Issue 67, 2013

Abstract

This study develops new insights into the profitability of trading rules through a synthesis of fractal geometry and technical analysis. The Hurst exponent (H) emerged from fractal geometry as a means of detecting longterm dependencies in a time series, the same dependencies that technical analysis should be able to identify and exploit to earn profits. Two tests of this synthesis are conducted. First, financial time series are classified into three groups based on their H to determine if a higher (lower) H results in higher returns to trending (contrarian) trading rules. Second, the relationship between H and profits from technical analysis are estimated through OLS regression. Both tests suggest that the fractal nature of a time series explains a significant portion of the profits from technical analysis.

I. Introduction

Technical analysis is a broad discipline that analyzes past price and volume data to identify patterns that predict future price movements. Identified patterns provide the basis for technical trading rules, which generate buy and sell signals. The efficacy of technical analysis has been researched extensively. The research results are mixed, providing support for (e.g., Brock, Lakonishok and LeBaron (1992), and Gençay (1999)), and against (e.g. Allen and Karjalainen (1999), Lo, Mamaysky and Wang (2000), and Bokhardi et al. (2005)) technical analysis’ ability to forecast security returns.

Recently, Hurst’s exponent (H) (Hurst, 1951) has emerged from fractal geometry into economics research as a means of classifying a time series based on its long-term dependencies (Peters, 1991 and Peters, 1994). A value of H of 0.50 indicates that a series exhibits Brownian motion.[1] 0<H<0.5 indicates an anti-persistent series, suggesting that the data set exhibits mean-reverting tendencies. 0.5<H<1 indicates a persistent series, suggesting the data is trend reinforcing. The strength of the trend increases as H approaches 1. The H thus provides a method of classifying time series, which may be beneficial for the discipline of technical analysis.

The purpose of this study is to develop new insights into the discipline of technical analysis through a synthesis with fractal geometry. The synthesis posits the following: fractal geometry provides a technique (H) that detects long-term dependencies (reinforcing or reverting trends) in the historical price data of a time series; these are the same trends that technical analysis purports to identify and utilize to predict future price movements. Therefore, trending trading rules should be more effective on trend-reinforcing time series, while contrarian trading rules should be more effective in anti-persistent, or mean-reverting, markets.

It is important to note that the discipline of technical analysis includes various technical trading rules. In addition, the technical trading rules can be specified differently, leading to a large number of trading rule variants. Accordingly, popular trending and contrarian rules will be used to test the synthesis. Specifically, trending trading rules will be represented by the filter rule, moving average crossover rule, and the trading range breakout rule, while contrarian trading rules will be represented by the Bollinger Band. In addition, the effectiveness of the technical trading rules is defined as the profits generated from the technical trading rule’s buy and sell signals.

Two empirical tests are conducted to evaluate the synthesis and the resulting relationship between H and profits from technical analysis. First, the financial series are classified into three groups based on their H value (H<0.5; 0.5<H<0.55; H>0.55) to determine if time series having a higher (lower) value H result in higher profits to trending (contrarian) trading rules. Second, OLS regression is used to estimate the relationship between H and profits from trending and contrarian trading rules.

The results suggest that the H is able to identify long-term dependencies in a time series and that these time series result in higher profits to technical trading rules. The classification analysis reveals that profits from trending trading rules are higher (average of 11%) for time series that exhibit long-term dependencies (high H) and lower (average of -16.8%) for time series that exhibit anti-persistent trends. The regression analysis results in a significant R2 of 0.31, revealing that the fractal nature of a time series explains a significant portion of the profits from technical trading rules. The results are consistent with the theory presented by the synthesis.

This study makes a significant contribution to the literature from both a theoretical and empirical perspective as this is the first known study to merge the disciplines of technical analysis and fractal geometry. The extant literature that investigates the fractal nature of financial data seeks only to determine market predictability (e.g. Qian and Rasheed, 2004, Corazza and Malliaris, 2002). This study extends the literature by investigating whether technical trading rules can exploit a time-series’s predictability to generate profits. The results reveal that the fractal nature of a time-series is related to the profitability of different types of trading rules (i.e., trending versus contrarian). Additionally, the results suggests that an investor, who understands the fractal nature of a timeseries, should alter their investment strategy by employing a contrarian trading rule on time-series that exhibits anti-persistence and a trending trading rule on time-series that exhibit long-term dependencies.

The remainder of the paper is organized as follows: Section II provides a review of the literature and develops the synthesis and hypotheses, Section III describes the data, Section IV discusses the methodology, Section V presents the results, and Section VI offers concluding thoughts.

