Fractal Analysis of Highly Volatile Markets:

An Application to Technology Equities

 

QUARTERLY REVIEW OF ECONOMICS AND FINANCE (2004) 44(1): 155-179

 

Robert F. Mulligan, Ph.D.

Department of Business Computer Information Systems & Economics

Western Carolina University

College of Business

Cullowhee NC 28723

Phone: 828-227-3329

Fax: 828-227-7414

Email: mulligan@wcu.edu

 

 

Acknowledgements

The author wishes to thank Patrick Hays, Gary A. Lombardo, and two anonymous referees for many helpful comments and suggestions.  I remain solely responsible for errors.  Financial support in the form of a visiting research fellowship from the American Institute for Economic Research and a summer research grant from the Western Carolina University College of Business is gratefully acknowledged. 

 

 

Abstract

 

This paper examines technology equity price series using five self-affine fractal analysis techniques for estimating the Hurst exponent, Mandelbrot-Lévy characteristic exponent, and fractal dimension.  Techniques employed are rescaled-range analysis, power-spectral density analysis, roughness-length analysis, the variogram or structure function method, and wavelet analysis.  Evidence against efficient valuation supports the multifractal model of asset returns (MMAR) and disconfirms the weak form of the efficient market hypothesis (EMH).  Strong evidence is presented for antipersistence of many technology equities, suggesting markets do not price all technology securities efficiently, or equally efficiently.

JEL classifications: C15, C22, G12, G14.

Keywords: long memory, fractal analysis, Hurst exponent, multifractal model of asset returns.

 

Introduction

Long memory in a sample of highly volatile technology equity price series is examined using five alternative methods for estimating Hurst exponent (1951), fractal dimension, and Mandelbrot-Lévy characteristic exponent (Lévy 1925).   Mandelbrot-Lévy distributions are also referred to as stable, Lévy-stable, L-stable, stable-Paretian, and Pareto-Lévy.  Samuelson (1982) popularized the term Mandelbrot-Lévy, but Mandelbrot avoids this expression and the other terms remain current.  A new characteristic exponent test for the extremely leptokurtic Cauchy distribution (Mulligan 2000b) is applied, examining Cauchy character in certain equities.  Fractal structure in equity prices indicates traditional statistical and econometric methods are inadequate for analyzing security markets, at least for the most volatile technology securities.  Findings have implications for the efficient market hypothesis (EMH), and for the multi-fractal model of asset returns (MMAR) of Mandelbrot, Fisher, and Calvet (1997).

 

The booming technology sector of the 1990s provides an interesting subject for analysis.  It was touted as the "New Economy" (Kelley 1998) not subject to standard economic laws.  The technology sector also presented a productivity paradox (Berndt and Malone 1995; Brynjolfsen 1993) in which increased use of computers and other advanced equipment, supposedly motivated by the improved productivity the new technology would provide, never resulted in any measurable productivity gains.  An essential and highly variable feature of the New Economy was receiver competence (Eliasson 1985, 1986, pp. 47 ff., 57 ff., 1990a) or absorbtive capacity (Cohen and Levinthal 1990), in which the technology firms' customers had to make intelligent use of the new equipment and software to realize gains in productivity.  Preexisting business strategies formulated for static environments proved inadequate  in dynamic environments calling for development of new strategies (Carpenter and Westphal 2001).  The extremely dynamic environment of the rapidly-changing, innovation-charged technology sector may explain why antipersistence in equity returns is observed for so many large, established technology firms.  Eliasson (1996) identifies the merger of computing and telecommunications technologies into the internet as the fifth generation of computers.  At the time the internet was emerging, many informed actors predicted the fifth generation would be heralded by the introduction of extremely fast and powerful supercomputers, and Japan targeted its industrial policy according to this assumption (Johansson 2001, p. 47).  The fact that expectations of informed individuals were so dramatically disappointed illustrates the dynamic nature of the technology sector, a truly experimentally-organized economy (Eliasson 2001a). 

 

The finding that some security prices may be Cauchy distributed offers new and extremely interesting information about the behavior of these securities, and the efficiency of the markets where they are traded.  Using the modified rescaled-range (R/S), which is robust against short-term dependence, Lo (1991) found no long memory in stock prices.  Technology stocks are of special interest, because they might be thought to be less likely to exhibit long memory than less volatile securities.  Nevertheless, the high volatility of these equities makes them attractive subjects for fractal analysis.  In applying his modified R/S analysis to equity prices, Lo overturned earlier results based on classical R/S methods finding long memory, but he did not examine the most volatile technology stocks.

 

Mandelbrot (1963a, 1963b) demonstrated all speculative prices can be categorized in accordance with their Hurst exponent H, also called the self-affinity index or scaling exponent (Mandelbrot et al 1997).   The Hurst exponent was introduced in the hydrological study of the Nile valley and is the reciprocal of the characteristic exponent alpha (Hurst 1951).  Some security prices are persistent with (0.50 < H < 1.00).  These less noisy series exhibit clearer trends and more persistence the closer H is to one, and investors should earn positive returns.   Neely, Weller, and Dittmar (1997) found technical trading rules, formalized with a genetic programming algorithm, provided significant out-of-sample excess returns.  However, Hs very close to one indicate high risk of large, abrupt changes, as H = 1.00 for the Cauchy distribution, the basis for the characteristic exponent test.

 

Many technology securities in the sample are anti-persistent or ergodic or mean-reverting with (0.00 < H < 0.50), indicating they are more volatile than a random walk.  This is one kind of business environment which promotes competition and innovation, and where firms respond to the uncertain business environment with experimental and dynamic resource allocation (Eliasson 1990a, 1996, p. 110).  If the highly volatile returns are uncorrelated across different assets, risk can be minimized by diversification.  Ergodicity, that is, H significantly below 0.50, strongly disconfirms the efficient market hypothesis, indicating market participants persistently over-react to new information, imposing more price volatility than would be consistent with market efficiency.  Hs significantly above 0.50 demonstrate stock prices are not random walks, shedding some doubt on weak market efficiency and indicating technical analysis can provide systematic returns. 

 

Any findings of non-normality or non-Gaussian character have severe implications for pricing financial derivatives.  Because the Black-Scholes (1972, 1973) option pricing model assumes normally-distributed prices for underlying securities, financial derivatives based on non-normal securities prices cannot be priced efficiently.  In this kind of highly-volatile market environment, advantage accrues to small adaptive firms which can react most quickly in response to market instability or rapid technological change (Piore and Sabel 1984).  The validity of agents' information assessments dates rapidly in a highly volatile market even if it is initially correct.  In addition, the finding of antipersistence or ergodicy suggests a more general phenomenon similar to that described in Mussa's (1984) disequilibrium-overshooting model for exchange rate determination. 

 

Long memory series exhibit non-periodic long cycles, or persistent dependence between observations far apart in time.  Short-term dependent time series include standard autoregressive moving average and Markov processes, and have the property that observations far apart exhibit little or no statistical dependence.  Mandelbrot's R/S or rescaled range analysis distinguishes random from non-random or deterministic series.  The rescaled range is the range divided (rescaled) by the standard deviation.  Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and non-random components.  Conventional statistical techniques lack power to distinguish unpredictable random components and highly predictable deterministic components (Peters 1999).  R/S analysis evolved to address this difficulty.  R/S analysis exploits the structure of dependence in time series irrespective of their marginal distributions, statistically identifying non-periodic long-run cyclical dependence as distinguished from short dependence or Markov character and periodic variation (Mandelbrot 1972a, pp. 259-260).  Mandelbrot likens the differences among the three kinds of dependence to the physical distinctions among liquids, gases, and crystals.

