## June 7th, 2006清谷子:long-term memory in the foreign exchange

[清谷子@copyright]The purpose of this report is to empirically test whether there is long-term memory or detectable persistence in the foreign exchange time series with Edgar Peters’ rescaled range (R/S) methodology. R/S test has already been applied in many financial assets and financial markets as a major methodology to detect whether particular financial assets and financial markets are in fact following a random walk and lognormally distributed as Efficient Market Hypothesis strongly suggested.

The Efficient Market Hypothesis states all information have been incorporated into the market price and furthermore, the market is following a random walk, which is to say, that the impact of new information is essentially unpredictable, it is as likely to be negative as positive. Therefore, there will not be any predictable patterns for market movements.

Financial and economic time series have often been modeled as random-walk processes (Cheng and Deets, 1971; Gogley, 1990). However, the random walk has been proven to fit real market data poorly in many cases. For example, see Mandelbrot (1963a); Fama (1965); Greene and Fielitz (1977); Helms and Martell (1985); Hsieh (1989); Baillie and Bollerslev (1994); Barkoulas et al. (1997). Other models with trends and cycles have been suggested, and one category of such models is characterized by long memory or long-term memory.

According to Jin and Frechette (2004), memory means that observations are not independent; each observation is affected by the events that preceded it. Even a simple autoregressive series exhibits memory, but autoregressive behavior is considered short memory. If a series exhibits long memory, then there is persistent temporal dependence even between distant observations.

A burst of research into long memory in the 1990s brought long memory models into the mainstream. The main conclusion was that most market data exhibit long memory, The R/S test, together with the fractional integration, has been frequently used in the literature for some years to detect long-term memory in time series (Jin and Frechette, 2004).

The ob

Corazza et al. (1997) suggest that in R/S test, when H ≠0.5, there exists fractal (though 0<H<0.5 has a different pattern with 0.5<H<1.0) and when H ＝0.5, the Market is efficient. In all seven pairs of data, all values of H are between (0, 0.5), which indicates that there exist fractals in the foreign currency exchange market.

This report has five sections. The first section gives a brief introduction on background of R/S methodology, section II describes the methodology used in this paper in step-by-step detail, section III describes the data, Section IV reports the empirical results and implications of the findings, and Section V concludes.

I: Rescaled Range Analysis (R/S)

History

Rescaled Analysis (R/S) methodology is originated from a hydrologist H.E. Hurst. He was inspired by the periodical cycles of Nile River flood and Einstein’s Theory, thus invented Rescaled Range Analysis to find the best solution to build a dam to preserve the best capacity of storage of water in all seasons. His methodology provides the essential tool to yield important insights about the EMH and behavior of security returns (Peters, 1989).

Appling R/S methodology, Hurst discovered that a number of natural phenomena exhibit behavior that can be characterized by a biased random process, as opposed to a pure random process during the 1950s and 1960s. A biased random process means that there is a long-term dependence, or a “memory”, between observations. The events of one period influence all the periods that follow. This dependence is called “persistence” (Peters, 1989).

Following Hurst’s footsteps, Mandelbrot and Van Ness (1968) introduced a generalization of Brownian motion called fractional Brownian motion (FBM). FBM has normally distributed but non-independent increments, which makes it an excellent way to model long memory. R/S analysis was subsequently refined and applied to economic time series in a series of studies by Mandelbrot (1963a, 1963b, 1972), Peters (1989, 1994) and Lo (1991). R/S analysis provides a valuable statistic called the Hurst (1951) exponent, H.

Principles

Basic concepts

The rescaled range is the range of partial sums of deviations from the mean of a time series, rescaled by its standard deviation. Consider a series of N sample returns X = ﹛Xt﹜and sub-time series Xi,T of length T, where T is less than or equal to N. Determine all possible nonoverlapping sub-time series Xi,T, and calculate the sample means Xm1T. Calculate cumulative sums of deviations for each sub-time series, Xsi,T, k = ∑ k j=1 (X j,i,T- Xmi,T), where Xj, i, T denotes the jth observation of sub-time series Xi,T. The rescaled range statistic is

(R/ S)i, T = [Max 1≤ k≤ T (Xsi, T,k) - Min1≤ k≤ T (Xsi, T,k )]/ Si, T (1)

Where Si,T is the standard deviation of sub-time series Xi,T.