II. Theory and Hypothesis Development

A synthesis of technical analysis and fractal geometry can provide fertile ground for new theory development, along with novel empirical tests, to enhance our understanding of the profitability of technical trading rules. The following literature review develops this synthesis. Section A discusses the technical analysis literature, Section B discusses fractal geometry, and Section C develops the synthesis and Hypotheses.

A.Technical Trading Rules

Early studies on technical analysis conducted by Alexander (1961 and 1964) and Fama and Blume (1966) suggested that excess returns cannot be realized by making investment decisions based on filter rules. However, Sweeney (1988) re-examined the data used by Fama and Blume (1966) and found that filter rules applied to 15 of the 30 Dow Jones stocks earned excess returns over buy-and-hold alternatives. Technical trading rules have also been extensively tested in the foreign exchange market (Dooley and Shafer (1983), Sweeney (1988), and Schulmeister (1988)).

The number of studies on technical trading rules significantly increased during the 1990s. Some of the most influential studies that provide indirect support for trading rules include Jegadeesh and Titman (1993), Blume, Easley, and O’Hara (1994), Chan, Jagadeesh, and Lakonishok (1996), Lo and MacKinlay (1997), Grundy and Martin (1998), and Rouwenhorst (1998). Stronger evidence can be found in the research of Neftci (1991), Neely, Weller, and Dittmar (1997), Chang and Osler (1994), Osler and Chang (1995), Lo and MacKinlay (1997), and Neely and Weller (1998). One of the most influential studies on technical analysis is that prepared by Brock, Lakonishok, and LeBaron (1992) who used bootstrapping techniques and two simple, yet popular, trading rules to reveal strong evidence in support of the predictive nature of technical analysis.

However, not all studies support the efficacy of technical analysis. For example, Allen and Karjalainen (1999) found no evidence that the rules were able to earn economically significant excess returns over a buy-and-hold strategy during the period 1970 – 1989. Furthermore, Lo, Mamaysky and Wang (2000) found that certain technical patterns can provide information when applied to a large number of stocks; however, the results do not imply that technical analysis can be used to generate excess trading profits. Finally, Bokhardi et al. (2005) investigated the effectiveness of simple trading rules and concluded that trading rules cannot be used profitably after adjusting for transaction costs.

B. Fractal Geometry

Fractal geometry has recently emerged into the world of mathematics as a complement to Euclidean geometry as an attempt to better explain and describe the objects and shapes of the real world. In the 1960s, Benoit Mandelbrot believed that Brownian motion[2] was not an adequate statistical description of the true stochastic process generating securities returns. To resolve this inadequacy, Mandelbrot worked in two perpendicular directions. One direction involved relaxing a Brownian motion assumption of finite variance, which introduces what Mandelbrot termed the “Noah Effect.” The other direction entailed relaxing an independence assumption, thereby allowing for a “Joseph Effect.” (Mandelbrot, 1972). The Noah Effect (recalling the Biblical account of the great deluge) refers to the tendency of various time series with presumably independent increments, especially speculative time series, to exhibit abrupt and discontinuous changes.

The Joseph Effect is named after the biblical story in which Joseph prophesied that the residents of Egypt would face seven years of feast followed by seven years of famine (Mandelbrot, 2004)[3]. The Joseph effect denotes the property of certain time series exhibiting persistent behavior (such as years of flooding followed by years of drought along the Nile River basin) more frequently than would be expected if the series were completely random but without exhibiting any significant short-term (Markovian) dependence. To describe such processes, Mandelbrot broadened the idea of Brownian motion into the class of stochastic processes called “fractional” Brownian motion (fBm).

A fractal time series is statistically self-similar regardless of the time frame over which the increments of the series are observed, aside from its scale. For example, a time series of daily, weekly, monthly, or yearly observations would exhibit similar statistical characteristics. Schroeder (1991) notes that the paradigm of random fractals can be described as Brownian motion, that is, a white-noise process, that exhibits these scaling time-series properties.

Fractional Brownian motions exhibit complicated long-term dependencies that can be characterized by the Hurst exponent (H). Mandelbrot developed a statistical technique called rescaled range analysis to measure the Hurst exponent, which generally ranges from 0 to 1 (Hurst et al., 1965)[4]. If 0.5<H<1, the series will exhibit persistence, with fewer reversals and longer trends than the increments of Brownian motion. In this case, the graph would appear smoother than that of a random walk. On the other hand, if 0<H<0.5, then the series will exhibit anti-persistence, as evidenced by a greater number of reversals and fewer and shorter trends than in a white-noise series.