 

Fractal analysis aims at distinguishing deterministic linear behavior from completely unpredictable nonlinear stochastic or probabilistic-chaotic behavior.  Somewhere in between lie nonlinear dynamic or chaotic behavior, predictable in the short run but not the long run, and complex processes, predictable in the long run but not the short run (Peters 1999, pp. 164-167).  Complex processes exhibit local randomness but global structure, in contrast with nonlinear dynamic processes.  Different classes of statistical processes are potentially predictable to different extents, but applying the fractal taxonomy (see Table 1) to correctly categorize the data under consideration is the necessary first step before we can forecast what can be forecast.  Respecting the limitations to predictability which inhere in different kinds of statistical behavior is a precondition for constructing meaningful forecasts. 

 

Long memory in equity prices would allow investors to anticipate price movements and earn positive average returns.  Fractal analysis offers an alternative to conventional risk measures and permits an evaluation of investment alternatives.  Fractal analysis can also identify ergodic or anti-persistent series, e.g., negative serial correlation.  Ergodic series should also have much shorter cycle lengths than random walks or trend-reinforcing series.  Five techniques are reported in this paper, Mandelbrot's (1972a) AR1 rescaled-range or R/S analysis, power spectral-density analysis, roughness-length relationship analysis, variogram analysis, and wavelet analysis.  Each method analyzes equity returns as self-affine traces, providing estimates of the Hurst exponent, fractal dimension, and Mandelbrot-Lévy characteristic exponent.  The characteristic exponent is then used as a test statistic for the Cauchy distribution.  The remainder of the paper is organized as follows.  A literature review is provided in the second section.  The data are documented in the third section.  Methodology and empirical results are presented in the fourth section.  Concluding remarks are provided in the fifth section.

 

Literature

This section describes first, the empirical literature applying fractal analysis to capital markets, then discusses a variety of theoretical justifications for fractal behavior in technology equities over the 1990s, including competence-incompetence mismatching, volatility associated with firm turnover, and high rates of innovation.  Finally the fractal taxonomy of time series, applied below to interpret the empirical results, is developed.

 

Empirical applications of fractal analysis

The search for long memory in capital markets has been a fixture in the literature applying fractal geometry and chaos theory to economics since Mandelbrot (1963b) shifted his attention from income distribution to speculative prices.  Fractal analysis has been applied extensively to equities (Greene and Fielitz 1977; Lo 1991; Barkoulas and Baum 1996; Peters 1996; Koppl et al 1997; Kraemer and Runde 1997; Barkoulas and Travlos 1998; Koppl and Nardone 2001), interest rates (Duan and Jacobs 1996; Barkoulas and Baum 1997a, 1997b), commodities (Barkoulas, Baum, and Oguz 1998), exchange rates (Cheung 1993; Byers and Peel 1996; Koppl and Leland 1996; Barkoulas and Baum 1997c; Chou and Shih 1997; Andersen and Bollerslev 1997; Koppl and Broussard 1999; Mulligan 2000a), and derivatives (Fang, Lai, and Lai 1994; Barkoulas, Labys, and Onochie 1997; Corazza, Malliaris, and Nardelli 1997).

 

Competent use of market information

A firm's endeavor should focus on its field of competence.  Firms seek to exploit their business environment as competent teams coordinating inputs in a dynamic process (Eliasson 1990a), thus production cannot be captured by a static production function (Johansson 2001, p. 15).  Coordination performed by firm-level decision makers adds value in each stage of production (Mises 1998 pp. 480-485; Rothbard 1970, pp. 323-332; Garrison 1985, p. 169, 2001, p. 46).  The firm's actions are experimental, responding to the uncertain business environment (Eliasson 1996, p. 110).  Just as a static environment leads to the implementation of existing strategies,  the dynamic environment of the technology sector strongly encourages the development of new strategies (Carpenter and Westphal 2001).  Firms face environmental uncertainty both because knowledge and information are always finite resources, and also because no individual or combination of individuals, such as a firm, can ever make use of all available information or knowledge.  Individuals necessarily filter out most of the information they encounter in order to make intelligent and effective use of a limited subset, forming what Eliasson (1990a) calls a competence bloc.  Project-oriented management, which facilitates the compartmentalized use of limited information, has long been the paradigm in information technology.   Piore and Sabel (1984) suggest market instability promotes competitiveness, and provides an advantage to small firms which can react more quickly in response to market volatility, high uncertainty, or rapid technological change.  Highly competent, highly innovative firms should contribute antipersistence or ergodicity to the market.  Their actions impose higher volatility on capital markets because they are engines of Schumpeterian (1934) creative destruction.

 

Firms act as, and interact in, competence blocs, to effect resource allocation in experimentally-organized economies.  Competent resource allocation is not a conventional optimization process, but a search activity which aims at uncovering an unrealizable optimum.  Using Eliasson's (1996) terminology, entrepreneurial managers seek to allocate resources found in the state space to the business opportunity set of profitable outcomes.  Entrepreneurs compete to reach the best optima within the partially unexplored business opportunity set.  Entrepreneurial incompetence can result in capital (in this case both financial and physical capital) being misallocated, that is, allocated toward unprofitable uses outside the business opportunity set.  Furthermore, the very activity of invention, innovation, learning, facilitating competent consumers, competent venture capitalists, etc., transforms the business opportunity set and makes better optima possible.  "Both the state space and the business opportunity set are, however, at each point in time bounded (but expanding through exploration)" (Johansson 2001, p. 18).

 

Johansson (2001) and Eliasson (1983, p. 274, 1990b) suggest a non-convergence property, characterized by instability of market equilibria, should be "expected in an economy where information use and communication activities dominate resource use and where technological change in computing and communications technology dominates total productivity change through constant systems reorganization" (Johansson 2001, p. 121).   This contrasts with less competent, less entrepreneurial firms, which are likely to exhibit persistency in equity returns, as opposed to antiperisitence or ergodicity.

 

Incompetent money

Transactions costs, identified by Coase (1937, pp. 38-46; 1988, p. 7) as the main reason the division of labor is organized in firms, include information costs.  This implies that the transactions costs avoided through organizing production in firms more than offset inefficiencies imposed by bureaucratic organization.  The mere existence of firms presumptively demonstrates it successfully minimizes transactions costs, at least over the long run.  Transactions costs are especially critical in the technology sector, where information obsoletes rapidly.  Competence blocs can only persist if resource allocation is flexible, ongoing, and competently informed.  Even if knowledge is embodied in the labor force as human capital, without augmentation through ongoing training, this human capital depreciates rapidly through the diffusion of invention and innovation.  If a firm's core practices remain unchanged in a dynamic context, lowered performance outcomes are likely (Schumpeter 1942; Hannan and Freeman 1984; Tushman and Anderson 1986; Levinthal 1994).

 

Venture capitalists fund the formation of new firms and expansion of existing firms.  In so doing they perform the vital function of recognizing and correctly valuing or pricing innovation (Eliasson and Eliasson 1996b; Eliasson 1997; Johansson 2001, p. 23).  Competent firms are alert to disequilibrium prices which signal opportunities for entrepreneurial discovery (Kirzner 1984a, p.146; 1984b, pp. 160-161; 1997) and exploit the information contained in disequilibrium prices to adjust the production structure.  Johansson notes that "incompetent money," that is, "capital not bundled with market knowledge, probably has a negative effect on firms, since the financial capital then confers power and authority to actors who do not understand the business (or the competence of the entrepreneur)" (emphasis in original).  Johansson suggests government as the primary supplier of incompetent money (Carlsson et al 1981; Bergstrom 1998) but during the nineties, it appears private sources supplied the U.S. technology sector with all the incompetent money it could absorb.  This incompetent money may have resulted from an expansionary monetary policy.

