Fractal Analysis of Highly Volatile Markets:
An Application to Technology Equities
QUARTERLY
REVIEW OF ECONOMICS AND FINANCE (2004) 44(1): 155179
Robert F. Mulligan, Ph.D.
Department
of Business Computer Information Systems & Economics
Phone:
8282273329
Fax:
8282277414
Email:
Acknowledgements
The author wishes to
thank Patrick Hays,
Abstract
This paper examines technology equity price series using five
selfaffine fractal analysis techniques for estimating the Hurst exponent,
MandelbrotLévy characteristic exponent, and fractal
dimension. Techniques employed are
rescaledrange analysis, powerspectral density analysis, roughnesslength
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,
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 MandelbrotLévy characteristic exponent (Lévy 1925). MandelbrotLévy distributions are also referred to as stable, Lévystable, Lstable, stableParetian, and ParetoLévy. Samuelson (1982) popularized the term MandelbrotLévy, 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 multifractal 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 rapidlychanging, innovationcharged
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
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 rescaledrange (R/S), which is robust against shortterm 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
Many technology securities in the sample are antipersistent or ergodic or meanreverting 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 overreact 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 nonnormality or nonGaussian character have severe implications for pricing financial derivatives. Because the BlackScholes (1972, 1973) option pricing model assumes normallydistributed prices for underlying securities, financial derivatives based on nonnormal securities prices cannot be priced efficiently. In this kind of highlyvolatile 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) disequilibriumovershooting model for exchange rate determination.
Long memory series exhibit nonperiodic long cycles, or persistent dependence between observations far apart in time. Shortterm 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 nonrandom 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 nonrandom 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 nonperiodic longrun cyclical dependence as distinguished from short dependence or Markov character and periodic variation (Mandelbrot 1972a, pp. 259260). 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 probabilisticchaotic 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. 164167). 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 antipersistent series,
e.g., negative serial correlation. Ergodic series should also have much shorter cycle lengths
than random walks or trendreinforcing series.
Five techniques are reported in this paper, Mandelbrot's (1972a) AR1
rescaledrange or R/S analysis, power spectraldensity analysis,
roughnesslength relationship analysis, variogram
analysis, and wavelet analysis. Each
method analyzes equity returns as selfaffine traces, providing estimates of
the
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 competenceincompetence 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 firmlevel decision makers adds value in each stage of production (Mises 1998 pp. 480485; Rothbard 1970, pp. 323332; 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. Projectoriented 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 experimentallyorganized 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 nonconvergence 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. 3846; 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. 160161; 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
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 selfallocating (Pelikan 1993; Eliasson 1996); incompetence, in contrast, may be described as selfmisallocating. 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) competencedifficulty (CD) gap hypothesis as a potential source of long memory in asset prices, offering a theoretical expectation of long memory. The CD 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 CD 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 nonperiodic 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 mediumsized 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, wellestablished 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 "firstmover 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) "puttyclay" model of investment, which emphasizes the distinction between highlyliquid, 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 innovationcharged 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 decisionmaking. 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),
Methodological approach
Mandelbrot (1972b, 1974) and Mandelbrot, Fisher, and Calvet (1997) have developed the multifractal model of asset returns (MMAR), which shares the longmemory 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, trendreinforcing series, also called biased random walks, random walks with drift, or fractional Brownian motion, 3.) random walks, or 4.) antipersistent, ergodic, or meanreverting series.
<<Table 1 about here>>
Table 1 provides
the taxonomy of time series identified through fractal analysis. Because the
Fractal analysis also gives an estimate of the average nonperiodic 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 onehalf years, were never long enough to provide reliable estimates of nonperiodic cycle length.
<<Table
2 about here>>
Data
The data are daily
closing prices reported by the exchanges for each traded
equity. A sample of fiftyfour
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
MandelbrotLévy
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
Approximately two cycle lengths of data are necessary for good estimates average nonperiodic 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 fiftyfour 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 exponentiallygrowing 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
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 splitadjusted 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
Rescaledrange 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 twotail 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 highlyestablished, 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)HewlettPackard (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 thirtyone of the fiftyfour equities in the sample, by at least the five percent significance level, twotailed. 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 weakform 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 BlackScholes (1972, 1973) option pricing model and its underlying assumption of normallydistributed asset prices. Financial derivatives based on nonnormal 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 shortterm 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.
RoughnessLength 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 roughnesslength 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 fiftyfour. 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), HewlettPackard (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.
MandelbrotLévy 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 MandelbrotLévy
characteristic exponent alpha generated by
<<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 roughnesslength 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 onetail level. The wavelet alpha rejects the Cauchy null at the 0.05 onetail level, but not the 0.01 level. Strong evidence of Cauchy character for any equity series would have been extremely surprising. Findings suggest nearCauchy character may be exclusively a smallsample property. Interesting, volatile stocks with the highest betas are not necessarily the most nearly Cauchydistributed. Although Aperian, which displays the strongest evidence of nearCauchy 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, lessentrepreneurial 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, lessestablished, moreinnovative, moreentrepreneurial firms should exhibit ergodic returns, and this result is generally observed. When small, lessestablished 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
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 nearCauchy 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). NearCauchy 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 twentynine of the fiftyfour equities sampled, by at least one technique (either R/S, variogram, or both) other than roughnesslength, 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
Rescaledrange 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
R/S
= an^{H}
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 antipersistent series, and is greater than one for a persistent or autocorrelated series. Mandelbrot, Fisher, and Calvet (1997) refer to H as the selfaffinity index or scaling exponent.
Power Spectral Density Analysis: This method uses the properties of power spectra of selfaffine 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 selfaffine, 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.
RoughnessLength Relationship Method: This method is similar to R/S, substituting the rootmeansquare (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) = w^{H}. As noted above, the roughnesslength 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
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 nonstationary 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 selfaffine traces also have selfaffine properties.
Consider n wavelet
transforms each with a different scaling coefficient a_{i},
where S_{1}, S_{2},..., S_{n} are
the standard deviations from zero of the scaling coefficients a_{i}. Then
define the ratio of the standard deviations G_{1}, G_{2}, ..., G_{n1} as: G_{1} = S_{1}/S_{2}, G_{2} = S_{2}/S_{3},
..., G_{n1} = S_{n1}/S_{n}. Then the average value of G_{i}
is estimated as G_{avg} = (G_{i})/(n – 1). The estimated
MandelbrotLévy Characteristic Exponent Test: The
MandelbrotLévy distributions are a family of
infinitevariance 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
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 MandelbrotLévy 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 MandelbrotLévy 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 Cauchydistributed random variables. This distribution is used to test estimated alphas for technology equities against the Cauchy null.
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Siegfried, J. & Evans, L. (1994). Empirical studies of entry and exit: a survey of the evidence. Review of Industrial Organization 9(2), 121155.
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Tushman, M.L. & Anderson, P. (1986). Technological discontinuities and organizational environments. Administrative Science Quarterly 31, 439465.
Table 1 Fractal Taxonomy of Time Series 