The vast majority of the research that investigates the Hurst exponent in financial market seeks to determine whether H can identify the predictability of the financial market (e.g., Corazza and Malliaris (2002) and Glenn (2007)). For example, Greene and Fielitz (1977) used the rescaled range analysis and found considerable evidence of temporal dependence in daily stock returns for the period December 23, 1963 to November 29, 1968, after accounting for short-term linear dependencies (autocorrelation) within the data. The most popular example is by Peters (1991), who estimates the Hurst exponent to be 0.778 for monthly returns on the S&P 500 from January 1950 to July 1988.

More recently, research has shifted to using the H as part of an investment strategy. For example, Hodges (2006) and Bender et al. (2006) began investigating the possibility of developing portfolios based on identifiable long-term dependencies. Hodges (2006) examined an investor’s ability to form arbitrage portfolios under realistic transactions costs for values of H very different from 0.5. Bender et al. (2006) seeked to develop a general theory of arbitrage portfolio building based on long-term dependent processes. However, neither of these papers specifically focused on investigating the efficacy of technical analysis in light of long-term dependencies.

The research on the H in financial settings has yet to incorporate any elements from the discipline of technical analysis. Specifically, the current body of literature does not use technical analysis to build on the identification of financial market predictability to determine whether profits can be generated after accounting for transaction costs.

C. A Synthesis of Technical Analysis and Fractal Geometry

The extant body of literature provides inconclusive evidence on the profitability of technical trading rules. Furthermore, the literature on the fractal nature of financial markets appears to stop once dependencies have been identified or refuted. New insights into technical analysis can be obtained by extending the literature through a synthesis of fractal geometry and technical analysis.

Technical trading rules are based on the premise that time series exhibit certain patterns in their past data that can be used to predict future movements. It can therefore be deduced that trending technical trading rules (e.g., filter rule, break-out, and moving average rules) should be more profitable on time series that exhibit long-term dependencies. Conversely, time series that are anti-persistent should not provide returns to trending trading rules, as there are no continuing patterns in the time series to identify and exploit. The H can be used to identify the dynamic processes of a time series. Therefore, based on this synthesis, identifying the dependencies in a time-series’ motion should be able to partially explain the inconsistent and conflicting results evident in the extant body of literature. The synthesis of fractal geometry and technical analysis provides the first hypothesis:

H1: The profitability of trending technical trading rules should be higher with time series that have higher Hurst exponents and lower with time series that have lower Hurst exponents.

As opposed to trending technical trading rules, investors may employ a contrarian trading rule. A contrarian rule attempts to exploit a reversal pattern in a time series and essentially sells into strength (expectation of price decline after an increase) and buys into weakness (expectation of a price increase after a decrease). It can therefore be deduced that contrarian technical trading rules should be more profitable on time series that exhibit anti-persistence. Conversely, time series that are persistent should not provide as profitable results as those brought by contrarian technical trading rules, as there are no continuing reversal patterns in the time series to identify and exploit. This reasoning leads to the second hypothesis:

H2: The profitability of contrarian trading rules should be higher with time series that have lower Hurst exponents and lower with time series that have higher Hurst exponents.

Hypotheses 1 and 2 investigate the relationship between H and the profits from trending and contrarian technical trading rules; however, accepting the hypotheses does not suggest that investors can successfully utilize technical analysis to earn abnormal profits by understanding the fractal nature of a time-series because both H and profits will be calculated on the same dataset. Therefore, an additional hypothesis, with a different test, is required to understand whether traders can successfully employ a trading strategy that uses an observed H to correctly employ a trading rule.

H3: The lagged H (Ht-1) can predict whether a contrarian or trending trading rule will be profitable for a time series.

Testing the first three hypotheses will make a significant contribution, as there are no other known studies that test the relationship between profits from technical analysis and long-term dependencies.

Aside from the main hypotheses, an ancillary hypothesis will be tested. Recall that a fractal characteristic is that it should be statistically self-similar regardless of the time frame over which the increments of the series are observed. There is very little empirical evidence on the effectiveness of the trading rules at various scales (e.g., daily, weekly, monthly). Brownian motion suggests that independent increments are identically normally distributed, whereas pure fractal Brownian motion suggests that a time series is statistically “self-similar” (apart from scale) regardless of the time frame over which the increments of the series are observed. There is a vast amount of literature that discusses the non-normality and lack of Brownian motion of stock returns (Cootner 1964, Fama 1965, Officer 1972). If the stock returns exhibit the characteristics of a fractal time series, rather than Brownian motion, there should be no difference in the effectiveness of technical trading rules with data at different scales. This leads to the development of the fourth hypothesis:

H4: There is no difference between the profitability of technical trading rules on the same data set when calculated for different time scales.  