 

The process of industrial innovation includes the allocation and combination of competencies for which no one understands the full extent or implications (Johansson 2001, p. 25).  In this connection, Eliasson (1994) describes the labor market as a "market for competencies."  The technology sector probably leads all others in the significance which attaches to competence blocs, the harm that can be effected by competence misalignments, and the difficulty in perfectly juxtaposing adjacent competencies.  Thus, misallocation is inevitable, and an essential part of economic progress.  It is necessary to contrast naturally unbalanced growth with the misallocation induced by an expansionary monetary policy.  Competence possesses the unique property of being self-allocating (Pelikan 1993; Eliasson 1996);  incompetence, in contrast, may be described as self-misallocating.  In a rapidly changing state space, due to technological change or other factors, competence obsoletes rapidly and becomes incompetence.  Where allocation is not sufficiently flexible, misallocation must result and must be persistent.

 

Cheung and Lai (1993) suggest Heiner's (1983) and Kaen and Rosenman's (1986) competence-difficulty (C-D) gap hypothesis as a potential source of long memory in asset prices, offering a theoretical expectation of long memory.  The C-D gap is a discrepancy between investors' competence to make optimal decisions and the complexity of exogenous risk, which is widely thought to be especially high for technology securities.  A wide C-D gap leads to investor dependency on deterministic rules, which can lead to persistent price movements in one direction - a crash or speculative bubble.  Due to irregular arrival of new information, Kaen and Rosenman argue persistent price movements may suddenly reverse direction, leading to non-periodic cycles.  Persistence in equity returns is thus expected for larger, more established, less entrepreneurial firms, in contrast to smaller firms with more effectively defined competence blocs.  Program trading introduces the same phenomenon of persistent returns, and interestingly enough, is more likely to be engaged in for large firms than small.   In addition, many technology investors rely more on perceived market sentiment, which is also subject to both persistence and unpredictable reversals.

 

Firm size, age, and innovation

Researchers have identified the importance of small and medium-sized firms, which typify the technology sector, in driving economic growth (Birch 1979, 1981, 1987; Davidsson et al 1994a, 1994b, 1996), as well as documenting negative relationships between firm growth and firm size and/or firm age (Evans 1987a, 1987b; Dunne et al 1987).  Kirchoff (1994) found that these growth effects were strongly amplified for the technology sector.  A related line of inquiry has documented decreasing shares of production and employment by large, old, well-established firms, being displaced by increasing shares to large numbers of newer, smaller firms, since about 1970 (Brock and Evans 1986, 1989; Carlsson 1989, 1992; Loveman and Sengenberger 1991; Acs 1996a; OECD 1996).  Thus, we should expect to observe ergodic returns for small firms, and persistent returns for larger ones.

 

Small firms, such as those that dominate the technology sector, act as agents of change (Acs 1992) and tend to be more innovative than larger firms, which often suffer from more bureaucratic organization (Acs and Audretsch 1987a, 1987b, 1988, 1993).  Small, innovative firms typically gain "first-mover advantages" (Thomas 1985), though large firms can also be first movers.  The level of bureaucratic inertia a firm experiences increases with age and size (Hannan and Friedman 1984).  Because firm age and firm size highly correlated, empirical examinations may have difficulty separating these two as causal factors. 

 

Small firms contributed the majority of innovations in the technology sector (Acs and Audretsch 1990a, 1990b), and in some cases the success of these innovations enabled the innovating firm to become a large one, sometimes a less innovative one.  Microsoft and Intel typify this evolutionary process.  Large, established firms, exploiting the comparative advantage that comes from being large and established, generally deepen existing innovations the firm may have pioneered (Almeida and Kogut 1997; Almeida 1999).  This kind of essential, though clearly less innovative activity, should result in persistent, rather than ergodic, returns for the larger firms.  Though large firms have comparative advantage in extending existing innovations they originally pioneered, eventually diminishing returns must set in.  Johansson (2001, p. 71) suggests large firms look for innovative processes, trying to improve what they already do well, whereas small firms look for innovative products, which are more important for long run growth (Acs et al 1999).   Lombardo and Mulligan (2003) note that established firms tend to allocate resources along historical, as opposed to dynamic, patterns.

 

As a firm grows or ages, it becomes increasingly difficult to alter the competence base of its research functions, or of the firm as a whole, an observation which supports the expectation of persistence of returns for large firms.  Leastadius (2000) suggests large established firms only embrace new technology that complements the organization's existing competence base.  New technology which challenges the organization's competence base, or renders it obsolete, will typically be resisted.  Because large firms have existing capital structures and knowledge bases to protect, they will be resistant to change which does not complement existing physical and human capital.  This distinction is similar to, though more general than, that underlying Bischoff's (1970) "putty-clay" model of investment, which emphasizes the distinction between highly-liquid, uninvested financial capital, such as venture capitalists provide to small, new firms, and highly illiquid, installed capital equipment, such as might be abundant in large established firms.  Here, the distinction is generalized to include human capital.  Small firms are freer to adapt than large firms because the small firms are not constrained by large illiquid stocks of human or physical capital. Also, an established firm may be more interested in protecting existing rents than creating new profits (Geroski 1995, p. 431), also supporting the expectation of persistence.  Small new firms will not have rents to protect.

 

Small firms' less bureaucratic organization enables them to better exploit new knowledge and information, compared with large firms (Link and Reese 1990; Link and Bozeman 1991).   Thus, the technology sector's dominance by small firms leads to a higher rate of innovation, which can be thought of as random exogenous shocks, thus explaining a high level of volatility among technology equities.  In reality, however, innovations are neither random nor exogenous, but result from firms' response to their environment, including uncertainty and technological change.  Acs et al (1997) suggest small firms contribute more innovation because they better respect and protect the property rights of innovators, compared with large firms.

 

Firm entry, exit, and innovation

Empirical investigations find firm age and size have negative impacts on firm growth rates, and conversely, firm youth and smallness have positive impacts (Davidsson et al 1994a, 1994b, 1996; Liu 1999; Heshmati 2001; Johansson 2001).  The microeconomic factor of high firm turnover (firm entry combined with firm exit, which frees up resources for better uses) has been found to contribute to macroeconomic growth (Davidsson et al 1994a, 1994b, 1996; Kirchhoff 1994; Reynolds 1994, 1997, 1999; Griliches and Regev 1995; Dunne et al 1998, 1999; Foster et al 1998; Callejón and Segarra 1999; Johansson 2001).  Audretsch (1995a) concludes that gross firm entry and exit are more important for generating jobs than net firm entry, and Johansson (2001, p. 169) concludes "macroeconomic stability requires microeconomic instability."

 

With a huge and complex state space, there are always opportunities for realizing large systems productivity effects through dynamic resource reallocation, most of which occurs through firm entry and exit (Eliasson 1991a, 1991b; Eliasson and Taymaz 2000; Johansson 2001, p. 119).  Allocative improvements are likely to be possible as long as the state space is sufficiently large and complex that market participants, intelligent consumers, skilled workers, inventors, entrepreneurial managers, venture capitalists, cannot marshal so much knowledge and information that they can acquire a dominant comparative advantage over their competitors.  It is especially noteworthy that the same situation could arise in a far smaller state space subject to rapid change, such as the innovation-charged technology sector.

 

A high rate of innovation, even if successfully diffused and adopted, results in rapid resource reallocation and high firm turnover (entry and exit), which may well be observable in equity returns series as ergodicity or antipersistence for smaller firms.  Rapid changes break down the effectiveness of price signaling in markets, resulting in lost profits through poor or incorrect decisions, and motivates a retreat from decision-making.  Eliasson (1990) documents increased search efforts face increasing negative returns.  When learnable information and knowledge are in rapid flux, there is less incentive for discovery and learning, and firms tend to retreat into established activities, lowering economic growth (Eliasson 1983, 1984, 1991b).   Because larger firms should be more successful in implementing this "retreat to habit," they are more likely to exhibit persistent returns.  The collapse of technology equities can thus be seen as a natural process of Schumpeterian creative destruction, rather than a process of correcting the malinvestment triggered by monetary overexpansion, though that phenomenon may also have contributed to a speculative bubble in the technology sector.