Term 
'Color' 

Fractal dimension 
Characteristic exponent 
Antipersistent, Ergodic, Meanreverting, Negative serial correlation, 1/f noise 
Pink noise 
0 ≤ H < ½ 
0 ≤ D < 1.50 
∞ ≤ a < 2.00 
Gaussian process, Normal distribution 
White noise 
H º ½ 
D º 1.50 
a º 2.00 
Brownian motion, Wiener process 
Brown noise 
H º ½ 
D º 1.50 
a º 2.00 
Persistent, Trendreinforcing, 
Black noise 
½ < H < 1 
1.50 < D < 1 
2.00 < a < 1 
Cauchy process, Cauchy distribution 
Cauchy noise 
H º 1 
D º 1 
a º 1 
Note: Brown noise or Brownian motion is the cumulative sum of a normallydistributed whitenoise 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 brownnoise 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 
08/14/9806/18/01 
712 
6.60 
Advanced Micro Devices 
AMD 
NYSE 
12/31/9306/18/01 
1882 
1.83 
Amazon 
AMZN 
NASDAQ 
05/15/9706/18/01 
1031 
3.20 
American Power Conversion 
APCC 
NASDAQ 
12/31/9306/18/93 
1882 
1.41 
Ameritrade 
AMTD 
NASDAQ 
04/01/9706/18/01 
1063 
2.93 
Andrew Electronics 
ANDW 
NASDAQ 
12/31/9306/18/93 
1882 
0.70 
Aperian 
APRN 
NASDAQ 
09/11/0006/18/01 
189 
10.60 
Apple Computer 
AAPL 
NASDAQ 
12/31/9306/18/93 
1882 
1.22 
Applied Micro Circuits 
AMCC 
NASDAQ 
11/25/9706/18/01 
896 
2.99 
AT&T 
T 
NYSE 
12/31/9306/18/93 
1882 
0.96 
Audiocodes Ltd 
AUDC 
NASDAQ 
05/28/9906/18/01 
514 
5.20 
Bridge Technology 
BRDG 
NASDAQ 
04/28/9906/18/01 
525 
6.80 
Broadcom 
BRCM 
NASDAQ 
04/17/9806/18/01 
799 
3.82 
Cisco Systems 
CSCO 
NASDAQ 
12/31/9306/18/93 
1882 
1.85 
Corel 
CORL 
NASDAQ 
12/31/9306/18/93 
1882 
1.20 
Datastream Networks 
DSTM 
NASDAQ 
03/30/9506/18/93 
1569 
2.08 
Dell Computer 
DELL 
NASDAQ 
12/31/9306/18/01 
1882 
2.18 
Digital Lava 
DGLV 
NASDAQ 
02/18/9906/18/01 
584 
5.90 
Doubleclick 
DCLK 
NASDAQ 
02/20/9806/18/01 
838 
3.70 
Drugmax 
DMAX 
NASDAQ 
10/19/9806/18/01 
666 
12.70 
E Bay 
EBAY 
NASDAQ 
09/24/9806/18/01 
688 
3.55 
Ericsson 
ERICY 
NASDAQ 
12/31/9306/18/93 
1882 
1.74 
Futurelink 
FTRLD 
NASDAQ 
07/09/9706/18/01 
974 
5.10 
General Electric 
GE 
NYSE 
12/31/9306/18/93 
1882 
1.19 