III. The Data

H and the profits from the trading rules are calculated for all 30 DJIA stocks (as of July 2008) for the 10-year period of July 1998 to July 2008. The time period was selected as it provides enough data to calculate both the H and the technical trading rules and ends just prior to the 2008 credit crisis. As the data set includes many differing events (e.g., Long-Term Capital Management issues and the Asian crisis in the late 90s, dot-com bubble in early 2000s, etc.), sub-period analysis is conducted to determine the sensitivity of the results to the time period selected.

Trading rules can be calculated at various data frequencies. The data frequency selected depends on different factors and preferences. This study utilizes daily and weekly data. Daily data is used because a typical off floor trader will most likely use daily data (Kaastra and Boyd, 1996). Furthermore, intraday time series can be extremely noisy. Along these lines, weekly data is also used as it is readily available to all traders.

The use of raw daily price data in the stock market has many problems, as movements are generally nonstationary (Mehta, 1995), which interferes with the estimation of the H. The market-index series are transformed into rates of return to overcome these problems. Given the price level P1, P2, … Pt, the rate of return at time t is transformed by:

where pt denotes the spot price (stock market indices or the exchange rate). The descriptive statistics for the 30 DJIA components are presented in Table 1.

IV. Methodology

The individual and average profits from 12 trading rules, along with H, are calculated for all 30 stocks. Profitability is defined as the returns from the trading rules less the buy-and-hold strategy returns, adjusted for transaction costs. Therefore, by definition, profits can also be negative.

A.Trading rules

The trending trading rules are the moving-average crossover rule (MACO), the filter rule, and the trading range break-out rule (TRBO). A MACO rule attempts to identify a trend by comparing a short moving average to a long moving average. The MACO generates a buy (sell) signal whenever the short moving average is above (below) the long moving average. This study tests the MACO rule based on the following signals:

where Ri,t is the log return given the short period of S, and Ri,t-1 is the log return over the long period L. The following short-long combinations will be tested: (1, 50), (1, 200) and (5, 150). The same MACO rules were tested in the seminal study by Brock, Lakonishok, and LeBaron (1992).

Filter rules generate signals based on the following logic: buy when the price rises by ƒ percent above the most recent trough and sell when the price falls ƒ percent below its most recent peak. This study tests the filter rule based on three parameters: 1%, 2%, and 5%, which is consistent with many seminal, prior studies (e.g., (e.g., Brock, Lakonishok and LeBaron (1992)).

The TRBO generates a buy signal when the price breaks above the resistance level and a sell signal when the price breaks below the support level. The resistance/support level is defined as the local maximum/minimum. The TRBO rule is examined by calculating the local maximum and minimum based on 50, 150 and 200 days, defined as follows:

where Pt is the stock price at time t. Again, these are the same TRBO rules tested by Brock, Lakonishok, and LeBaron (1992). In order to maintain consistency, all of the contrarian trading rules are defined in the exact same fashion as in Brock, Lakonishok, and LeBaron (1992).

The contrarian trading rule will be represented by a variant of the Bollinger Bands (BB). Bollinger Bands are a tool that can be used in a variety of trading rules, not necessarily contrarian in nature. However, this study will employ a contrarian strategy using Bollinger Bands in an attempt to profit from anti-persistent trends in financial time-series. Bollinger Bands require two parameters: the moving average and the standard deviation. Traditionally a 20-period moving average is used. The BB strategy tested will generate a sell signal when the price of the security exceeds the 20-day moving average plus two standard deviations (i.e., the market is said to be overbought). A buy signal is generated when the price of the security is less than the 20-day moving average minus two standard deviations (i.e., the market is said to be oversold). This strategy is denoted by BB(20, 2). In addition to using the traditional parameters, two variants are tested: BB(30,2) uses a 30-day average to determine whether information from longer time frame can generate more informative signals. BB(20,1) uses the traditional 20-day moving average, but uses only +/-1 to generate signals to determine whether a narrower band can generate more precise signals. These are the same BB parameters tested in prior literature (e.g. Lento, Gradeojevic and Wright, 2007).

