] C. Langton, Computation at the Edge of Chaos, Ph.D Thesis, University ofMichigan, 1990.

[10] Blake LeBaron, "The changing structure of stock returns" (working paper,Wisconsin Univ., 1988).

[11] Blake LeBaron, private communication.

[12] E.g., W. Li, "Power spectra of regular languages and cellular automata,"Complex Systems, 1, 107-130 (1987).

[13] W. Li, Problems in Complex Systems, Ph.D Thesis, Columbia University,1989 .. (Available from University Microfilm International, Ann Arbor, MI.)

[14] W. Li, "Expansion-modification systems: another model for l/f spectra"(SFI preprint, 1990).

[15] W. Li, N. Packard, and C. Langton, "Transition phenomena in cellular au­tomata rule space," to be published in Physica D, 1990.

[16] Andrew Lo, "Long-term memory in stock market process (Sloan Schoolworking paper 3014-89-EFA, MIT, 1989).

(17] Andrew Lo and C. MacKinlay, "Stock market prices do not follow randomwalks: evidence from a simple specification test," Review of Financial Stud­ies, 1,41-66 (1988).

Dow Jones 12

[18] B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).

[19] B. Mandelbrot, and J. Van Ness, "Fractional Brownian motions, fractionalnoises and applications," SIAM Review, 10, 422-437 (1968).

[20] P. Manneville, "Intermittency, self-similarity and I/f spectrum in dissipativedynamical systems," Le Journal De Physique, 41(11), 1235-1243 (1980).

[21] Robert M. May, "Simple mathematical models with very complicated dy­namics," Nature, 261, 459-467 (1976).

[22] T. Musha, and H. Higuchi, "The l/f fluctuation of a traffic current on anexpressway," Japanese Journal of Applied Physics, 15(7), 1271-1275 (1976).

[23] Charles R. Nelson and Charles I. Plosser, "Trends and random walks inmacroeconomic time series: some evidence and implications," Journal ofMonetary Economics, 10, 139-162 (1982).

[24] P.L. Nolan, et al. "Rapid variability of 10-140 keV X-rays from Cygnus X-I,"The Astrophysical Journal, 246,494-501 (1981).

[25] N. Packard, "Adaptation toward the edge of chaos," in Complexity in Bio­logical Modeling, eds. S. Kelso and M. Shlesinger (World Scientific, 1988).

[26] R. Palmer, D. Stein, E. Abrahams, and P. Anderson, "Models of hierarchi­cally constrained dynamics for glassy relaxation," Physical Review Letters,53(10), 958-961 (1984).

[27] W. Press, B. Flannery, S. Teukolsky, and Wo Vetterling, Num.erical Recipesin C: the Art of Scientific Computing (Cambridge Dniv. Press, 1988).

[28] Kurt Thearling, private communication.

[29] A. Van der Ziel, "On the noise spectra of semiconductor noise and of flickereffect," Physica, 16, 359-372 (1950)0

[30] R. Voss, and J. Clarke, "l/f noise in music and speech," Nature, 258,317­318 (1975).

[31] E.g.,. M. Weissman, "1/f noise and other slow, non-exponential kinetics incondensed matter," Reviews of Modern Physics, 60 (2), 537-571(1988).

[32] E ..g., K.G. Wilson, "Problems in physics with many scales of length," Scien­tific American, 241, 158-179 (1979).

Dow Jones

PRICE3000

price; trend; price-trend

13

2000

1000

o

-1000o 1000 2000 3000 4000 5000 6000

DAY

Figure 1: Daily Dow Jones Industrial average from October 14, 1974to November 3, 1989. There are 5346 data points, but only 4096points starting from the day No. 1200 (marked by an arrow) are usedto calculate the power spectrum. Also shown is the linear fitting ofthe data: met) = 445.9 + 0.31t. The curve at the lower part of thefigure is the de- trended price.

Dow JonesLog10(P(f»

14

(a)

Logl0CP(f»

10.0 1000.,0

Log10 (f)

(b)

1. 0 10., 0 100., 0 ' 1000 'e 0

LOBiO (f)

Figure 2: Power spectra (in log-log scale) of the time series in Figure1:(a) spectrum of the original daily price X(t). The best-fit straightline is 1/ f1079 (this exponent is estimated from point 8 to point 2000).(b) spectrum of tIle de-trended (by lillear trend) price Xl(t)o TIlebest-fit straight line is 1/ f1.76 (the exponent is estimated from 1=8to 1=2000).

Dow Jones 15

PRICE3000 -r------+----+-----4-----+---....;..---~

(a)

-1000 -#----....-.of-----+a----.........;..----+--o----+-----J.

