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University of Amsterdam
Finance Faculty
Master Thesis
Varying market efficiency
through scaling long memory
Author: Dylan John van Haaften 5981050 Supervisor: Dr. Liang Zou
Abstract
We investigate whether markets possess long range dependence. We employ the somewhat forgotten but tried and tested rescaled range analysis which is adjusted for possible biases
such as small sample bias. We also aim to research and estimate historical levels of long memory and whether these have varied over our horizon of 1951 until 2015 for the S&P 500,
and 2000 until 2015 for the MSCI World, EM and Sector Indices. We also look at market crashes in regards to the long memory quality and sectors and whether this would be more consistent to be linked to behavioral theory through the fractal market hypothesis. We believe
our study shows modest long memory characteristics that point to past market inefficiencies, moreover our analysis on market crashes appears to show long memory does indeed decrease after large stock swings or more endemic changes such as the 2008 global financial crisis.
Table of Contents
Abstract ... 3 Introduction ... 4 1. Literature review ... 5 Summary ... 19 2. Empirical Methodology ... 20 3. Results ... 21 4. Conclusion ... 29 References ... 31 Appendices ... 36Introduction
Market efficiency remains a heavily contested subject. In the efficient market hypothesis (EMH) asset prices reflect all available information, and new information cannot be predicted (Fama, 1970, Samuelson 1965) and therefore follows a random walk. An area of criticism is that there are discernible trends and patterns in stock markets over time and time scales. This anomaly of trend persistence is referred to as long range dependence or long memory. If long memory is present this implies strong market sign predictability, and markets are inefficient. In weak form market efficiency, analysis of past prices and information may not hold
abnormal returns (Fama, 1970). However price return long memory is not easily exploitable, as other factors such as transaction size and liquidity offset long memory making the market more efficient (Lillo and Farmer, 2004). Nevertheless the long memory property in time series may provide insights into market efficiency and behavioral market dynamics.
Our approach differs from existing literature in our sample size of daily returns, application of R/S methodology to sector indices and tying the long memory phenomenon to existing
behavioral theory within the frame of market crashes. The key authors for this thesis, Peters (1991), Lo (1991) and Kristoufek (2012) applied rescaled range analysis to financial time series with varying results. Peters showed that across assets and time frames long memory was present, however Lo believed this long memory was in fact short memory that was misallocated as long memory. Kristoufek took it a step further and indicated that markets should let go of the EMH on basis of market long memory, and embrace the Fractal Market Hypothesis (Peters, 1994). All main authors came to the same conclusion namely that financial markets are relatively inefficient on basis of either persistent short or long interdependence. Moreover Lo, Peters and Kristoufek agree that rescaled range analysis is reasonable measure for long memory properties.
We include sector indices in our determination of long memory because though study has been done in indices and individual stocks, sector benchmarks have been largely omitted from existing literature. Moreover market collapses have we believe become more sector driven idiosyncrasies, as demonstrated by the 2000 dot-com bubble , the US housing crisis in 2007 and the global financial crisis of 2008. We believe that as the last study was performed on sectors in 1996 (Barkoulas and Baum) perhaps new insights can be obtained in light of more recent crashes. We will also apply a rolling window rescaled range analysis to show whether long memory properties have changed in stock market following market crashes. And finally
we believe it is plausible based on behavioral economic literature that behavioral biases are reflected in the long memory statistic through the fractal market hypothesis.
We aim to answer (i) whether long memory pervades (sector) index returns consistently, (ii) whether equity markets collapses affect the long memory characteristic of markets and (iii) whether this conforms to market efficiency. We will structure answering these questions as follows, first by citing relevant literature and weighing approaches, second by laying out our methodology, followed by a discussion of results and conclusion. We expect long memory is consistently present in financial returns on the basis that we do not believe market prices follow a random walk due to behavioral biases and heterogeneous investors in the market.
1. Literature review
In order to accurately assess the properties of long memory we will perform a full literature review on long memory in financial markets. We will first discuss the definition of long memory; we will then discuss a methodology for deriving long memory through rescaled range analysis, after which we will link long memory to market efficiency and behavioral theory and discuss previous literature empirical conclusions. We will end with a formal summary of our conclusions based on literature. We focus on six publications in particular Lo (1991), Peters (1991,1994), Mandelbrot (1972), Kristoufek(2012) and Taqqu et al. (1999).
Long memory definitions
Definition 1.1
A stationary process
has the long memory property, if for its
autocorrelation function:
(1)
Equation (2) holds:
(
2)If the process satisfies this, the autocorrelations decay to zero so slowly their sum does not converge (Beran, 1994) implying infinite interdependence. To connect definition 1.1 to long memory processes we introduce self similarity.
Definition 1.2 A stochastic process
is self similar with self similarity
parameter
, if for any positive stretching factor , the rescaled process
has
the same distribution as the original process
. If the increments
are stationary the autocorrelation function is given by:
(3)
Beran (1994) describes that using a Taylor expansion of :
(4)
And as spectral density is of approximately with a constant , if then the autocorrelation function is with , implying the
long
memory property (Beran, 1994).
Long memory would translate to Figure 1 so as 1 lag relates to all future lags, causing the autocorrelation function to never hit zero, hence we speak of asymptotic decay.
Figure 1
Autocorrelation lag plot for S&P 500 for 2008-2015
Long memory estimation
The long memory property was first discovered by hydrologist Harold Edwin Hurst in 1951, as he discovered that water level fluctuations that were assumed random, in fact had some correlation and displayed a trend, which signified long range statistical dependence. During the Aswan dam prelude he described these fluctuations through the R/S statistic; which is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. Regressing log(time) on log(R/S) and a constant allowed Hurst to determine whether a time series demonstrated persistence, anti-persistence or a random walk. The beta of this regression is therefore still referred to as the Hurst exponent. And though Hurst’s statistic did not help in determining the eventual height of the dam, future applications of his analysis would range from Physics to Geography to Bioscience.