The statistical significance of the trading rules is determined through a bootstrapping methodology as developed by Levich and Thomas (1993). The bootstrap approach does not make any assumptions regarding the distribution of the generating function. Rather, the distribution of the generating function is determined empirically through numerical simulations. The data sets of raw closing prices, with the length N + 1, correspond to a set of log price changes of length N. M = N! separate sequences can be arranged from the log price changes with a length of N. Each element of the sequence (m = 1, …, M) will correspond to a unique profit measure (X [m, r]) for each variant trading rule (r for r = 1, … , R.) used in this study. Therefore, a new series can be generated by randomly rearranging the log price changes of the original data set.

This simulation can generate one of the various notional paths that the security could have taken from time t (original level) until time t + n (ending day). The simulation process of randomly mixing the log price returns is repeated 10,000 times for each data set, resulting in 10,000 identically and independently distributed (i.i.d.) representations from the m = 1 , … , M possible sequences. Each technical trading rule (MACO, filter rule, and TRBO) is then applied to each of the random series, and the profits X[m, r] are measured. This process generates an empirical distribution of the profits. The profits from the original data set are then compared to the profits from the randomly generated data sets. If the profits resulting from the original data set are greater than the α-percent threshold level of the empirical distribution, then the null hypothesis will be rejected at the α percent level.

B. The Hurst Exponent (H)

The basis of the rescaled range analysis was laid by Hurst et al. (1965). Mandelbrot and Wallis (1968, 1969a, 1969b, and 1971) examined and further elaborated the method. The following is a brief discussion of the rescaled range analysis and the Hurst exponent calculation[5].

The stochastic process of fractional Brownian motion (fBm) occurs when the second-order moments of the increments scale as follows:

with H [0, 1]. The Brownian motion is then the particular case where H = 0.5. The exponent H is called the Hurst exponent. The H measures dependencies in time series’ non-stochastic motion and is calculated through rescaled range analysis (R/S analysis). For a time series where X = X1, X2, …, Xn, R/S analysis can be calculated by first determining the mean value m, followed by the mean adjusted series Y:

The H can be estimated through ordinary least squares regression. Between Log(N) and Log (R/S). Figure 1 graphically presents the rescaled range analysis that is used to estimate the Hurst exponent on the Dow Jones Industrial Average time series.

The H has been estimated on all thirty Dow components. Table 2 presents the descriptive statistics for the H calculated on all thirty time series.

There are various scholars who rebut the ability of H to identify long-term dependencies by arguing that the rescaled range analysis is skewed. Specifically, issues with the sensitivity of the H to short-term memory, the effects of pre-asymptotic behavior on the significance of the H estimate and the problems with structural changes (e.g., Ambrose et al. (1993), Chueng (1993), Jacobsen (1996)) have been raised. The most significant rebuttal was offered by Lo (1991), who reports that the rescaled range analysis could confuse long-term memory with the effects of short-term memory. However, since Lo’s publication, many economists have reported that his tests were potentially flawed (Mandelbrot, 2004). Additionally, a number of new contributions have suggested alternative techniques of estimating a pathwise version of H to eliminate the lack of reliability in the H (Bianci, 2005 and Carbone et al., 2004). Recently, Qian and Rasheed (2004) concluded that the H provides a measure for predictability, especially for H that are larger than 0.50.

C. Testing the Synthesis

Two empirical tests are conducted to evaluate the synthesis. A classification test will group each DJIA component based on its estimated H. Based on the descriptive statistics in Table 2, three groups have been developed in order to capture differences in the H. Recall that H values below 0.5 suggest anti-dependence. Therefore, one category will be defined as 0.5 >H. Ten of the thirty stocks, or one third of the sample, fall into this category for both the daily and weekly observations. Time series with an H in excess of 0.5 suggest longterm dependencies. Because 20 of the remaining 30 stocks have an H in excess of 0.5, time series with long term dependencies are broken down into the following two components: 0.5 <H< 0.55; and 0.55 <H. A total of 16 (ten) of the time series fall into the 0.5 <H< 0.55 category with the daily (weekly) observations, while the remaining four (ten) time series fall into the 0.55 <H category. Additional sensitivity tests will be conducted on the categories used for the analysis. The synthesis postulates that technical analysis should generate higher profits for group three and lower profits for group one. A quasi-contingency table will present the average profits for each group.

The second test uses OLS regression to estimate the statistical relation between profits from technical analysis and H. To test whether data sets that exhibit higher H result in higher profits from technical analysis, the following equation is estimated:

where Profitsi represents the returns in excess of the buy-and-hold trading strategy for DJIA component i, and Hi represents the Hurst exponent for DJIA component i. The intercept is expected to be positive for Hypothesis 1 and negative for Hypothesis 2.