 

Empirical and theoretical studies of firm turnover include Orr (1974), Du Rietz (1975), Baldwin and Gorecki (1989), Acs and Audretsch (1989), and Johansson (2001).  Siegfried and Evans (1994) propose the stylized fact that entry increases and exit decreases with firm profitability and growth of local markets.  However, Audretsch (1995b) finds firm survival rates lower in highly-innovative markets, such as the technology sector, than in less-innovative markets, though surviving firms have higher growth rates, an outcome also observed by Baldwin (1995).

 

Methodological approach

Mandelbrot (1972b, 1974) and Mandelbrot, Fisher, and Calvet (1997) have developed the multifractal model of asset returns (MMAR), which shares the long-memory feature of the fractional Brownian motion (FBM) model introduced by Mandelbrot and van Ness (1968).  The statistical theory necessary to identify empirical regularities and local scaling properties of MMAR processes with local Hölder exponents is developed by Calvet, Fisher, and Mandelbrot (1997) and applied by Fisher, Calvet, and Mandelbrot (1997).  Mandelbrot's (1972a, 1975, 1977) and Mandelbrot and Wallis's (1969) R/S or rescaled range analysis characterizes time series as one of four types: 1.) dependent or autocorrelated series, 2.) persistent, trend-reinforcing series, also called biased random walks, random walks with drift, or fractional Brownian motion,  3.) random walks, or 4.) anti-persistent, ergodic, or mean-reverting series. 

<<Table 1 about here>>

 

Table 1 provides the taxonomy of time series identified through fractal analysis.  Because the Hurst exponent H is the reciprocal of the Mandelbrot-Lévy characteristic exponent alpha, estimates of H indicate the probability distribution underlying a time series.  H = 1/alpha = 1/2 for normally-distributed or Gaussian processes.  H = 1 for Cauchy-distributed processes.  H = 2 for the Lévy distribution governing tosses of a fair coin.  H is also related to the fractal dimension D by the relationship D = 2 - H.  In fractal analysis of capital markets, H indicates the relationship between the initial investment R and a constant amount which can be withdrawn, the average return over various samples, providing a steady income without ever totally depleting the portfolio, over all past observations.  Note there is no guarantee against future bankruptcy.

 

Fractal analysis also gives an estimate of the average non-periodic cycle length, the number of observations after which memory of initial conditions is lost, that is, how long it takes for a single outlier's influence to become immeasurably small.  If equity series are random walks with H = 0.50, returns are purely random and should lead to investors' breaking even over the long run.  It was found that the series used here, up to 1800 daily observations over seven and one-half years, were never long enough to provide reliable estimates of non-periodic cycle length. 

<<Table 2 about here>>

 

Data

The data are daily closing prices reported by the exchanges for each traded equity.  A sample of fifty-four technology firms was selected as shown in Table 2.  Some are familiar names; others are known chiefly, if at all, for their spectacular and ultimately infamous performance volatility.  Many series date from relatively recent initial public offerings (IPOs), and therefore are not available for the whole sample period.  Extremely high volatility, as indicated by large changes, high standard deviation, high beta, or Hurst exponent and Mandelbrot-Lévy characteristic exponent close to one, may well be exclusively small sample properties.  These features are not observed for any of the more established firms which traded at the beginning of the sample range.  The maximum sample period is December 31, 1993 to June 18, 2001 – seven and one-half years of daily data.  For some equities, data is only available for a subsample.  Geneva Saint Lawrence (GNVH) has only 109 observations in the period studied.  Twenty-three equities are available for substantially the whole sample: Advanced Micro Devices (AMD), American Power Conversion (APCC), Andrew Electronics (ANDW), Apple Computer (AAPL), AT&T (T), Cisco Systems (CSCO), Corel (CORL), Dell Computer (DELL), Ericsson (ERICY), General Electric (GE), Hewlett-Packard (HWP), Intel (INTL), IBM (IBM), Microsoft (MSFT), Nokia (NOK), Nortel (NT), Novell (NOVL), Novellus (NVLS), Oracle (ORCL), Solectron (SLR), Sun Microsystems (SUNW), Texas Instruments (TXN), and Xerox (XRX).  Seven more, Amazon (AMZN), Ameritrade (AMTD), Datastream Networks (DSTM), Futurelink (FTRLD), Greystone Digital (GSTN), Lucent Technologies (LU), SAP AG (SAP), and Yahoo (YHOO) are available for more than half the sample period.  Most of the remainder are observed for over one year, though several are only available for briefer periods. 

 

Approximately two cycle lengths of data are necessary for good estimates average non-periodic cycle length using classical R/S techniques (Mandelbrot 1972a; Peters 1994, 1996).  Since the average cycle length, if it exists, is not known, this time period offers little potential of including a sufficient number of cycles to allow average cycle length to be definitively measured.  Some of the greatest volatility is observed in firms with recent IPOs, after the tech stock slump.  Table 2 presents the sample of fifty-four technology equities examined, giving the sample range, number of daily price observations, and the standard beta measure of relative volatility.

 

The data are adjusted for stock splits.  Stock splits introduce discontinuities because investors often take them to herald increased potential for future growth, and many exponentially-growing technology firms seem to have competed to see who would split most often and in the most exotic manner.  In the sample range considered, for example, Microsoft (MSFT) and Texas Instruments (TXN) each split 2:1 four times, and Dell Computer (DELL) split 2:1 six times, but Amazon (AMZN) split 2:1 twice and 3:1 once, and Yahoo (YHOO) split 2:1 twice, 3:2 once, and 3:1 once.   The opposite extreme is represented by Greystone Digital (GSTN) which reverse split 1:42 once only.

 

Failure to adjust raw price series for splits would have introduced discontinuities unrelated to any fundamentals.  Such shifts in the mean can introduce biases toward finding fractal dynamics (see, for example, Cheung (1993), Granger and Hyung (1999), and Diebold and Inoue (2000)).  It is essential to test for fractal behavior using split-adjusted prices to reflect the actual return on the equities and ensure the findings are not a statistical artifact due to the presence of nonstationarities in the mean of the process, but instead reflect genuine features of the underlying data generating process.

 

Empirical Results

This section discusses and interprets the results of five alternative fractal analysis methods for measuring the Hurst exponent H presented in table 3.  All data are converted to logarithmic returns, losing one observation.  Standard errors are given in parentheses.  H ranges from 1.00 to 0.50 for persistent series, is exactly equal to 0.50 for random walks, ranges from zero to 0.50 for anti-persistent series, and is greater than one for a persistent or autocorrelated series.  Mandelbrot, Fisher, and Calvet (1997) refer to H as the self-affinity index or scaling exponent.    

 

Rescaled-range or R/S Analysis:  R/S analysis is the traditional technique introduced by Mandelbrot (1972a) and improved by Lo (1991).  Hs estimated by this method are mostly around 0.50, superficially suggesting Gaussian processes and supporting the Efficient Market Hypothesis.  Only a few are greater than 0.50, indicating persistent price movement for Advanced Micro Devices (AMD), Digital Lava (DGLV), E Bay (EBAY), Intel (INTC), Intraware (ITRA), Nokia (NOK), Nortel (NT), and Red Hat (RHAT).  Of these potentially persistent series, Advanced Micro Devices and Intel reject the null hypothesis of normality at the 0.01 two-tail significance level, and Intraware rejects the null at the 0.05 level.   It is interesting that Intel, today one of the more established technology firms, was organized in 1968 by former employees of Fairchild Semiconductor who desired a more entrepreneurial environment (Audretsch 1999, p. 28).  The finding of persistence supports the characterization of Intel and Advanced Micro Devices as highly-established, less innovative, firms.  These results formally support the multifractal model of asset returns, which is more general, by disconfirming the weak form of the efficient market hypothesis for those three securities.