Benoit Mandelbrot’s research on fractal and self-similar processes in nature, and the ability for the Hurst exponent to be used as a fractal dimension estimator prompted Mandelbrot to publish a series of papers on long memory through Hurst’s methodology. The first paper
"Noah, Joseph, and operational hydrology"(Mandelbrot and Wallis; 1968) built a bridge between the “Noah effect”1
and the “Joseph effect”2 which they noted was often diagnosed ex-post, and therefore often missed in predictive modeling. The bridge to finance would be made by Mandelbrot (1963) as he had already published a paper on that he believed markets did not follow Bachelier’s (1900) random walk. In 1972 Mandelbrot was the first author to apply Hurst’s R/S analysis to financial time series concluding that financial asset time series were non-Gaussian, but more fat-tailed conforming to a Lévy stable or Paretian distribution.
Rescaled Range Analysis
To estimate whether long memory is present in financial market the most often used method is rescaled range analysis, also called range over variance. Other methods such as the
Absolute Value method, Wavelet transform method, Spectral Regression method or
Detrended Fluctuation Analysis are often; similar to the R/S method derived from other exact fields that model self similarity and statistical interdependence. Other methods have not been researched to the same extent as R/S analysis when judged by the number of publications. We prefer the R/S method because (1) it is the most broadly used model, (2) has been researched
1 Time series tend to have abrupt and discontinuous changes, referencing the great flood in the Old Testament. 2 Finite time series tend to be part of larger trends (cycles) more often than being random, referencing Egypt
experiencing 7 years of feast after 7 years of famine. Cycles on aggregate last 4,5, and 10 years (Peters, 1996; Nawrocki, 2008)
extensively, and (3) most biases and estimation issues regarding R/S analysis have been discovered. Moreover R/S analysis provides an attractive graphical way to estimate long memory properties and is intuitive in its applications and interpretation.
To estimate the Hurst exponent through the rescaled range method one has to follow a few steps discussed ahead, the picture below3 (Figure 2) shows an example of rescaled range analysis. We will use a similar approach as used by Qian and Khaled (2004), Peters (1991) and Weron (2002), Kristoufek (2012) and Lo (1991).
Figure 2
Graphical representation of RS analysis in regards to water level and dam height
Source: Hurst 1951, Liv and Jens Feder, 1990
Calculating R/S
For a time series :
1. Convert stock price into log returns
)
(5)2. Calculate mean value m:
(6)
3. Calculate the mean adjusted series Y
(7)
4. Calculate the cumulate deviate series Z
4
(8)
3
We use a different notation for the variables to Figure 2
5. Calculate range series R
(9) For
6. Calculate the standard deviation series S
(10)
For
7. Calculate the Rescaled range statistic
(11)
8. As R/S scales as a power law as time increases indicating:
(12)
The power law relation (Weron, 2002) between time and R/S is not necessarily helped by calculating R/S in each iteration of t, The method Hurst, Mandelbrot, Peters and Kriftoufek use and which is most commonly done is to create logarithmically spaced sub vectors (d)5 which satisfies a) that the number of sub vectors is at least 5, setting , and b)
(Rose, 1991, Kristoufek, 2012). Rescaled range is first calculated for the whole
sample (Figure 3; RS(0)), after which the sample is split in two and the two rescaled ranges are averaged, until eventually d6 is reached. In this thesis we will use a default value of estimation of 50 individual d parameters which is also used by Peters (1991) and Weron (2002) as a baseline.
5
For instance 2, 4, 8, 16, 32…1024 (Qian and Rasheed, 2004)
Figure 3
The interplay of ranges and sub space vectors d
Source: Kristoufek, 2012
9. To attain the Hurst exponent or the coefficient of the logarithm of Rescaled ranges of variance for d7 values. Perform an OLS regression of time on (R/S) values (Taqqu et al, 19995, Di Matteo 2007):
(13)
10. Make a pox plot of OLS regression as in Figure 4 on Log-log axes, to demonstrate power law scaling.
Figure 4
Pox plot of R/S statistics versus time and corresponding OLS regression to determine Hurst for the MSCI Healthcare Index
Source: Bloomberg, Matlab
7 d is the n of regression if no results are cut.
} R/S average 0 } R/S average 1 } R/S average 2 RS(0) RS(1) RS(2) RS(3) RS(4) RS(5) RS(6) RS(7) } R/S average 3 RS(0) RS(0) RS(1) RS(0) RS(1) RS(2) RS(3) 0,044+0,5T 1 10 100 1 10 100 1000 R /S t
The obtained Hurst exponent can then be interpreted as indicating the underlying time series is persistent or trend reinforcing. At the underlying time series is mean reverting; or anti-persistent. When the Hurst exponent is equal to there is neither persistence nor anti-persistence, and sign expectation is zero, which is consistent with a Brownian motion or random walk, as increments are then independent as in Definition 1.3. When , the time series is mean reverting which becomes stronger as H approaches zero. The Hurst exponent can take any value in the domain .