Between (R/S)i,T and Ti, there is a mathematical relationship

(R/S)i,T ~THi (2)

ba

Ln (R/S)i,T = ln (a)+Hln(Ti)+ εi,T (3)

Where the intercept of the regression, ln(a), is a constant that has no particular meaning in this specific case andεi,T is a error term with zero expected value. By using several different starting points, average values of R/S may be computed for different values of T. The steps are explained more fully by Mandelbrot and Taqqu (1979), Lo (1991), and Corazza et al. (1997) (Jin and Frechette, 2004).

What does the value of H imply?

According to the equation and definition, H ranges between (0, 1), and H = 0.5 is the dividing value.

H = 0.50: denotes a random and statistically independent (uncorrelated) series— a random walk. The present does not influence the future. The correlation coefficient is 0. Its probability density function is normal. Such process increases with the square root of time (Chu, 2003)

The relation between the Hurst exponent and the fractal dimension is simply D = 2-H. that means that for statistically independent fractional Brownian movement, with D = 1.5, the Hurst exponent should be H = 0.5. For the ‘various’ natural phenomena with H being about 0.72, the fractal dimension D is about 1.28 (according to Peters, many natural phenomenon have an H value of 0.72), which means this is a ‘rather smooth’ profile-like curve from fractal dimension perspective (Vanouplines, 2006).

0.50 < H <1> 0.5, then the series is less lagged than a random walk would be, implying that the series is less turbulent than a random walk (Jin and Frechette, 2004).

Peters (1989) mentioned the higher the Hurst exponent, the stronger the persistence and the less “white noise” there is in a time series. With a 1>H>0.5, it is possible to detect the trend, as it does for many nature phenomena with H = 0.72 where fractals have been successful in explaining behavior to some extent.

0< H < 0.50: denotes an “anti-persistent”, or ergodic series. That is, the data has a tendency to reverse the current trend. This process is said to be mean reverting. For example, if the system has been up (down) in the previous period, it is more likely to be down (up) in the next period. The strength of the anti-persistence depends on how close H is to zero and the correlation coefficient approaches –1 as H approach 0. The frequent reversal results in less distance covered by the process than would occur given a random process (Chu, 2003).

If H< 0.5, as it approaches 0, the length and strength of the anti-persistence behaviour increases, and a series plot becomes more lagged than a random walk, implying that the series is more turbulent than a random walk (Jin and Frechette, 2004). The anti-persistence behaviour has a rather high fractal dimension (1.5<D<2), corresponding to a very ‘noisy’ profile-like curve (Vanouplines, 2006).

Fractals: Corazza et al. (1997) suggest that Market Efficiency Hypothesis only holds when H = 0.5, with a D = 1.5, there is no fractals. However, if any value of H ≠ 0.5, there exists fractal. The pattern is of different types dividing by the value of 0.5. When 0<H<0>H>0.5, it shows persistence, an upward or downward price trend will continue for a while till some sudden event stops it. It is less volatile, and has much less “white noise”.

Advantages and Disadvantages of R/S analysis

The most acceptable paradigm is that the stock returns are normally distributed, which means that asset prices follow lognormal distributions. The efficient market hypothesis (EMH) assumes that all investors immediately react to new information, so that the future price movement of a stock is unrelated to new information, and the future price movement of a stock is unrelated to the past or present patterns of price movements. However, these two paradigms are challenged by numerous studies.

Actually, do people behave in this matter? Peters (1991a) mentioned that most people wait for confirming information and do not react until a trend is clearly established. Consequently, there will be an uneven assimilation of information. This will cause the stock price movement to follow a biased random walk, instead of random walk, implies that there is memory underlying in the series.