 

Many more series have low Hs, suggesting antipersistence or ergodicity (e.g., negative serial correlation): Applied Micro Circuits (AMCC), Audiocodes (AUDC), Bridge Technology (BRDG), Broadcom (BRCM), Drugmax (DMAX), General Electric (GE), Geneva St Lawrence (GNVH), Greystone Digital Technology (GSTN), Hear Me (HEAR), Log on America (LOAX), SAP AG (SAP), Sprint (PCS), Star Scientific (STSI), Time Warner Telecom (TWTC), and Voicenet (VTC), reject the null hypothesis of normality at the 0.01 level; Ameritrade (AMTD), AT&T (T), Cisco (CSCO), Dell (DELL), Doubleclick (DCLK), Ericsson (ERICY)Hewlett-Packard (HWP), Juno (JWEB), Oracle (ORCL), Novell (NOVL), Solectron (SLR), and Texas Instruments (TI) reject the null at the 0.05 level.  These results provide further difficulty for weak form efficiency.   R/S analysis formally rejects weak form efficiency for thirty-one of the fifty-four equities in the sample, by at least the five percent significance level, two-tailed.  This measurable antipersistence or ergodicity demonstrates market participants habitually overreact to new information, and never learn not to.  It also suggests these firms are competent and entrepreneurial, even though many are large, established firms, including AT&T, Ericsson, General Electric, and SAP.

 

Normality or Gaussian character is a sufficient condition for weak market efficiency, but not a necessary condition.  This result is generally interpreted as support for the more general multifractal model of asset returns and disconfirmation of the weak-form efficient market hypothesis; however, it really does not prove that the market is inefficient.  More importantly, findings of H < 1 strongly reject weak market efficiency because they demonstrate ergodicity or antipersistence.  These findings are absolutely fatal to the Black-Scholes (1972, 1973) option pricing model and its underlying assumption of normally-distributed asset prices.  Financial derivatives based on non-normal asset prices cannot be priced efficiently.  Thus even if the equity markets for technology stocks are efficient, in spite of substantial empirical evidence against efficiency, the derivatives markets clearly are not efficient.  Hs different from 0.50 demonstrate the return series have not been random walks, shedding significant doubt on weak market efficiency and indicating technical analysis could have provided systematic returns.  Nevertheless, this finding may be due to short-term dependence still present after taking AR1 residuals, or systematic bias due to information asymmetries, or both.

<<Table 3 about here>>

 

Power Spectral Density Analysis:  Hs estimated by this technique are predominantly in the antipersistent range (H < 0.50).  Those in the persistent range (H > 0.50) are Advanced Micro Devices (AMD), Aperian (APRN), Audiocodes (AUDC), Hear Me (HEAR), Nortel (NT), Red Hat (RHAT), Sagent Technology (SGNT), and 24/7 Media (TFSM).  The most dramatically antipersistent equities are Datastream Networks (DSTM) with H = 0.208 and General Electric (GE) with H = 0.383, again suggesting General Electric behaves more like a small, entrepreneurial firm.  Most are fairly close to 0.50.  Note these results often flatly contradict those provided by other techniques.  Spectral density does not provide a standard error for H, and thus cannot be used for formal hypothesis testing.

 

Roughness-Length Relationship Method:  This method provides the most consistent rejection of weak market efficiency.  Formal hypothesis tests reject the Gaussian null for all series except Geneva St Lawrence (GNVH), for which too few observations are available to measure H.  Persistence (H > 0.50) is indicated for Digital Lava (DGLV), Futurelink (FTRLD), Intraware (ITRA), Red Hat (RHAT), Sagent Technology (SGNT), 24/7 Media (TFSM), and Voicenet (VTC), with antipersistence indicated for all others, including all the large and established firms.   One difficulty in applying the roughness-length method is that the standard errors are so low the null hypothesis of H = 0.500 is nearly always rejected no matter how nearly normal the asset returns.  The seemingly unambiguous rejection of weak market efficiency provided by this technique is best viewed cautiously.

 

Variogram Analysis:  Variogram Hs indicate persistence (H > 0.50) for Aperian (APRN), Red Hat (RHAT), and 24/7 Media (TFSM) at the 0.05 level; and Broadcom (BRCM)at the 0.01 level.  Series with low variogram Hs include Drugmax (DMAX), Greystone Digital Technologies (GSTN), Log on America (LOAX), Star Scientific (STSI), and Time Warner Telecom (TWTC), indicating antipersistence at the 0.01 level, and Andrew Electronics (ANDW), Futurelink (FTRLD), Solectron (SLR), Usurf America (UAX), and Voicenet (VTC), indicating antipersistence at the 0.05 level.   It is particularly surprising to see Andrew Electronics, an established manufacturer of coaxial cable, antennae, and signal transmission equipment, yield an outcome of antipersistence.  Variogram analysis disconfirms weak market efficiency for fourteen equities out of the sample of fifty-four.  The interpretation is that most technology equities are valued efficiently, but clearly not all.

 

Wavelet Analysis:  This method was developed by Daubechies (1990), Beylkin (1992), and Coifman et al (1992).  Wavelet H estimates indicate antipersistence (H < 0.50) for American Power Conversion (APCC), Applied Micro Circuits (AMCC), AT&T (T), Corel (CORL), Doubleclick (DCLK), Drugmax (DMAX), Ericsson (ERICY), General Electric (GE), Geneva St Lawrence (GNVH), Greystone Digital (GSTN), Hewlett-Packard (HWP), Intel (INTC), Intraware (ITRA), Juno Online Services (JWEB), Sprint (PCS), Star Scientific (STSI), Sun Microsystems (SUNW), Time Warner Telecom (TWTC), Usurf America (UAX), and Voicenet (VTC), indicating persistence (H > 0.50) for all other equities.  Again it is surprising to see some of the largest firms have antipersistent returns.  As with power spectral density analysis, the wavelet method does not provide a standard error for H and cannot be used for hypothesis testing.

 

Mandelbrot-Lévy Characteristic Exponent Test:  Various statistics are available to test the null hypothesis of normality, but not for the Cauchy distribution, the other extreme.  Mulligan (2000b) provides tables of percentages of the Mandelbrot-Lévy characteristic exponent alpha generated by Monte Carlo experiments with 1,000 iterations for different sample sizes.  These critical values can be used to evaluate estimated alphas for the Cauchy null; the null should be rejected if the estimated characteristic exponent lies outside the critical bounds.   Dispersion of alpha around the theoretical value of 1 varies greatly with the sample size. 

<<Table 4 about here>>

 

The equity series with the highest estimated Hs, implying the lowest estimated alphas, is Aperian (APRN).  Estimated Hs are 0.602 by power spectral analysis, and 0.712 by wavelets, implying alphas of 1.661 and 1.404 respectively.  It should be noted that the variogram H for Aperian is closer to 0.500, and the R/S and roughness-length estimates of H are both less than 0.500, strongly suggesting nearly normal character.  Critical values of alpha vary with sample size.  Critical values for N = 250 and N = 100 are interpolated to obtain appropriate critical values corresponding to the sample under consideration with N = 190.  The power spectrum alpha rejects the Cauchy null at the 0.01 one-tail level.  The wavelet alpha rejects the Cauchy null at the 0.05 one-tail level, but not the 0.01 level.  Strong evidence of Cauchy character for any equity series would have been extremely surprising.  Findings suggest near-Cauchy character may be exclusively a small-sample property.   Interesting, volatile stocks with the highest betas are not necessarily the most nearly Cauchy-distributed.  Although Aperian, which displays the strongest evidence of near-Cauchy character, has an extremely high beta of 10.60, many equities with higher betas but longer sample ranges unambiguously reject the Cauchy null.