Figure 5
A persistent series, the S&P 500 E(H>0.5) An anti-persistent series, the VIX, E(H<0.5)
Source: CBOE, Matlab
Mean reversion, Trend persistence and the random walk
Essentially the Hurst exponent can take two directions that correspond to either mean reversion or trend persistence (Figure 5). Mean reversion indicates a move away from the mean over time is likely to be followed by a move towards the mean. A trend persistent or reinforcing series, however, is likely to follow up a sign move with a similar sign move. This self-similarity describing property is why R/S analysis and the Hurst exponent are associated with Fractals and Benoit Mandelbrot; who believed many processed that were deemed natural or random followed self-similar underlying processes. The link between R/S analysis and the random walk or Brownian motion is that a random walk is neither persistent nor anti
persistent and is likely to converge over t to infinity, there may be persistence or anti-persistence in parts of the Brownian motion; but this would be a coincidence.
0 500 1000 1500 2000 2500 1990 1995 2000 2005 2010 2015 0 10 20 30 40 50 60 70 80 90 1990 1995 2000 2005 2010 2015
Definition 1.3
A stochastic process is a standard Brownian motion if: 11. =012. It has continuous sample paths
13. It has independent, normally distributed increments
If that would imply the analyzed process does not have independent increments. Moreover before Bachelier, Fama and Samuelson, a French stockbroker named Jules Regnault (1863) noted that the price deviation of a stock is equal to the square root of time, moreover Regnault verbally commented that investors should have no directional expectation, providing the base for Bachelier’s “Theorie de la speculation” (1900) which would form the basis for the Efficient Market Hypothesis (EMH), which we will discuss ahead. A Hurst exponent that is neither persistent nor anti persistent, allows a random walk and an efficient market.
Figure 6
Relative peakedness (leptokurtosis) of market returns
Source: Fama 1965
If prices show sign interdependence such as when there is an underlying long memory process, volatility can drift away from Brownian motion, as autocorrelation; if persistent, is not independent. Therefore if there is long memory in markets this could imply markets have a more leptokurtic distribution than a random walk such as the normal distribution would imply (Figure 6).
Notes on R/S analysis
Because R/S analysis is the most frequently used method to obtain the Hurst exponent and determine Long memory properties, there has also been discussion regarding its accuracy as an estimator. There are three known issues with the Hurst exponent estimation through R/S namely: (i) potential short range dependence that is misinterpreted as long range dependence, (DiMatteo, 2007; Lo and Mackinlay, 1999; Alfi et al, 2008), (ii) the R/S method is known to overestimate the long memory factor even at large samples (Lo, 1991; Taqqu et al.), and (iii) R/S analysis is inaccurate at low samples (Peters, 1991; Anis and Lloyd, 1951). Offsetting these factors is that R/S estimation is robust against changes in the marginal distribution, even for long-tailed or skewed distribution (Rose, 1996).
We believe (i) and (iii) are not applicable to this thesis as R/S as we use a large sample and we elect to use R/S over Lo’s modified R/S as Taqqu et al. (1999) showed Lo’s R/S method suffered from (i) a negative long memory bias, (ii) difficult to pick short range dependence exclusion element q, and (iii) is susceptive to large swings in predicted values for a given q. However to deal with (ii) we employ the expected rescaled range statistic, and calculate:
(14)
For given t values in our sample (Kristoufek, 2012). To determine the E(R/S) for a process with an assumed underlying Brownian motion (H=0.5) Feller (1951) suggested:
(15)
However this is only valid for infinite samples (Qian and Rasheed, 2004; Kristoufek, 2012), and as samples in finance are finite Fellers E(R/S) underestimates the small sample bias. Peters (1991) suggested expansion (16), which creates a higher expected Hurst coefficient at lower samples as is visible in Figure 7.
(16)
Where r is the t-1 lag operator in Peters expected rescaled range statistic. Equation (15) has become the benchmark for testing long memory significance in a finite setting. But is in our view slightly heuristic in the absence of formal confidence and await scientific review of Kristoufek’s (2012) method of confidence interval and variance estimation for a future alternative.
Figure 7
Application of Peters E(R/S) versus time for H=0.5
Source: Peters(1991), Excel, Matlab
We expect we will not have too many issues with overestimation of the Hurst exponent through R/S, as we have a large sample in all our datasets and employ Peters E(R/S) statistic to adjust the rolling R/S statistic. Moreover as our sample will be greater than 300; which is the bottom end for reliable R/S estimation without adjustments, Feller and Peter E(R/S) estimates will converge towards for large samples.
Empirical results of Hurst estimation in literature
Though R/S analysis is by no means popular in finance there are numerous sources in a variety of conditions and data sets that confirm long memory. Apart from Mandelbrot empirical evidence for long memory in stock price returns was given by Greene and Fielitz (1977). Based on RS analysis (Hurst, 1951; Mandelbrot 1972, Mandelbrot 1975), they concluded that if statistical dependence is not arbitraged away then markets may be inefficient. Aydogan and Booth (1988) examine the same US stock returns and find no evidence of long memory, and signal that Greene and Fielitz may have misinterpreted the long memory they found as they found evidence of quickly decaying short memory. Lo (1991) explains the presence of long memory often found in markets as miss-specified quickly decaying short correlations and suggest a modified R/S statistic that accounts for a certain short correlation by removing covariance between and over q lags. Peters (1994) however argues that adjusting for short correlation along with the possibility of overshooting correction mutes actual long memory effects as measured by Lo’s methodology. Lo and MacKinlay (1988, 1990) built upon the idea that long range statistical dependence was caused
0,1 1 10 100 1000 1 10 100 1000 10000 100000 E (R /S ) t
by serial short correlation further as to provide a solid base for rejection of a random walk process underlying stock prices. In 1999 Taqqu, Teverovski and Willinger argue to Peters point and find that Lo’s methodology is biased negatively in regards to determining long memory, as the adjustment factor is unknown and cannot be determined precisely. As Kristoufek (2012) has continued Peters work as far as Rescaled Range analysis, we believe consensus literature points to Rescaled Range analysis as an acceptable method for measuring long memory processes.