V. Results

A. Profits from Technical Analysis on the DJIA Components

The profits from the technical trading rules and the H for each DJIA component are presented in Table 3 with daily data and Table 4 with weekly data. Calculated with daily data, H ranges from a low of 0.452 to a high of 0.587. Weekly data result in a wider range of H (0.431 to 0.629).

The trending trading rules were profitable for 7 of the 30 components when calculated with daily data and on 13 of the 30 components with weekly data. It is interesting to note that the MACO and TRBO were much more effective than the filter rules. The filter rules generated an average negative profit of 10.59% (daily) and 22.6% (weekly). The filter rules’ poor performance is consistent with prior studies (Szakmary, Davidson, and Schwarz, 1999; Wong, C., 1997; Nelly and Weller, 1998).

The contrarian trading rules (BBs) generated average profits of 2.19%, 3.37%, and 2.52% for all 30 stocks with daily data. The BBs generated profits on 23 of the 30 stocks with daily data. The results with weekly data are more volatile, with average profits of 15.5%, 15.2%, and 17.0% from the trading rules, with only 21 of the 30 stocks being profitable.

B. Hurst Exponent and Profits from Trending Trading Rules (Hypotheses 1)

Hypothesis 1 postulates that stocks with higher H should yield higher profits from trending trading rules. To test this relation, each DJIA component was grouped according to its H value. The first group includes stocks with values of H less than 0.5. The second group includes stocks with a value of H that is greater than 0.5 but less than 0.55. The final group is for stocks with a H greater than 0.55.

Table 5 presents the results of the classification analysis in the form of a contingency table. Panel A presents the results for the combined weekly and daily data, while Panel B and Panel C present the results for daily and weekly data, respectively.

The results provide strong evidence that profits from trending trading rules are partially explained by the longterm dependencies, as identified by the H estimation. All three panels reveal that profits are lowest for stocks with values of H that are less than 0.5 and increase in H. The trading rules earned returns of 11.5% in excess of the buy-and-hold strategy for stocks with a value of H greater than 0.55, while technical analysis underperformed by 16.7% for stocks with values of H less than 0.5.

The robustness of the results is tested though sub-period analysis. The H is estimated over three sub-periods (n.b.: the sub-periods divide the data set into three equal periods) and the profits from the technical trading rules are also calculated on the same sub-periods (untabulated). The results of the sub-period analysis for the association between H and the profits generated by technical analysis are presented in Table 6. The sub-period analysis confirms the robustness of the results in Table 5, as profits are higher for stocks that exhibit long-term dependencies as identified by the H. Sub-periods 1 and 3 exhibit consistent patterns of increasing returns in conjunction with increasing H; however, the weekly data on sub-period 2 do not reflect the synthesis.

These results are consistent with the synthesis and tend to corroborate the proposition that a value of H less than 0.5 exhibits anti-persistent trends that limit trending trading rules’ ability to identify patterns, whereas time series with a value of H greater than 0.55 exhibit persistent trends that are identified by the trending trading rules to earn profits. However, the results partially explain, as opposed to completely explaining, the profits from the trading rules because there are anomalies. For example, Pfizer’s weekly time series exhibited an anti-persistent nature (H of 0.431), yet technical analysis was able to earn an average profit of 14.2%.

In addition to the sub-period analysis, additional sensitivity testing is conducted on the categories selected for the analysis. The tables have been recalculated with only two categories: 0.5 >H and 0.5 <H. The results (not presented) are the same as the main results in Table 5. In addition, the tables have been recalculated with the following three categories: (1) 0.5 >H; (2)0.5 <H< 0.54; and (3) 0.54<H. This categorization results in 10 observations, or one-third, in each category. Again, the results (not presented) are the same as the main results in Table 5. The results also hold with the following classification: 1) 0.5 >H; (2)0.5 <H< 0.52; and (3) 0.52<H. Accordingly, the results are robust to the boundaries selected for the categories. Note that there are not enough observations in excess of H> 0.60 in order to create such a category. 

Table 5 and Table 6 provide evidence to support Hypothesis 1; however, the classification system does not offer the precision of statistical rigor. Therefore, additional tests of the relationship between the H and profits from technical analysis are provided through regression analysis. The results of the regression of Equation 12 are presented in Table 7. Panel A presents the estimation using average returns for all three trading rules as a proxy for Profiti, while Panel B presents the results of the estimation using the average of each trading rule (MACO, filter, and TRBO) as a proxy for Profiti.