 

Conclusion

Large, established, less-entrepreneurial firms should exhibit persistence in equity returns.  This expectation is frequently supported by the empirical examination.  Sometimes, however, large, established firms are found to have ergodic returns.  This result supports the multifractal model of asset returns (MMAR) and disconfirms the weak form of the efficient market hypothesis.  It also suggests that some large technology firms behave in a highly entrepreneurial and innovative manner.  Smaller, less-established, more-innovative, more-entrepreneurial firms should exhibit ergodic returns, and this result is generally observed.  When small, less-established exhibit persistence in equity return, the interpretation suggested is that either

(a) information deficits prevent market participants from valuing these equities properly, imposing persistence, or

(b) these small firms are not innovative or entrepreneurial, but are mistakenly perceived as such, attracting "incompetent money."  If so, these firms, which proliferated during the high tech build up, served the useful function of liquidating incompetent money and moving that capital into more competent hands.

Equities traded in efficient markets should have Hurst exponents approximately equal to 0.50, indicating prices change in a purely random, normally-distributed manner.  Securities with significant secular trends and non-periodic cycles should display time persistence with H > 0.50, unless market efficiency imposes randomness and normality anyway.  Many securities display H approximately equal to 0.50 by one or more of the four techniques.  However, often results indicate H is approximately equal to 0.50 by one or more methods, and indicate H is not equal to 0.50 by others.  For example, Aperian (APRN) shows H close to 0.50 indicating price efficiency, by rescaled-range and roughness-length relationship methods, but shows H > 0.50 indicating price persistence, by power spectral-density, variogram and wavelet methods.  Formal hypothesis tests indicate Aperian is significantly antipersisent by rescaled-range and roughness-length methods (i.e., H is significantly less than 0.50, though not greatly less in magnitude), and significantly persistent by the variogram method, (i.e., H is significantly greater than 0.50.)  Furthermore, the characteristic exponent test fails to reject the Cauchy null at the 0.01 level, a result highly inconsistent with weak market efficiency or Gaussian character.

 

Equities with regime shifts such as splits, mergers, and acquisitions, present special difficulties of interpretation.  Though their price series may have different Hs and cycle lengths for different behavioral regimes, it may be impossible to measure H accurately when the duration of the regime is too small compared to the cycle length. 

 

At least one technology equity appears to exhibit Cauchy or near-Cauchy behavior.  This finding, along with widespread evidence of antipersistance or ergodicity, tends to disconfirm the efficient market hypothesis and support the more general multifractal model of asset returns (MMAR).  Near-Cauchy behavior is only observed for one short price series, that is, an equity with a recent public offering, suggesting the phenomenon may be of only recent vintage, or that it does not generally last very long, and thus is not observed in price series of more established firms.

 

Many technology securities yield evidence of antipersistence, ergodicity, or negative serial correlation.  Formal hypothesis tests indicate H is significantly less than 0.50, though often not by a great magnitude, for twenty-nine of the fifty-four equities sampled, by at least one technique (either R/S, variogram, or both) other than roughness-length, which always rejects the null of normality.  This contradicts the efficient market hypothesis in its weak form, and suggests these equities are not efficiently priced.  The conclusion suggested is that market participants are incapable of efficiently valuing some technology equities, though not necessarily all.  Disconfirmation of the efficient market hypothesis in its weak form suggests possibilities for constructing nonlinear econometric models for improved price forecasting and option valuation.


Appendix

Statistical Methodology

 

Rescaled-range or R/S Analysis:  R/S analysis is the conventional method introduced by Mandelbrot (1972a).  Time series are classified according to the estimated value of the Hurst exponent H, which is defined from the relationship

R/S = anH

where R is the average range of all subsamples of size n, S is the average standard deviation for all samples of size n, a is a scaling variable, and n is the size of the subsamples, which is allowed to range from an arbitrarily small value to the largest subsample the data will allow.   Putting this expression in logarithms yields

log(R/S) = log(a) + H log(n)

which is used to estimate H as a regression slope.  Standard errors are given in parentheses.  H ranges from 1.00 to 0.50 for persistent series, is exactly equal to 0.50 for random walks, ranges from zero to 0.50 for anti-persistent series, and is greater than one for a persistent or autocorrelated series.  Mandelbrot, Fisher, and Calvet (1997) refer to H as the self-affinity index or scaling exponent.

 

Power Spectral Density Analysis:  This method uses the properties of power spectra of self-affine traces, calculating the power spectrum P(k) where k = 2p/l is the wavenumber, and l is the wavelength, and plotting the logarithm of P(k) versus log(k), after applying a symmetric taper function which transforms the data smoothly to zero at both ends.  If the series is self-affine, this plot follows a straight line with a negative slope –b, which is estimated by regression and reported as beta, along with its standard error.  This coefficient is related to the fractal dimension by:  D = (5 - beta)/2.  H and alpha are computed as H = 2 – D, and alpha = 1/H.   Power spectral density is the most common technique used to obtain the fractal dimension in the literature, although it is also highly problematic due to spectral leakage.

 

Roughness-Length Relationship Method:  This method is similar to R/S, substituting the root-mean-square (RMS) roughness s(w) and window size w for the standard deviation and range.  Then H is computed by regression from a logarithmic form of the relationship s(w) = wH.   As noted above, the roughness-length method provides standard errors so low the null hypothesis of H = 0.500 is nearly always rejected no matter how nearly normal the asset returns.

 

Variogram Analysis:  The variogram, also known as variance of the increments, or structure function, is defined as the expected value of the squared difference between two y values in a series separated by a distance w.  In other words, the sample variogram V(w) of a series y(x) is measured as:  V(w) = [y(x) – y(x+w)]2, thus V(w) is the average value of the squared difference between pairs of points at distance w . The distance of separation w is also referred to as the lag.  The Hurst exponent is estimated by regression from the relationship V(w) = w2H. 

 

Wavelet Analysis:  Wavelet analysis exploits localized variations in power by decomposing a series into time frequency space to determine both the dominant modes of variability and how those modes vary in time.  This method is appropriate for analysis of non-stationary traces such as asset prices, i.e. where the variance does not remain constant with increasing length of the data set.  Fractal properties are present where the wavelet power spectrum is a power law function of frequency.  The wavelet method is based on the property that wavelet transforms of the self-affine traces also have self-affine properties.     

 

Consider n wavelet transforms each with a different scaling coefficient ai, where S1, S2,..., Sn are  the standard deviations from zero of the scaling coefficients ai.  Then define the ratio of the standard deviations G1, G2, ..., Gn-1 as: G1 = S1/S2,  G2 = S2/S3, ..., Gn-1 = Sn-1/Sn.  Then the average value of Gi is estimated as Gavg = (Gi)/(n – 1).  The estimated Hurst exponent H is computed as a heuristic function of Gavg.  The Benoit software computes H based on first three dominant wavelet functions, i.e., n is allowed to vary up to 4, and i for the scaling coefficient ai is allowed to vary from i = 0, 1, 2, 3. 