Most other studies looked at either local markets or currency indices and broader studies of multiple indices of countries have only been done by a few authors. Henry (2002) found long memory in German, Japanese, South Korean and Taiwanese markets out of 9 markets tested. Chen (2000) found that long memory pervaded all APAC index returns. Contrastingly Chow, Pan and Sakano (1996) find no evidence of long memory across 22 indices. We conclude rescaled range analysis results in developing markets (Wright 2001) and smaller markets (Tolvi, 2003; Limam, 2003) appear to have consistent long memory. However currency markets (Baum, Barkoulas and Caglyan; 1999, Barkoulas and Baum; 1997 and 1996,
Embrechts, Cader and Deboeck, 1994) appear to have no such long memory. A Sector index analysis as we aim to do of long memory has only been published by Barkoulas and Baum (1996) on S&P sector indices and their results were not supportive of long memory in stock market returns. However they did find individual stock’s that showed evidence of long memory. Using spectral regression they found that there was no consistency in long memory properties. Further differentiating factors between research papers were returns intervals; where most estimates used daily returns. However Lillo and Farmer (2004) in their research to offsetting (trading and liquidity) factors for long memory used tick by tick data to find long memory persistence.
A meta-analysis on available literature on long memory in returns shows that twenty-three sources confirm long memory in financial markets (Mandelbrot; 1972, Greene and Fielitz; 1977, Ding, Granger and Engle; 2001, Taqqu, Teverovski and Willinger; 1995 and 1999, Peters; 1991 and others) in the form of a Hurst exponent above . Seventeen sources deny long memory is present in financial markets most notably Lo (1991). Five studies show
inconclusive results and two studies show mean reversion (e.g. H< ). We therefore conclude on the basis of the largest body of evidence pointing towards long memory, as well as the
rebuttal of Taqqu et a.l on Lo and Aydogan et al. regarding short term memory in long term memory that literature is supportive of long range dependence in financial markets.
Application of R/S in a finite setting
In general, publications (Mandelbrot, 1972; Ding, Granger and Engle, 2001; Lo, 1991) have been aimed at proving markets do or do-not possess a long memory characteristic ad
infinitum, and aim to prove markets are inefficient by definition. Proving a long memory characteristic of a process that is natural, such as in economics is difficult as there are finite bounds, in our case the existing sample. In this setting long memory must be used over different time intervals and cycles. Peters (1991) argues R/S analysis is very apt in this regard as it can consider distributions with infinite variance, as opposed to other methods such as spectral regression. Peters also indicates cycles over historical time series are often 4, 5 or 10 years corresponding to T values of at least 1000 trading days to obtain an accurate estimate. However Mandelbrot (1972) notes that even if the length cycle exceeds T, the R/S method is accurate at picking up long memory signals and trend behavior. Peters (1996) noted small samples may display Markovian8 dependence in the R/S statistic. Moreover Mandelbrot and Wallis (1969) noted that observations far removed from each other can be considered
independent, and that R/S analysis will asymptotically approach a random walk process Hurst exponent for growing samples. We conclude from this that the rolling Hurst exponent over a large T is an accurate measure if small samples are ignored, but that this estimation technique may display relatively low long memory towards the present as N becomes larger.
We believe that markets do not necessarily need to have a long memory characteristic all the time, as factors such as memory decay (Barberis et al, 2001), irrational exuberance (Schiller, 2000) and investment horizons (Peters, 1994) can vary over cycles. Moreover as Lo (1991) comments there are cycles and scale effects that over what we perceive as a long time frame imply persistence, momentum or positive interdependence, which in even a setting that is not infinity can be insignificant and incorrect.
8 Markovian dependence, is not a long memory property as it decays and does not persist, however this relatively
short not asymptotic decay of the autocorrelation function is according to Lo(1991) and Aydogan and Booth(1988) a key reason for (false) past inferences of long memory in stock market returns.
Long memory, inefficient markets and market collapses
The debate on whether the EMH holds remains a hot topic, particularly in the field of long memory. Mandelbrot (1963) already hypothesized Bachelier’s (1900) random walk
assumption regarding stock prices was incorrect on the basis of historical data. In spite of his publication Fama (1965, 1970) and Samuelson (1965) formed the EMH, which is widely accepted as a cornerstone of modern financial economics. The crux of the theory (weak-form efficiency) is that prices reflect all available information, which implies historical data analysis cannot consistently generate returns above market returns long term. Contra arguments come from a range of angles such as that markets are partially predictable (Malkiel, 2003), there is serial short range dependence (Lo and MacKinlay, 1999), there is long memory in markets (Mandelbrot 1972 ,Peters 1991), investors can display valuation biases (Schiller, 2000; Kahneman and Tversky, 1979), and the incidence of market crashes (Stanley, 2003). The link between predictive power of the Hurst exponent, long memory, behavioral theory and market crashes (global financial crisis 2007-2012)was first made by Kristoufek (2012), he found that during markets crashes investor horizons shrunk (through market liquidity9 and trading size effects. which shifted investors from long memory to shorter memory, and could propel markets into general mean reversion during a crisis. Moreover Kristoufek found that markets as a result of market collapses can be mean reverting, trending or efficient, and within the scope of currently available data, analysis of the long memory property for markets holds some predictive power10.