Peters (1989) proposed that Hurst’s R/S analysis has provided an important tool for the study of natural processes and, in turn, contributed to the development and application of fractal geometry. Fractal geometry provides a means of describing the organizing structures underlying irregular shapes or series generated by a non-linear dynamic system by identifying the consistent patterns that operate across scales of size or time. Therefore, The R/S analysis is an ideal statistical tool for analyzing the occurrence of rare events and is robust to possible nonlinear process that normality assumptions may not be needed. Whether the stock price movement follows a random walk or not can be detected by R/S analysis (Chu, 2003).

The behavior of the values of the estimates for different time-scaled distributions leads to the formulation of the hypothesis: the stochastic-process underlying futures returns are characterized by a fractal structure, as proposed by Peters (Corazza et al. 1997). With this hypothesis, Peters applies R/S analysis to test periodical and non-periodical cycles, which are not consistent with the two underlying assumptions of EMH: lognormal distribution and random walk (without memory).

However, Peter’s methodology has obvious flaws. R/S test is very sensitive to short-term dependence. Therefore, it is not distinguishable whether the value of H is the result of short-term memory or long-term memory. Because of such sensitivity, the long-term memory results from the classical R/S method can merely be due to short-term memory. Lo (1991) propose a modification of the classical R/S analysis. The modified R/S is robust to both short-term dependence and highly non-normal innovations and its behavior is invariant over a general class of short-term memory processes but deviates for long-term memory processes. Moreover, unlike the classical R/S statistics, it has well-defined distribution properties (Corazza et al. 1997).

Important Reforms by Edgar Peters

A: Logarithmic Returns

In his 1989 examination on S&P 500 returns with R/S analysis, Edgar Peters used the monthly percentage change in prices as the basis for calculation. Later on, he realized that the percentage change in prices is not appropriate for R/S analysis and highly understates the results.

R/S analysis measures the cumulative deviation from the mean for various periods of time and examines how the range of this deviation scales over time. Because the percentage change in prices does not sum to its cumulative equivalent, it is not appropriate for R/S analysis. By redid the same test using logarithmic returns (Log Returns), Peters finds that value of H increases from 0.61 to 0.78, a more obvious persistence level in the trend. Peters claims that logarithmic returns have the same statistical properties as percentage change, but they sum to their cumulative equivalent.

B. Number of A and n

In his 1994 book, Peters adds two more important rules of thumb: A must be a number that will lead to N/A as an integer number and value of n must be ≥ 10.

The first n must be larger and equal to 10 because small values of n produce unstable estimates when sample sizes are small. The subsequent n must be increased to the next integer value because it will include the beginning and ending points of the times series (Peters, 1994).

C. Ordinary Least Squares Regression

In the past, the calculation of H must be performed with very complicated mathematical functions. However, with the computer software development, Peters applies the basic regression function to perform the calculation of H. This greatly increases the volume of usage on R/S into many financial assets and financial market analysis.

II: Methodology

Peters (1994) and Corazza et al. (1997) describe R/S in step-by-step guide, using one pair from the samples tested in this article: USD/JPY weekly data, the algorithm used can be summarized as follows:

Step 1 Select the original time series, with data series of (1, 2, 3, ………..M). In this case, select USD/JPY weekly data with M = 997. Covert this into a time series of length N = M- 1 = 996 of logarithmic ratios:

Log is calculated in Excel with “Ln” function.

Step 2 Determine a nonoverlapping subtime series, (M-1) /N = A. For example, N is the total number in each group, as N=n ≥ 10, let N1 be 10, the nearest number to make A as integer is 12, A= 996/12 = 83. That means there are total 83 groups with each group having 12 data. Compute mean and standard deviation for each group. For each group, the average value is defined as:

Step 3 For every subtime series, total 83 groups, calculate the sample mean, X, and calculate the difference between each data and the subseries sample X, then sum the nearest two cumulative deviation into Yn, the time series of accumulated departures from the mean value is defined as:

Step 4 Find the Max and Min in partial cumulative departures from the mean, then using Max value minus the Min value to get R, finally rescale the data by divided R with standard deviation of each group. For example, find max and min value in group 1 Y series, using max minus min to get the R1 for group 1, using R1 divided by S1 to get the (R/S) 1 for group 1.