 

Mandelbrot-Lévy Characteristic Exponent Test: The Mandelbrot-Lévy distributions are a family of infinite-variance distributions without explicit analytical expressions, except for special cases.  Limiting distributions include the normal, with finite variance, and the Cauchy, with the most extreme platykurtosis or fat tails.  Paul Lévy (1925) developed the theory of these distributions.  The Hurst exponent H introduced in the hydrological study of the Nile valley is the reciprocal of the characteristic exponent alpha (Hurst 1951).  The characteristic function of a Mandelbrot-Lévy random variable is:

log f(t) = i(δ)t – (γ)|t|α[1 + i(β)(sign(t)(tan[(α)(π/2)])],

where delta is the expectation or mean of t if alpha > 1; gamma is a scale parameter; alpha is the characteristic exponent; and i is the square root of -1.  Gnedenko and Kolmogorov (1954) showed the sum of n independent and identically distributed Mandelbrot-Lévy variables is:

n log f(t) = in(δ)t – n(γ)|t|α [1 + i(β)(sign(t)(tan[(α)(π/2)])],

and thus the distributions exhibit stability under addition.  Many applications of the central limit theorem only demonstrate Mandelbrot-Lévy character.  The result of normality generally depends on an unjustified assumption of finite variance.  Mandelbrot (1972a) introduced a technique for estimating alpha by regression, further refined by Lo (1991).  Mulligan (2000b) estimates the distribution of alpha for Cauchy-distributed random variables.  This distribution is used to test estimated alphas for technology equities against the Cauchy null.

 


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Table 1

Fractal Taxonomy of Time Series

 

Term

 

'Color'

 

Hurst exponent

 

Fractal dimension

 

Characteristic exponent

Antipersistent,

Ergodic,

Mean-reverting,

Negative serial correlation,

1/f noise

Pink noise

0 ≤ H < ˝

0 ≤ D < 1.50

∞ ≤ a < 2.00

Gaussian process,

Normal distribution

White noise

H ş ˝

D ş 1.50

a ş 2.00

Brownian motion, Wiener process

Brown noise

H ş ˝

D ş 1.50

a ş 2.00

Persistent,

Trend-reinforcing,

Hurst process

Black noise

˝ < H < 1

1.50 < D < 1

2.00 < a < 1

Cauchy process,

Cauchy distribution

Cauchy noise

H ş 1

D ş 1

a ş 1

Note:  Brown noise or Brownian motion is the cumulative sum of a normally-distributed white-noise process.  The changes in, or returns on, a Brownian motion, are white noise.  The fractal statistics are the same for Brown and white noise because the brown-noise process should be differenced as part of the estimation process, yielding white noise.

 


 


 Table 2

Technology Equity Price Series

Firm

Ticker

Exchange

Dates

N

beta

24/7 Media

TFSM

NASDAQ

08/14/98-06/18/01

712

6.60

Advanced Micro Devices

AMD

NYSE

12/31/93-06/18/01

1882

1.83

Amazon

AMZN

NASDAQ

05/15/97-06/18/01

1031

3.20

American Power Conversion

APCC

NASDAQ

12/31/93-06/18/93

1882

1.41

Ameritrade

AMTD

NASDAQ

04/01/97-06/18/01

1063

2.93

Andrew Electronics

ANDW

NASDAQ

12/31/93-06/18/93

1882

0.70

Aperian

APRN

NASDAQ

09/11/00-06/18/01

189

10.60

Apple Computer

AAPL

NASDAQ

12/31/93-06/18/93

1882

1.22

Applied Micro Circuits

AMCC

NASDAQ

11/25/97-06/18/01

896

2.99

AT&T

T

NYSE

12/31/93-06/18/93

1882

0.96

Audiocodes Ltd

AUDC

NASDAQ

05/28/99-06/18/01

514

5.20

Bridge Technology

BRDG

NASDAQ

04/28/99-06/18/01

525

6.80

Broadcom

BRCM

NASDAQ

04/17/98-06/18/01

799

3.82

Cisco Systems

CSCO

NASDAQ

12/31/93-06/18/93

1882

1.85

Corel

CORL

NASDAQ

12/31/93-06/18/93

1882

1.20

Datastream Networks

DSTM

NASDAQ

03/30/95-06/18/93

1569

2.08

Dell Computer

DELL

NASDAQ

12/31/93-06/18/01

1882

2.18

Digital Lava

DGLV

NASDAQ

02/18/99-06/18/01

584

5.90

Doubleclick

DCLK

NASDAQ

02/20/98-06/18/01

838

3.70

Drugmax

DMAX

NASDAQ

10/19/98-06/18/01

666

12.70

E Bay

EBAY

NASDAQ

09/24/98-06/18/01

688

3.55

Ericsson

ERICY

NASDAQ

12/31/93-06/18/93

1882

1.74

Futurelink

FTRLD

NASDAQ

07/09/97-06/18/01

974

5.10

General Electric

GE

NYSE

12/31/93-06/18/93

1882

1.19

Geneva St Lawrence

GNVH

NASDAQ

01/05/01-06/18/01

108

9.10

Greystone Digital

GSTN

NASDAQ

07/16/97-06/18/01

983

23.80

Hear Me

HEAR

NASDAQ

04/29/99-06/18/01

535

5.60

Hewlett-Packard

HWP

NYSE

12/31/93-06/18/01

1882

1.34

Intel

INTC

NASDAQ

12/31/93-06/18/93

1882

1.56

International Business Machines

IBM

NYSE

12/31/93-06/18/93

1878

1.30

Intraware

ITRA

NASDAQ

02/26/99-06/18/01

578

5.10

Juno Online Services

JWEB

NASDAQ

05/26/99-06/18/01

516

6.00

Log on America

LOAX

NASDAQ

04/22/99-06/18/01

540

5.00

Lucent Technologies

LU

NYSE

04/04/96-06/18/93

1312

2.06

Microsoft

MSFT

NASDAQ

12/31/93-06/18/01

1881

1.80

Motorola

MOT

NYSE

12/31/93-06/18/93

1882

1.34

Nokia

NOK

NYSE

12/31/93-06/18/93

1882

2.00

Nortel

NT

NYSE

12/31/93-06/18/01

1882

2.16

Novell

NOVL

NASDAQ

12/31/93-06/18/93

1882

2.28

Novellus

NVLS

NASDAQ

12/31/93-06/18/93

1882

2.40

Oracle

ORCL

NASDAQ

12/31/93-06/18/93

1882

1.97

Red Hat

RHAT

NASDAQ

08/11/99-06/18/01

467

2.91

Sagent Technology

SGNT

NASDAQ

04/14/99-06/18/01

546

6.00

SAP AG

SAP

NYSE

01/15/96-16/18/01

1364

1.32

Solectron

SLR

NYSE

12/31/93-06/18/93

1882

2.19

Sprint

PCS

NYSE

11/23/98-06/18/01

646

2.96

Star Scientific

STSI

NASDAQ

07/01/99-06/18/01

491

115.0

Sun Microsystems

SUNW

NASDAQ

12/31/93-06/18/01

1882

1.91

Texas Instruments

TXN

NYSE

12/31/93-06/18/01

1882

1.80

Time Warner Telecom

TWTC

NASDAQ

05/12/99-06/18/01

530

3.20

Usurf America

UAX

AMEX

10/22/97-06/18/01

909

6.70

Voicenet

VTC

AMEX

07/11/00-06/18/01

232

18.40

Xerox

XRX

NYSE

12/31/93-06/18/93

1882

1.50

Yahoo

YHOO

NASDAQ

04/12/96-06/18/01

1307

3.70

Note:

The sample of fifty-four technology firms represented here consists predominantly of NASDAQ stocks.  Fourteen firms are NYSE listed and two are AMEX listed.  All price series are adjusted for stock splits and converted to logarithmic returns.