Further evidence to long memory regarding behavioral biases that may contribute to long memory are the long-shot bias of Thaler and Ziemba (1988), prospect theory by Kahneman and Tversky (1978), and investor memory half life in regards to index sluggishness by
Barberis, Huang and Santos11 (2001). Behavioral biases can both act as an encouraging factor such as in Shiller’s “Irrational Exuberance” paper (2000), where valuations created an investor bandwagon. This “bandwagon” effect has been linked to serial autocorrelation and momentum, as earnings surprises lead abnormal returns in financial assets (Ball and Brown, 1968; Rendleman, Jones and Latané, 1982). However there is also offsetting a possibility of
9 This links back to Peters Fractal Market Hypothesis, which is an alternative generalization of the EMH, but
discussing it would be beyond the scope of this thesis.
10 Predictive power in a controlled ex-post allocated setting.
11 Barberis, Huang and Santos (2001) find that in application of prospect theory to asset prices there is a
sluggishness factor in the speed at which the benchmark levels adjust equal to 11 which can take values from
zero (not-sluggish) to one (very sluggish). If benchmarks move slowly as Shiller described in his book
mean reversion (Poterba and Summers, 1988; Fama and French, 1988) at longer holding periods. In all there is a strong behavioral element to long memory as we believe the heterogeneity in behavioral actors; versus homogenous EMH actors; i.e. all actors are expected to have the ability to perform behavioral, fundamental and technical analysis,
reflects the work of Peters (1994) and Kristoufek(2012) regarding a fractal market hypothesis.
The Fractal Market Hypothesis and Long Memory
The Fractal Market Hypothesis (FMH, Peters, 1994)) is a result of Peters (1991) criticism of the EMH and is an expansion on EMH conditions. The FMH is based upon the idea that investors are heterogeneous in investment horizons, which causes them to interpret information differently causing heterogeneous decision making. The existence of
heterogeneous investors guarantees that markets operate smoothly in stable markets as given prices and preference enough investors of each investment horizon participate in markets so demand and supply are smoothly cleared. However in a crisis, horizons clash, which causes markets to not clear so efficiently. One such example is when market correct suddenly longer term (fundamental) investors stay in the market as their horizon extends past for instance an economic cycle, however day traders and speculators often reliant on technical analysis and sentiment, flee the market reducing liquidity and market efficiency. For a FMH the underlying process must be fractal as well.
Definition 1.4
A process is considered multifractal if it has stationary increments which scale:(17)
For Integer and all q (Calvet and Fisher, 2008). Hurst is then related to asymptotically hyperbolic decaying autocorrelation function:
( 18)
For the Hurst values the same conclusions follow as in the long memory definition
(Kristoufek, 2012). The FMH is effectively an EMH with allowed autocorrelation as market efficiency and randomness is still underlying, but liquidity in crashes is a disruptive for market efficiency. We believe the FMH implications are attractive if our sample shows long memory and markets in crisis correct as hypothesized. This was proven by Kristoufek(2012) for the global financial crisis.
Summary
We believe long memory in literature points towards long memory consistently being present in markets, be it at a relatively low Hurst exponent of 0.6 on aggregate. We believe the relatively low measure of long memory found in markets versus the relatively high historical levels may point to discovery of a way to arbitrage long memory; however we have not found a source to corroborate this claim. If there is long memory in markets as defined by this thesis, over the sample period markets are not efficient12. Though the Hurst exponent holds some predictive power, it is estimated over a historical period, and market inefficiency is concluded after the fact. This is why we are not aware of any parties that have generated abnormal returns from employing some Hurst exponent estimation, which is the crux to disproving market efficiency in our view. Long memory does appear similar to momentum strategies abnormal returns (Jegadeesh and Titman, 2011). But in the absence of an investment strategy that arbitrages away long memory and the fact long memory has persevered since it has been discovered in the 1970’s by Mandelbrot, we cannot rule out that the EMH still holds and long memory is an anomaly.
This thesis deviates and adds to previously existing literature on a number of aspects. We apply R/S analysis to (sector) indices, research the rolling window Hurst exponent in a
situation of market collapse market. We base our market crash study on the graphical analysis approach of Kristoufek’s (2012) rolling window Hurst exponent, but apply the R/S method as opposed to the Detrended Fluctuation Analysis (DFA) used in the paper, moreover we double the time window (T=500 in Kristoufek, to T=1000) as we have more observations than in 2013, and extend the historical time frame. Moreover Mandelbrot indicated in his book “The (Mis) Behavior of Markets” (2004) that a lot of issues arise from “picking” samples which is why we aim to use all available data that is freely available13. Most papers on finite long memory processes apply this either on shuffled real data or on randomly generated data, we believe this approach is not suitable for this thesis because of the behavioral aspect of market (in) efficiency. Removing the sequence of actions removes the underlying basis for long memory by definition and the potential for a fractal relation in market efficiency.
12
This is subject to sample picking bias and time frame.
2. Empirical Methodology
The aim of this thesis is to prove long memory is present in markets; our approach is to use the rescaled range methodology as described in the literature review. We will use the rescaled range approach that is most used in accordance with Peters, Mandelbrot, Hurst, Kristoufek and Taqqu et al. We will apply steps 1) through 8)14 on a rolling (sliding window) time series with observation T at 1000(which is approximately 4 trading years). The time series will be S&P 500 and MSCI Index log returns as Rose (1991) indicated R/s analysis is robust. For the rolling observations we will calculate the ratio of observed Hurst exponent to the expected Rescaled Range which would indicate if there is long memory which is the method Kristoufek (2012) used to determine long memory. Moreover we truncate the series on both ends where we focus on the series between (
) and ( ) as at large N, upper level Ranges tend to
cause overestimation of the Hurst exponent (Peters, 1991), and at low end there is a small sample bias as discussed previously. We will take d=50 sub vectors as a default bucket, guaranteeing we will have at least 30 observation to perform an OLS regression. We will estimate the Hurst exponent for the entire time window (N=16501 for S&P, and 4172 for the ten MSCI indices), performing 47210 regressions in all to obtain the rolling Hurst windows seen below in Figure 8 until Figure 10. If we find long memory in the range of ; we conclude long memory in line with previous authors. As testing Hurst significance is still in the process of being developed15, we have no formal testing procedure except comparisons with previous research, and Peters Expected(R/S) for a given t and H. However as we estimate a rolling Hurst exponent, and have a very large N we believe any level above the expected RS for a Brownian motion ( ) will support our hypothesis of long memory in (sector) indices, also because we believe our largest T allows for larger significance but slightly lower H values at larger N values in the present.