Step 5 Set new subtime series length, N by Nser = N + S (S＞0) and N = Nser. The next n can give a integer of A is 83, all possible combinations of n giving A as integer number are: N×A, 12 ×83, 83×12, 166×6, 249×4, 332×3, 498 ×2, 996×1.

Step 6 Compute the average value of R1/S1, R2/S2, to Rj/Sj. Normalize each range R by dividing the corresponding Standard deviation to it. It is defined as:

Step 7 Apply Hurst equations by performing an ordinary least squares (OLS) regression on log (n) as the independent variable and log (R/S) n as the dependent variable. The intercept is the estimate for log C, the constant. The slope of the equation is the estimate of the Hurst exponent, H.

III: Data

The data is selected from the latest (2006) available foreign currency exchange data pool.

In order to test whether there is long term memory or any detectable persistence pattern, among the total data pool, seven pairs of data are being selected: NZD/USD monthly data with the smallest sample size of 96, EUR/USD weekly and USD/JPY weekly with largest sample size of 996, EUR/USD 50 minutes, EUR/USD 120 minutes, NZD/USD 10 minutes, NZD/USD daily with modest sample size of 700.

The data ranges from different frequency: from 10 minutes, 50 minutes, 120 minutes, to daily, monthly, and weekly; the data selection also covers all three available currency pairs: NZD/USD, ERU/USD, and USD/JPY. The selection of the data pairs is to avoid the bias that may be caused by different frequency and different currency pairs.

IV: Results

A. NZD/USD Monthly

This data has a sample size of 96, the smallest sample among all available data. Because of the smallest sample size, 6 and 8 are selected for n in spite of the rule of n must be larger or at least equal to 10. The H value is 0.295757 with a standard error 0.024392. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of NZD/USD Monthly

N R/S Log(R/S) Log (N)

6 3.3787 0.52875 0.778151

8 3.920134 0.593301 0.90309

12 4.440503 0.647432 1.079181

16 5.112435 0.708628 1.20412

24 5.782602 0.762123 1.380211

32 6.273152 0.797486 1.50515

48 6.852347 0.835839 1.681241

96 7.557372 0.878371 1.982271

R Square 0.96079

C 0.330314

H 0.295757

Standard Error 0.024392

B. EUR/USD 50 Minutes

This data has a sample size of 700 with relative modest sample size among all available data. The H value is 0.357096 with a standard error 0.013078. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of EUR/USD 50 Min

N R/S Log(R/S) Log (N)

10 3.806224 0.580494 1

14 4.537029 0.656772 1.146128

20 5.288957 0.72337 1.30103

25 5.645716 0.751719 1.39794

28 6.111039 0.786115 1.447158

35 6.709583 0.826696 1.544068

50 7.518512 0.876132 1.69897

70 8.48407 0.928604 1.845098

100 10.25943 1.011123 2

350 14.85683 1.171926 2.544068

700 17.20992 1.235779 2.845098

R Square 0.988073

C 0.258746

H 0.357096

Standard Error 0.013078

C. EUR/USD 120 Minutes

This data has a sample size of 700 with relative modest sample size among all available data. The H value is 0.262364 with a standard error 0.018976. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of EUR/USD 120 Min

N R/S Log(R/S) Log (N)