Log10(P{f»

o 1000 2000 3000 4000 5000 6000DAY

(b)

'1000'.0

Logl0 (f)

1000010.,01e+O Ol--..-,.............,......................,.;.--...................,.......,...r-"'!""'I"',...,..;--......-...............,...,.........,...........-;.....;.,l

~1o 0

Figure 3: (a) The daily price with the series de-trended from aquadratic trend: m( t) = 971.7- Oo28t + O.OOOllt2 .

(b) Power spectrum of the de-trended series in (a). The best-fit lineis 1//1.74 (the exponent is estimated from 1=8 to 1=2000).

Dow Jones

lagCp)

LOG PRICE3.5-t----t----+----+-----+-----+-----4

16

3.,0

(a)

o 1000 2000 3000 4000 5000 6000

DAY

, log(p)~linear-trend

Log10CP(f»

1e-0

1e.....O

leO 10.0 10000 l~dQ~o

Logl0(f)

Figure 4: (a) The daily log-price and tIle linear trend: m( t) = 2.8 +0.000097.(b) Power spectrum of the de-trended series in (a). The best-fit lineis 1/ f1.77 (tIle exponent is estimated from 1=8 to 1=2000).

(b)

Dovv Jones

PRICE200

100

17

o (a)

-100

-200o 1000 2000 3000 4000 5000 6000

DAY

le+O

le+O

1e+O

Logl0CP(f»

10'.0 100.0 iOOOeO

L9g10 (f)

(b)

Figure 5: (a) The difference series X2(t) = X(t +1) - X(t). Note tilatsome values of X2(t) (e.g., -508) are too large to be inside the plot.(b) Po\ver spectrum of the difference series-in (a). The best-fit straigiltline in log-log plot has tIle slope 0.045, or, the fitting function is fO.045

(the exponent is estimated from 1=1 to 1=2048).

Dow Jones

diff-log(p)

LOG PRICE DIFF0.10 -r----t----;.----t----+-----f----L.

-0.05

-0 .. 10 --t-----t----t-----t----+--4---+-----l

18

(a)

o 1000 2000 3000 4000 5000 6000DAY

diff-Iog(p)Logl0(P(f»

ie-OE

(b)

1 .. 0 100.0 lOO~"O

. Log10 (f)

Figure 6: (a) The difference series X2( t) = lOglO X(t +1) -loglO X( t).(b) Power spectrum of the difference series in (a). The best-fit straightline in log-log plot has the slope 0.0046, Of, tIle fitting functioll is/0.0046 (the exponent is estimated from /==1 to /==2048).

DO\;V]Olles

PRICE

19

100

75

50

25(a)

o

-25o 1000

Logl0<P(f»

2000 3000 4000 5000

DAY

le+O

le+O

le+O

1e-O

le-O

le-O100 100.0 ..too·6~·o

. ,. ,"

.. LogiO (f)

(b)

Figure 7: (a) The daily average bond price, tIle lillear trelld lTt(t) ==77.87 + O.OO.08t, and the series de-trended from this linear trendX2(t) = ~Y(t) - met).(b) Power spectrum of tIle de-trended series in (a). TIle best-fit lineis 1//2.2 (the exponent is estim'ated from 1==1 to 1=600).

Dow Jones 20

1000 0 up~

X down.Q1)0

r---i

""---'"

Q1) 100 '"~ "'"

.,.-.1 ""~

'"~~0 10 ""u

""H ""0

01 '"~

""

'"'"

.1 l-LL~ I I I I I I ! I I I I

20 40 60 80 100daily change

Figure 8: Histogram of the daily price change ~~ Each bin ill tIlehistogram represents the price change \vithin 10i < ~ ~ 10(i + 1)(i =0,1, ~ .., 10)~ TIle rise (squares) and tile fall (crosses) of price arecounted separately.(a) the histogram in semi-log scale. The straight line represents anexponential function: h(~) = 500e-O.08~.

Dow Jones 21

1000 0 up~ downOD x

0,......--; '"~ "'-

OD 100 "'-

~ "'-0,...".-1 ""~. "'-~::J

"'-

0 10 "-u "'-

~

0

0~ 1~

.110 20 50

daily change (log)100

Figure 8: (b) the histogram ill log-log scale. The straigllt line repre~

sents a power-law function: h(6.) = 106 /6..3•

Dow Jones

"-"-

~

Q[) "-0 100 "-r--i

"-' "-Q[) "-~ "-

.. ,..--i

"--+-J

~

~0u

10~

0

0~

1

00005 0001 0.015change of log-price

Figure 9: Histogram (in semi-log scale) of the daily change of log-priceLl .. Each bin in the histogram represents the log-price change withinD.001i < ~ :5 O.OOl(i + 1) (i =0,1, ... ,17). The rise (squares) alldthe fall (crosses) of price are counted separately. TIle straigllt linerepresents an exponential function: h(11) == 400e-400~.

22