For the crash study we will split up the samples and look at the changes in Long memory as denoted by the Hurst exponent on the S&P 500 Index and MSCI indices in the time frame of 2006 until 2015 to determine whether markets mean revert in a crisis across sectors and whether we see a sharp decline in the Hurst exponent followed by slow returns to long memory as investment horizons re-extend according to Peters FMH.
14
We will not be showing a pox plot for all indices as, we believe it is merely illustrative of the scaling property and not necessary per se.
15 Detrended Fluctuation Analysis by Kristoufek (2012), appears to be the primary modern estimation method,
but little is written and no other authors have tested his claims. Moreover that method is significantly more complicated than the R/S method.
The data sets used in this thesis are the daily closing prices for the S&P 500, MSCI16 sector indices, MSCI World and MSCI Emerging markets index. The S&P 500 is chosen as a base as it is well known, well researched in regards to R/S analysis and has a large sample size. The MSCI indices are chosen because they are similarly well known and are by market standards the most broadly used sector benchmarks in the world. Data on the S&P is
downloaded from CBOE and Bloomberg, and data on the MSCI Indices is downloaded from Bloomberg as well. The date ranges are from the first of January 1951 until the first of January 2016 for the S&P 500 and for the MSCI the first of January 2000 until the first of January 2016, both consistent with the earliest recorded publicly available historical data points. For our crash study we will use the Black Monday event on the nineteenth of October 1987 for the S&P 500; as the MSCI did not exist and look at the global financial crisis in 2007 and 2008 .
3. Results
The first pass over the entire horizon and sample as seen in Table 1 shows no significant long memory, though in most cases, which does indicate that market are not efficient. Further analysis (Figure 8, Figure 9, and Figure 10) show that historically Hurst coefficients have been volatile and even adjusting Hurst for Peters expected R/S coefficient long memory remains. We believe our results do not show any particularly strong long memory apart from historical highs in the S&P in the 1960’s. However we do conclude that long memory as measured in our study is at levels not consistent with efficient markets. Moreover as volatility in Hurst towards H=0.7 and consistent levels of Hurst above Peters Expected R/S for efficient markets show, markets have some long memory properties and display trending behaviors not consistent under the EMH.
Our sector analysis does not show any particular differentiation in long memory
characteristics, but significant long memory nonetheless. We expected sectors such as IT, Financials and Energy, which have all had independent booms and busts across the cycles between 2000 (2003 if corrected for T=1000 lost lags) and 2015 to perhaps show more trend reinforcement as expected under a FMH where for instance Financial firms may suffer from investor changed horizons due to regulation, litigation overhang and possible disruptive technologies. However our results show that the MSCI Financial index at the end of 2015 had H=0.53 which is higher than the E(H)=0.52 at this sample size, indicating markets are
relatively efficient. Though the results show levels consistent with a long memory process, we believe visible levels over the whole time frame are moderately unconvincing for the sector MSCI indices as we expected sectors to be harder to analyze than the S&P 500, which would translate to lower market efficiency. Regressing the E(R/S) by Peters in equation 14 on time supplies a Hurst coefficient of 0.52, which does show that on a relative basis some sectors do possess more long memory. Though longest range results over the full window may be indicative of a long memory property, as Mandelbrot mentioned some convergence towards Table 1
‡OLS regression statistics for MSCI and S&P indices Constant Hurst exponent †Lower 95% confidence interval †Upper 95% confidence interval P-value† MSCI World 0.04 0.52 0.52 0.52 0.00
MSCI Emerging markets 0.04 0.52 0.50 0.53 0.00
MSCI Energy (0.04) 0.58 0.57 0.60 0.00 MSCI Information Technology (0.05) 0.56 0.55 0.57 0.00 MSCI Consumer Disrectionary 0.01 0.54 0.53 0.55 0.00 MSCi Financial 0.02 0.53 0.52 0.55 0.00
MSCI Consumer Staples (0.05) 0.56 0.54 0.57 0.00
MSCI Materials 0.01 0.55 0.53 0.57 0.00
MSCI Utilities 0.05 0.50 0.49 0.51 0.00
MSCI Healthcare 0.04 0.51 0.50 0.52 0.00
S&P 500 (0.05) 0.55 0.54 0.56 0.00
Notes:
†All values were significant at the 1% level, however as we make no underlying assumptions regarding the distribution and in general significance for the Hurst exponent is not tested, we include these statistics for illustrative purposes only.
‡These are the regression estimations done with OLS n=31 in accordance with Peters cutting of non linear areas of R/S analysis; see methodology, whereas the rolling Hurst is done with n-=50, to optimally use data for the rolling window.
H=0.5 happens for a large sample. We look towards the rolling window Hurst exponent for further indication on market efficiency and historical Hurst performance.