The results are consistent with and corroborate the results presented in Tables 4 and 6, and further support the synthesis in Hypothesis 1: H is able to identify long-term dependencies in time series data that are exploited by technical trading rules to generate profits. The estimation has an R2 of 31%, with virtually all of the explanatory power resulting from the H variable. Panel B presents additional insight, revealing the resulting R2 of 37% when using the MACO and TRBO proxies for profits. The estimation with the filter rule as a proxy for profits does not yield strong results. This is a function of the aforementioned lack of profitability for filter rules. The sub-period analysis conducted in Section 5.3 provides further data to test the robustness of the estimation in Equation 12. The results of the estimation with the sub-period data are presented in Table 8.

The sub-period estimation with the daily data results in an R2 of 0.334 to corroborate the estimation results of Equation 12. The estimation results with weekly data reveal a weaker relationship (R2 of 0.084). Overall, the empirical evidence provides strong support for the acceptance of Hypothesis 1, as trending trading rules are more profitable with stock with higher long-term dependencies.

C. Hurst Exponent and Profits from Contrarian Trading Rules (Hypotheses 2)

Hypotheses 2 postulates that stocks with lower values of H should yield higher profits from contrarian trading rules. To test this relation between H and profits from the contrarian trading rules, each DJIA component was grouped according to its H value. The groupings are the same as in the test of Hypothesis 1. Table 9 presents the results of the classification analysis for the contrarian trading rules. Panel A presents the results for the combined weekly and daily data, while Panel B and Panel C present the results for daily and weekly data, respectively.

The results provide evidence that profits from contrarian trading rules are partially explained by the long-term anti-dependencies in a time series. Panel A reveals that profits are highest for stocks with values of H that are less than 0.5 and decrease in H. Contrarian trading rules were able to earn returns of 11.3% in excess of the buy-andhold trading strategy for stocks with values of H less than 0.5. The results from Panel B (daily data) and Panel C (weekly data) are not as consistent.

Again, regression analysis between H and profits from contrarian rules is conducted. The results of the regression of Equation 12 are presented in Table 10. The results are consistent with Table 9, as the regression provides some evidence of the synthesis in Hypothesis 2. The estimation produces a low R2 of 5%. However, the intercept for the H variable is negative and significant at the 10% level.

Overall, the empirical evidence provide some support for the acceptance of Hypothesis 2, as contrarian trading rules appear to be more profitable on stocks with lower values of H.

D. Lagged H and the Profits from Technical Analysis (Hypothesis 3)

Hypothesis 3 was proposed to determine whether investors can use the H in one period to predict which stocks will provide the most profit from technical analysis in the following period. A test is conducted through OLS regression between the profits from the trending trading rules in sub-periods 2 and 3 and the lagged H (sub-periods 1 and 2, respectively). For example, can Caterpillar’s H of 0.587 in the first sub-period be used to forecast profits through a trending trading rule for Caterpillar in the second period? If so, then investors can use information about a time-series dependence to earn profit. The results of the regression estimation are presented in Table 11 (Panel A).

The estimation suggests that the lagged H is not able to forecast future profits. The results are influenced by the H instability across sub-periods. This has important implications for investors. If H is changing for a time series, investors will be required to make subjective decisions and forecasts to develop the expectation of time series’ long-term dependencies to earn profits from technical analysis. Corazza and Malliaris (2002) also note that H is not fixed but changes dynamically over time.

To mitigate the impacts that an unstable H can have on forecasting profits in future periods, an additional exploratory test is proposed. The same OLS regression proposed regarding the lagged H was conducted with the daily and weekly time series that exhibit the lowest standard deviation of H across the sub-period. The low standard deviation is used as a proxy for a stable H. The results are presented in Table 11 – Panel B.The estimation results with both weekly and daily data are consistent with Table 11 – Panel A, rejecting Hypothesis 3 by suggesting that the lagged H is not able to forecast future profits on a time series. However, there does appear to be some predictive ability for traders, as the estimation that utilizes only the daily data result in an R2 of 0.213, suggesting that there may be profit potential with daily data.