10 3.927017 0.594063 1

14 5.391995 0.731749 1.146128

20 5.207214 0.716605 1.30103

25 5.643698 0.751564 1.39794

28 5.9437 0.774057 1.447158

35 6.514987 0.813914 1.544068

50 7.23311 0.859325 1.69897

70 8.113937 0.909232 1.845098

100 8.843686 0.946633 2

350 11.89965 1.075534 2.544068

700 12.35537 1.091856 2.845098

R Square 0.955035

C 0.394552

H 0.262364

Standard Error 0.018976

D. NZD/USD 10 Minutes

This data has a sample size of 700 with relative modest sample size among all available data. The H value is 0.227826 with a standard error 0.018655. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of NZD/USD 10 Min

N R/S Log(R/S) Log (N)

10 3.890995 0.590061 1

14 4.447361 0.648102 1.146128

20 5.167406 0.713273 1.30103

25 5.695645 0.755543 1.39794

28 5.582669 0.746842 1.447158

35 6.258182 0.796448 1.544068

50 6.95365 0.842213 1.69897

70 7.422759 0.870565 1.845098

100 7.852824 0.895026 2

350 9.826231 0.992387 2.544068

700 10.45547 1.019344 2.845098

R Square 0.943089

C 0.417601

H 0.227826

Standard Error 0.018655

E. NZD/USD Daily

This data has a sample size of 700 with relative modest sample size among all available data. The H value is 0.194155 with a standard error 0.013864. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of NZD/USD Daily

N R/S Log(R/S) Log (N)

10 4.040714 0.606458 1

14 4.54404 0.657442 1.146128

20 5.201541 0.716132 1.30103

25 5.608532 0.748849 1.39794

28 5.653812 0.752341 1.447158

35 5.924151 0.772626 1.544068

50 6.57484 0.817885 1.69897

70 6.896221 0.838611 1.845098

100 7.20831 0.857833 2

350 8.680364 0.938538 2.544068

700 9.714818 0.987435 2.845098

R Square 0.956121

C 0.459086

H 0.194155

Standard Error 0.013864

F. EUR/USD Weekly

This data has a sample size of 996, the largest sample size among all available data. The H value is 0.234238 with a standard error 0.006047. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of EUR/USD Weekly

N R/S Log(R/S) Log (N)

12 4.159883 0.619081 1.079181

83 6.787136 0.831687 1.919078

166 7.766766 0.89024 2.220108

249 8.644194 0.936724 2.396199

332 8.854052 0.947142 2.521138

498 9.939319 0.997357 2.697229

996 11.97792 1.078381 2.998259

R Square 0.996679

C 0.370335

H 0.234238

Standard Error 0.006047

G. USD/JPY Weekly

This data has a sample size of 996, the largest sample size among all available data. The H value is 0.278765 with a standard error 0.015496. The following tables shows the R/S results and the graph shows the comparison between actual value of H and when H = 0.5.

R/S Analysis of USD/JPY Weekly

N R/S Log(R/S) Log (N)

12 4.246963 0.628078 1.079181

83 6.949146 0.841931 1.919078

166 8.104948 0.90875 2.220108

249 9.349679 0.970797 2.396199

332 9.586783 0.981673 2.521138

498 11.8362 1.073212 2.697229

996 15.01492 1.176523 2.998259

R Square 0.984785

C 0.309682

H 0.278765

Standard Error 0.015496

Findings: There is anti-persistence (0<H<0.5) in all seven pairs of data. They are all statistically significant. The evidence shows that in foreign currency exchange market of the seven pairs of data, the pure random walk, the Market Efficiency Hypothesis does not apply. The foreign currency exchange market follows a more turbulent walk. We also have learned from H value that the foreign currency exchange market is influenced by the past. The biased Hurst phenomenon shows that there is a theoretical basis for market timing and tactical asset allocation.

V: Conclusion

Fractal has been found in many natural phenomena and in many financial assets and financial markets. This report finds that it also exists in foreign currency exchange market and thus rejects the notion that Efficient Market Hypothesis can apply to foreign currency exchange market. However, this conclusion is only applicable to the seven examined pairs of data. Further research with more complete data is needed to finalize that the conclusion from this examination.

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