The MSCI Emerging markets index has shown the strongest return and volatility, but has a weak Hurst exponent of H=0.52 in Table 1, but has shown relatively high Hurst exponents historically. This is consistent with our expectation on the basis of Tolvi (2003) and Limam (2003); who indicated developing market suffer from more inefficiency. What we also find striking is the gap between the MSCI World and the S&P 500. The MSCI world contains around 1653 constituents versus the S&P 500, and is more diversified geographically, the unexpected deviation of the comparative values could be attributable to the US market, the longer time frame, though one would expect a converging Hurst exponent in the presence of market efficiency, or due to the amount of constituents. In retrospect perhaps comparing the sector indices with the individual stocks would have been a better way to determine long memory in sector indices as current level of long memory are too weak (0.5<H<0.6), to make definitive point. Within the sectors the MSCI Energy index appears to have the highest Hurst exponent over our horizon and appears to be closely related to the economic cycle in Figure 10. We find this odd in a way as under the FMH long memory properties deviate as a result of changes of investor horizons, and the economic GDP cycle is one of multiple years and easily one of the longer investment horizons. Previous research on oil consumption (Mohn and Osmundsen, 2008; Lean and Smyth, 2009) and Prices (Serletis, 1992, Weron and
Przybylowicz, 2000) indicated that energy may be mean reverting, however we see no evidence of this in our estimate. The MSCI World, Emerging Markets, Information
Technology, Consumer Discretionary, Energy and Materials Indices appear to show a Hurst exponent course consistent with a FMH. As the (2008) crisis decreases investment horizons, reflecting lower long memory characteristics. The remaining MSCI Healthcare, Utilities, Consumer Staples and Financial indices show a path that we cannot link to the FMH regarding the 2008 global crisis. However we do note all 4 sectors offer primary goods to consumers, perhaps making them less prone to cycles; however we were not able to find literature on this. We believe sectors do show consistent long memory characteristics but fail to support the sector divergences with underlying behavioral theory in the lack of available research on specific sector performance and long memory.
Below are the graphical representations for the rolling Hurst exponent, though our estimated Hurst exponent only signals modest long memory on average, we believe our approach does prove markets vary in their degree of inefficiency, are on aggregate moderately inefficient, and signify long range dependence leaving the possibility open for advanced models to forecast markets to a degree violating the EMH.
Figure 8
S&P 500 rolling Hurst estimate with window of 1000 observations
Figure 9
S&P Rolling Hurst exponent over Peters expected R/S statistic.
In our estimation we fall short of the expected long memory target on the basis of our
literature review of . However as the S&P is broadly traded and is well diversified, it is probably an easier financial instrument to understand than sector indices, which would limit persistent market inefficiency.
Figure 10
Estimates for MSCI Indices
MSCI World MSCI Emerging markets
MSCI Energy MSCI Information Technology
MSCI Consumer Discretionary MSCI Financial
MSCI Consumer Staples MSCI Materials
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Rolling Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Rolling Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Rolling Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Rolling Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Hurst Exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Hurst exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
MSCI Utilities MSCI Healthcare
Source: Bloomberg, Matlab, Excel
Figure 11
Summary statistics for rolling Hurst estimation
Max Min Mean Median
Standard
Deviation N MSCI World
0,66 0,51 0,58 0,58 3,0% 3171
MSCI Emerging markets
0,71 0,52 0,62 0,62 3,5% 3171 MSCI Energy 0,68 0,47 0,57 0,57 3,9% 3171 MSCI Iinformation Technology 0,67 0,50 0,58 0,58 3,4% 3171 MSCI Consumer Discretionary 0,67 0,51 0,59 0,59 2,9% 3171 MSCi Financial 0,69 0,51 0,58 0,58 3,4% 3171
MSCI Consumer Staples
0,70 0,50 0,59 0,59 3,3% 3171 MSCI Materials 0,65 0,48 0,56 0,56 2,9% 3171 MSCI Utilities 0,70 0,49 0,59 0,59 4,3% 3171 MSCI Healthcare 0,71 0,43 0,56 0,56 4,8% 15500 S&P 500 0,66 0,51 0,58 0,58 3,0% 3171
Source: Bloomberg, Matlab, Excel
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
Hurst exponent Hurst exponent over expected hurst exponent
0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 0,4 0,5 0,6 0,7 0,8
We believe in a crash scenario the FMH holds as long memory on aggregate declined towards the trough of the crisis in the summer of 2007 recovering towards 2010 in a distinct v shape consistent with the findings of Kristoufek (2012). The MSCI World, Energy, Financial Consumer Discretionary, Consumer Staples, Materials, Utilities and Healthcare indices contained less long memory, consistent with short memory i.e. short term correlation based trading strategies exiting the market (Peters, 1994). The MSCI Financial shows particularly strong decline before the market collapse indicating there may be some predictive power in the Hurst exponent as similarly noted by Kristoufek in his crash analysis. Though on
aggregate we believe Figure 12 proves not only that long memory was present, but also that long memory moves in accordance with behavioral based expectations consistent with fractal markets as opposed to efficient markets. We do note that the MSCI emerging markets index and the MSCI Information Technology Index demonstrated different behaviors than expected, which we cannot explain.
Figure 12
MSCI Indices during the Global Financial Crisis 2007-2008 Estimates for MSCI Indices
MSCI World MSCI Emerging markets
MSCI Energy MSCI Information Technology
MSCI Consumer Discretionary MSCI Financial 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009
MSCI Consumer Staples MSCI Materials
MSCI Utilities MSCI Healthcare
Source: Bloomberg, Matlab, Excel
To test whether the fractal market hypothesis holds for other market collapses and there exists no MSCI data before 2000 we look to the S&P 500 in regards to the 19th of October 1987 Black Monday market collapse. In which we see a similar pattern arise as in Kristoufek’s analysis in 2012 indicating a strong link between market inefficiency, long memory and behavioral theories such as the Fractal Market Hypothesis. Moreover we note that though the intraday decline of -21% is the strongest to date the pattern is relatively consistent with our sector analysis and earlier literature.