E. Profits from Technical Analysis on Different Scales (Hypothesis 4)

Table 12 presents comparative summary statistics between profits from the trading rules calculated at different scales (daily and weekly). The empirical evidence reveals that technical trading rules result in different profit levels when calculated at different scales and therefore rejects the Hypothesis 4. Overall, the average excess return for all trading rules is negative 2.58%. Moving averages result in negative 2.24% profits, while filter rules, Bollinger Bands and trading range break-out rules result in profit levels of negative 10.59%, 2.7% and negative 0.22%, respectively. Calculating the trading signals with weekly data results in an average profit for all trading rules of 0.25%. Moving averages result in negative 3.10% profits, while filter rules, Bollinger Bands and trading range break-out rules result in profit levels of negative 22.6%, positive 15.9% and positive 10.8%, respectively.

Table 12 reveals that the trading rules are more profitable when calculated with weekly data, as 201 of the 360 variants (30 stocks x 4 trading rules x 3 variants of each rule) are profitable, or 55.8%, while only 161, or 44.7%, variants are profitable with daily data. As discussed earlier, all stocks that result in profits from technical analysis when calculated with daily data are also profitable with weekly data. However, the weekly data result in much more variation in the profits, made evident by wider ranges and higher standard deviations.

VI. Conclusion

Currently, much research is being conducted on capital markets and technical analysis. Research on the application of fractal geometry to capital markets has been much more limited. More specifically, there are no known studies that investigate the relationship between technical analysis and fractal geometry. The extant literature solely investigates the ability of H to identify predictability in the financial market. This paper makes a significant contribution by extending the literature to determine whether technical analysis is able to generate abnormal returns (after accounting for transaction costs) on time series that exhibit long-term dependencies or anti-dependence.

Two tests are conducted to evaluate the relationship between H and profits from technical analysis. Both tests provide evidence that H is able to identify long-term dependencies and anti-dependence that result in higher (lower) profits from trending (contrarian) trading rules. Therefore, profits from trending (contrarian) trading rules are higher for time series that exhibit long-term dependence (anti-dependence). This is consistent with the main postulate of the synthesis (Hypotheses 1 and 2).

There are some limitations that must be addressed. First, the substantial data requirements skew the sample towards the larger Dow Jones component firms. Therefore, the results may not necessarily be generalizable to a broader population that includes smaller firms. Future research could focus on testing a more diverse sample of firms (e.g., all 500 stocks of the S&P 500) and for a longer period of time. Second, this study implicitly assumesthat the Hurst exponent is constant for each time series over the time period analyzed.It is possible that the H may vary within sub-periods of the data set. However, the results from Hypothesis 3 provide indirect support that this assumption does not invalidate the results as the H in one period can be lagged to predict future profits from technical trading rules. Thirdly, the ten year period analyzed may not be sufficiently long enough to allow for the emergence of the long-term dependencies to emerge. In addition, the ten year period covers a trading range market versus a secular uptrending market. Accordingly, future research could focus on testing long time periods which include secular bull and bear markets.

There are a many future research opportunities to further develop the synthesis between fractal geometry and technical analysis. The first priority is to further test the relationship between profits and H with more robust data, likely making use of global equity markets and various firm sizes. Utilizing time series from the global equity market will provide future researchers with a larger range of H in their sample and also provide access to a larger pool of securities. Researchers should also seek to understand what causes anomalies in the synthesis. For example, Pfizer’s weekly time series exhibited an anti-persistent nature (H of 0.431), but technical analysis was able to earn an average profit level of 14.2%. Finally, and most importantly for investors, researchers should investigate how H in one sub-period can be used, ex ante, to identify which time series will yield the most fruitful results from technical analysis. This paper has made a contribution in this area by suggesting that the sub-periods with daily data are able to predict future sub-period profits from technical trading rules. Researchers are encouraged to continue research in this area to provide a significant impact to traders and investors.

Footnotes

Acknowledgements: The author would like to thank the anonymous reviewers for their comments that helped improve this manuscript.

1. Brownian motion is a mathematical concept used to describe a random process.
2. Brownian motion is a well-known paradigm in finance that can be described as a white-noise process for which the independent increments are identically normally distributed. A white-noise process is the statistical paradigm against which the sequence of increments from a chaotic dynamical process is typically contrasted. White noise traditionally refers to a sequence whose increments are independently and identically distributed with zero mean and finite variance. Brownian motion underlies most of modern finance theory’s most important contributions.
3. Mandelbrot (2004) provides a detailed history and discussion of the Hurst exponent, the Joseph effect, and Noah effect.
4. Additionally, the power spectrum of the increments of fractional Brownian motion is proportional to fβ, where β= 2H-1, so that Brownian motion as a white-noise paradigm has a flat spectrum (Feder, 1988).

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