GNVH 
NASDAQ 
01/05/0106/18/01 
108 
9.10 
Greystone Digital 
GSTN 
NASDAQ 
07/16/9706/18/01 
983 
23.80 
Hear Me 
HEAR 
NASDAQ 
04/29/9906/18/01 
535 
5.60 
HewlettPackard 
HWP 
NYSE 
12/31/9306/18/01 
1882 
1.34 
Intel 
INTC 
NASDAQ 
12/31/9306/18/93 
1882 
1.56 
International Business Machines 
IBM 
NYSE 
12/31/9306/18/93 
1878 
1.30 
Intraware 
ITRA 
NASDAQ 
02/26/9906/18/01 
578 
5.10 
Juno Online Services 
JWEB 
NASDAQ 
05/26/9906/18/01 
516 
6.00 
Log
on 
LOAX 
NASDAQ 
04/22/9906/18/01 
540 
5.00 
Lucent Technologies 
LU 
NYSE 
04/04/9606/18/93 
1312 
2.06 
Microsoft 
MSFT 
NASDAQ 
12/31/9306/18/01 
1881 
1.80 
Motorola 
MOT 
NYSE 
12/31/9306/18/93 
1882 
1.34 
Nokia 
NOK 
NYSE 
12/31/9306/18/93 
1882 
2.00 
Nortel 
NT 
NYSE 
12/31/9306/18/01 
1882 
2.16 
Novell 
NOVL 
NASDAQ 
12/31/9306/18/93 
1882 
2.28 
Novellus 
NVLS 
NASDAQ 
12/31/9306/18/93 
1882 
2.40 
Oracle 
ORCL 
NASDAQ 
12/31/9306/18/93 
1882 
1.97 
Red Hat 
RHAT 
NASDAQ 
08/11/9906/18/01 
467 
2.91 
Sagent Technology 
SGNT 
NASDAQ 
04/14/9906/18/01 
546 
6.00 
SAP AG 
SAP 
NYSE 
01/15/9616/18/01 
1364 
1.32 
Solectron 
SLR 
NYSE 
12/31/9306/18/93 
1882 
2.19 
Sprint 
PCS 
NYSE 
11/23/9806/18/01 
646 
2.96 
Star Scientific 
STSI 
NASDAQ 
07/01/9906/18/01 
491 
115.0 
Sun Microsystems 
SUNW 
NASDAQ 
12/31/9306/18/01 
1882 
1.91 

TXN 
NYSE 
12/31/9306/18/01 
1882 
1.80 
Time Warner Telecom 
TWTC 
NASDAQ 
05/12/9906/18/01 
530 
3.20 
Usurf 
UAX 
AMEX 
10/22/9706/18/01 
909 
6.70 
Voicenet 
VTC 
AMEX 
07/11/0006/18/01 
232 
18.40 
Xerox 
XRX 
NYSE 
12/31/9306/18/93 
1882 
1.50 
Yahoo 
YHOO 
NASDAQ 
04/12/9606/18/01 
1307 
3.70 
Note: The sample of fiftyfour 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 (Standard Errors in Parentheses) 

Firm 
R/S 
Power
Spectrum 
Roughnesslength 
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 

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 
HewlettPackard 
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 
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 

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 
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 MandelbrotLévy
characteristic exponent alpha is the reciprocal of the 2. Hs computed by R/S, Roughnesslength, 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 twotail significance level is indicated with *, at the 0.01 level with **. 
Table 4 Summary Cauchy Distribution Tests MandelbrotLévy Characteristic Exponent Test 

Firm Technique 
Est alpha 
Onetailed critical alpha N = 100 
Onetailed critical alpha N = 250 
Interpolated critical alpha N_{sample} = 190 
Outcome 
APRN Power spectrum 
1.661 
alpha_{10%} = 1.256 alpha_{5%} = 1.354 alpha_{1%} = 1.634 
alpha_{10%} = 1.157 alpha_{5%} = 1.215 alpha_{1%} = 1.324 
alpha_{10%} = 1.1966 alpha_{5%} = 1.2706 alpha_{1%} = 1.4480 
Rejects Cauchy H_{0} at 1% significance level 
APRN Wavelets 
1.404 
alpha_{10%} = 1.256 alpha_{5%} = 1.354 alpha_{1%} = 1.634 
alpha_{10%} = 1.157 alpha_{5%} = 1.215 alpha_{1%} = 1.324 
alpha_{10%} = 1.1966 alpha_{5%} = 1.2706 alpha_{1%} = 1.4480 
Rejects Cauchy H_{0} at 5% significance level but not 1% level 
Note: Critical values from Mulligan 2000b. N_{sample} is 190. Under the Cauchy null, the distribution of the MandelbrotLévy characteristic exponent varies with the sample size. 