0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009 0,40 0,50 0,60 0,70 2-1-2006 2-1-2007 2-1-2008 2-1-2009
Figure 13
S&P 500 with Black Monday on the 19th of October 1987
Source: CBOE, Matlab
4. Conclusion
With this thesis we aimed to answer whether long memory pervades sector index returns consistently, whether market collapses affect the long memory characteristic and whether this conforms to market efficiency in the EMH sense. We believe our results have proved across many observations that long memory is present in markets at levels that are not consistent with Brownian motion and an underlying random walk. Moreover this effect was visible on the level of market indices and even in emerging markets and well known indices such as the S&P 500 and MSCI World. Moreover we have discussed several behavioral biases that so far have not been explicitly linked statistical measures such as long memory. Employing the FMH analysis on rolling Hurst exponents during a crash showed that markets acted relatively consistently in that Long memory declines reflecting a shift in the aggregate investor horizon over the crash time period. Moreover our large sample size and safe approach to avoid biases show consistent Hurst exponents of above which does provide a basis for concluding market are rarely ever efficient as measured by the Hurst exponent.
Comparing our results to Kristoufek who used the same graphical analysis we used on
detrended fluctuation analysis in addition to R/S analysis, we conclude we get lower estimates for the rescaled range on aggregate. However our analysis of the S&P 500 does seem
comparable in values, trend, and crash behavior validating our approach17. Moreover as
17
To test our method we have also run multiple random fractal Brownian motion tests with an H=0.5, and done a check on the mean reversion of the VIX which proves our model works to determine mean reversion as well.
0,4 0,45 0,5 0,55 0,6 0,65 0,7 2-1-1986 2-1-1987 2-1-1988 2-1-1989
Mandelbrot indicated higher samples may converge, we believe the fact we still estimated values of greater than 0.5 reflects the robustness of the R/S statistic, even in underlying markets such as the S&P that have materially changed over 1950 till 2015. Comparing our results to Peters expected R/S Hurst exponent has also yielded results that are consistent at long memory. However we must note the Hurst exponent in our estimations rarely ever crosses values of , which indicates that though markets appear inefficient, this
inefficiency is not very large, and not as large as we expected based on previous experiments. On the basis of our results we believe markets operate at moderate inefficiency, and both our long run estimate, rolling window Hurst exponent and the rolling window Hurst exponent over expected Hurst exponent show that there is a persistent long memory effect. Moreover we find some moderate evidence for the FMH as some sectors and the S&P index respond to crashes through shorter time horizons. Our estimate of the Hurst exponent across sectors, and indices over a long range yield an aggregate H of 0.55, which falls short of the 0.6 target we set ourselves, to prove a long memory effect. We believe literature and our results harmonize in that the efficient market hypothesis does not hold at all times. Moreover after performing our research we reflect that fundamentally, short memory may be more relevant than we previously assumed, as after Lo (1991) and Taqqu et al. (1999) we dismissed Lo’s statistic as negatively long memory biased without including it in our study. In retrospect, the long memory property would be fundamentally easier to arbitrage away than short memory making short memory a more attractive research field in light of our perceived declining Hurst
exponent values over 1950 until 2015.
As we are not aware of any strategy to generate excess returns, nor any definitive proof of long memory in finance in an infinite setting to disprove the EMH, we believe the EMH is still the most attractive and plausible theory, and as we have seen long memory decrease over time in our estimates, we believe markets may become more efficient. Furthermore we find it ironic that in a fractal market hypothesis setting, market collapses; which have served as a primary argument against efficient markets, actually make markets more efficient in our horizon of consistent . We believe the application of R/S analysis on time series deserves further research as both long and short term dependence has only limited sources, and the applications may hold insights for many fields from physics to finance.
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Appendices
Appendix A: Summary statistics for S&P and MSCI Indices
Table 2
Summary statistics for MSCI World Index, MSCI Emerging markets Index, MSCI Sector Indices and S&P 500 index
Max Min Mean Median Standard Deviation
N Start End
MSCI World 1810.8 688.6 1263.7 1250.3 268.7 4172 3-1-2000 31-12-2015 MSCI Emerging markets 1338.5 245.6 751.9 842.1 298.9 4172 3-1-2000 31-12-2015 MSCI Energy 338.3 94.4 199.3 212.7 61.5 4172 3-1-2000 31-12-2015 MSCI Information Technology 232.0 39.5 94.0 85.2 33.8 4172 3-1-2000 31-12-2015 MSCI Consumer Disrectionary 203.8 53.7 115.3 107.4 34.9 4172 3-1-2000 31-12-2015 MSCI Financial 167.3 35.8 100.3 99.2 25.2 4172 3-1-2000 31-12-2015 MSCI Consumer Staples 212.5 63.4 123.8 118.2 40.6 4172 3-1-2000 31-12-2015 MSCI Materials 340.2 80.6 187.4 203.3 64.2 4172 3-1-2000 31-12-2015 MSCI Utilities 170.1 55.5 104.7 104.2 23.0 4172 3-1-2000 31-12-2015 MSCI Healthcare 228.7 67.1 115.8 103.8 36.6 4172 3-1-2000 31-12-2015 S&P 500 2130.8 16.7 474.4 138.5 544.8 16501 3-1-1950 31-12-2015 Source: CBOE. Bloomberg