 


 


Table 3

Fractal Analyses of Technology Securities

Estimated Hurst Exponent H, Various Methods

(Standard Errors in Parentheses)

 

Firm

 

 

R/S

Power Spectrum

Roughness-length

 

Variogram

 

Wavelet

24/7 Media

0.499

(0.012)

0.541

0.507

(0.001)**

0.524

(0.009)**

0.514

Advanced Micro Devices

0.529

(0.007)**

0.506

0.462

(0.007)**

0.548

(0.135)

0.524

Amazon

0.486

(0.018)

0.479

0.435

(0.003)**

0.525

(0.043)

0.544

American Power Conversion

0.489

(0.027)

0.443

0.436

(0.002)**

0.477

(0.032)

0.491

Ameritrade

0.484

(0.012)*

0.456

0.437

(0.002)**

0.551

(0.078)

0.540

 

Andrew Electronics                        

0.482

(0.029)

0.432

0.470

(0.005)**

0.442

(0.040)*

0.558

Aperian

0.448

(0.006)**

0.602

0.403

(0.000)**

0.584

(0.003)**

0.712

Apple Computer

0.479

(0.039)

0.499

0.452

(0.003)**

0.499

(0.037)

0.647

Applied Micro Circuits

0.461

(0.005)**

0.489

0.365

(0.002)**

0.526

(0.043)

0.497

AT&T

0.476

(0.022)*

0.432

0.398

(0.008)**

0.538

(0.189)

0.458

Audiocodes Ltd

0.450

(0.018)**

0.502

0.392

(0.002)**

0.517

(0.030)

0.500

Bridge Technology

0.386

(0.013)**

0.366

 

0.338

(0.001)**

0.462

(0.062)

0.519

Broadcom

0.475

(0.012)**

0.470

0.376

(0.005)**

0.554

(0.041)*

0.610

Cisco Systems

0.476

(0.012)*

0.431

0.382

(0.001)**

0.553

(0.213)

0.531

Corel

0.496

(0.051)

0.499

0.486

(0.005)**

0.504

(0.101)

0.496

Datastream Networks

0.479

(0.046)

0.208

0.482

(0.003)**

0.428

(0.248)

0.656

Dell Computer

0.427

(0.063)*

0.448

0.426

(0.008)**

0.524

(0.270)

0.530

Digital Lava

0.516

(0.021)

0.439

0.506

(0.002)**

0.410

(0.197)

0.520

Doubleclick

0.488

(0.009)*

0.496

0.387

(0.002)**

0.547

(0.068)

0.493

Drugmax

0.374

(0.021)**

0.296

0.385

(0.004)**

0.162

(0.244)**

0.482

E Bay

0.501

(0.007)

0.460

0.445

(0.003)**

0.501

(0.058)

0.529

Ericsson

0.440

(0.051)*

0.483

0.389

(0.005)**

0.544

(0.175)

0.482

 

Futurelink

0.467

(0.053)

0.472

0.553

(0.002)**

0.312

(0.108)*

0.543

General Electric

0.447

(0.020)**

0.383

 

0.373

(0.007)**

0.376

(0.143)

0.490

Geneva St Lawrence

0.319

(0.001)**

n/a

n/a

n/a

0.383

Greystone Digital

0.314

(0.056)**

0.332

0.349

(0.003)**

0.357

(0.047)**

0.341

Hear Me

0.453

(0.006)**

0.514

0.469

(0.002)**

0.487

(0.022)

0.564

Hewlett-Packard

0.445

(0.032)*

0.424

0.413

(0.005)**

0.490

(0.085)

0.491

Intel

0.522

(0.010)**

0.468

0.449

(0.006)**

0.531

(0.131)

0.492

International Business Machines

0.483

(0.058)

0.420

0.469

(0.003)**

0.369

(0.505)

0.557

Intraware

0.538

(0.022)*

0.112

0.574

(0.002)**

0.486

(0.090)

0.452

Juno Online Services

0.458

(0.037)*

0.425

0.456

(0.001)**

0.302

(0.202)

0.491

Log on America

0.417

(0.016)**

0.443

0.375

(0.002)**

0.432

(0.055)**

0.506

Lucent Technologies

0.484

(0.037)

0.490

0.412

(0.005)**

0.563

(0.262)

0.593

Microsoft

0.460

(0.040)

0.433

0.423

(0.010)**

0.460

(0.271)

0.569

Motorola

0.495

(0.014)

0.433

0.406

(0.004)**

0.495

(0.060)

0.561

Nokia

0.503

(0.022)

0.484

0.441

(0.002)**

0.521

(0.058)

0.588

Nortel Networks

0.506

(0.043)

0.541

0.436

(0.005)**

0.579

(0.121)

0.533

Novell

0.439

(0.054)*

0.457

0.413

(0.010)**

0.475

(0.114)

0.534

Novellus

0.496

(0.025)

0.414

0.459

(0.004)**

0.424

(0.178)

0.551

Oracle

0.437

(0.051)*

0.495

0.430

(0.004)**

0.531

(0.068)

0.575

Red Hat

0.511

(0.015)

0.549

0.513

(0.005)**

0.541

(0.006)**

0.590

Sagent Technology

0.497

(0.025)

0.539

0.600

(0.002)**

0.484

(0.076)

0.528

SAP AG

0.439

(0.020)**

0.480

0.396

(0.003)**

0.481

(0.030)

0.532

Solectron

0.456

(0.027)*

0.458

0.390

(0.004)**

0.431

(0.052)*

0.590

Sprint

0.463

(0.022)**

0.483

0.460

(0.002)**

0.490

(0.010)

0.488

Star Scientific

0.366

(0.007)**

0.347

0.167

(0.003)**

0.396

(0.024)**

0.399


 

Sun Microsystems

0.482

(0.022)

0.463

0.409

(0.003)**

0.545

(0.200)

0.474

Texas Instruments

0.484

(0.014)*

0.451

0.408

(0.002)**

0.517

(0.174)

0.534

 

Time Warner Telecom

0.443

(0.010)**

0.420

0.362

(0.002)**

0.459

(0.014)**

0.490

Usurf America

0.494

(0.011)

0.394

0.404

(0.003)**

0.425

(0.048)*

0.461

Voicenet

0.456

(0.005)**

0.480

0.584

(0.001)**

0.493

(0.004)*

0.353

Xerox

0.457

(0.064)

0.491

0.438

(0.006)**

0.535

(0.095)

0.534

Yahoo

0.490

(0.027)

0.499

0.445

(0.003)**

0.590

(0.221)

0.518

Notes:

1.  The Mandelbrot-Lévy characteristic exponent alpha is the reciprocal of the Hurst exponent H, thus alpha = 1/H.  The fractal dimension D = 2 – H.

2.  Hs computed by R/S, Roughness-length, and Variogram methods are used for conventional hypothesis tests where the null hypothesis is H = 0.500, (i.e., equivalently alpha = 2, D = 1.500, or normality of the asset returns).  Rejection of the null at the 0.05 two-tail significance level is indicated with *, at the 0.01 level with **.

 


 

Table 4

Summary Cauchy Distribution Tests

Mandelbrot-Lévy Characteristic Exponent Test

Firm

Technique

Est alpha

One-tailed critical alpha

N = 100

One-tailed

critical alpha

N = 250

Interpolated critical alpha

Nsample = 190

Outcome

APRN

Power spectrum

1.661

 

alpha10% = 1.256

alpha5% = 1.354

alpha1% = 1.634

alpha10% = 1.157

alpha5% = 1.215

alpha1% = 1.324

alpha10% = 1.1966

alpha5% = 1.2706

alpha1% = 1.4480

Rejects Cauchy H0 at 1% significance level

APRN

Wavelets

1.404

alpha10% = 1.256

alpha5% = 1.354

alpha1% = 1.634

alpha10% = 1.157

alpha5% = 1.215

alpha1% = 1.324

alpha10% = 1.1966

alpha5% = 1.2706

alpha1% = 1.4480

Rejects Cauchy H0 at 5% significance level but not 1% level

Note:

Critical values from Mulligan 2000b.  Nsample is 190.  Under the Cauchy null, the distribution of the Mandelbrot-Lévy characteristic exponent varies with the sample size.