A Fractal Analysis of Foreign Exchange Markets
(JEL G15)

International Advances in Economic Research 6:1 February 2000

ROBERT F. MULLIGAN*

Department of Economics, Finance, & International Business
Western Carolina University College of Business

Long memory in foreign exchange markets is examined for the post-Bretton Woods period using Lo's [1991] modified rescaled range (R/S). Conventional rescaled range techniques are presented for comparison. Unlike conventional  techniques, Lo's analysis is robust to short-term dependence and conditional heteroskedasticity. Significant long memory and fractal structure are conclusively demonstrated for all twenty-two countries studied, indicating traditional econometric methods are inadequate for analyzing foreign exchange markets. However, short-term dependence and conditional heteroskedasticity are also present, making it difficult to describe the nature of the long memory process or processes in foreign exchange markets. The average non-periodic cycle ranges from seven months for Canada and the U.K., to approximately twenty months for Austria, Finland, France, Germany, Ireland, Japan, Malaysia, the Netherlands, Sweden, and Switzerland.  No support is found for the efficient market hypothesis. Results broadly agree with those provided by less sophisticated, less robust R/S methodologies, and suggest the possibility that traditional technical analysis should be able to achieve systematic positive returns. (JEL G15)

Long memory series exhibit non-periodic long cycles, or persistent dependence between observations far apart in time.  Short-term dependent time series include standard autoregressive moving average and Markov processes, and have the property that observations far apart exhibit little or no statistical dependence.

R/S or rescaled range analysis distinguishes random from non-random or deterministic series.  The rescaled range is the range divided (rescaled) by the standard deviation. Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and non-random components. Conventional statistical techniques lack power to distinguish random and deterministic components. R/S analysis evolved to address this difficulty.

R/S analysis exploits the structure of dependence in time series irrespective of their marginal distributions, statistically identifying non-periodic cyclic long run dependence as distinguished from short dependence or Markov character and periodic variation [Mandelbrot, 1972a, pp. 259-260].  Mandelbrot likens the differences among the three kinds of dependence to the physical distinctions among liquids, gases, and crystals.

Long memory in exchange rates would allow investors to anticipate price movements and earn positive average returns.  Fractal analysis offers an alternative to conventional risk measures and permits an evaluation of central banks' foreign exchange policies.  Countries with effective, well-administered pegs should have random-walk dollar exchange rates.  This occurs because central banks are intervening in the foreign exchange market to support the peg on a day to day basis, and the volume of their trading is relatively low and relatively stable.

Biased random walk exchange rates are characterized by abrupt and unusual central bank interventions of extraordinary volume compared to pegged currencies.   Thus fractal analysis also indicates the extent intervention characterizes the series.
Fractal analysis can also identify ergodic or anti-persistent series, e.g., negative serial correlation.  The more ergodic an exchange rate, the less stable the economy.  Ergodic exchange rates should also have much shorter cycle lengths than random walks or trend-reinforcing series.   One source of ergodic behavior is sub-optimal policy rules that delay intervention, overstate the amount required, or both.

Four techniques are reported in this paper, Hurst's [1951] empirical rule, Mandelbrot and Wallis's [1968] classic, naive R/S, Mandelbrot's [1972a] AR1 R/S, and Lo's [1991] modified R/S. A related technique, Peters's [1996] Vn, was used in an unsuccessful attempt to identify cycle length.  Naive R/S is shown to be highly biased, but gives an indication of the cycle length, though in this paper, traditional R/S analysis spuriously suggested a cycle too long to be measured.  The empirical rule and AR1 R/S give less biased measures of the Hurst exponent H.  AR1 R/S also indicates the cycle length, but again gave a spurious indication of a cycle too long to measure.  Lo's modified R/S does not measure H but gives a definitive unbiased measure of the cycle length, which turns out to be much shorter than indicated by traditional techniques.  The empirical rule and AR1 R/S give measures of H which may be biased but have no alternative.

Hurst's empirical rule performs surprisingly well compared to the classic and AR1 R/S technique, which are not robust to short-term dependence.  Using the modified R/S, which is robust against short-term dependence, Lo found no long memory in stock prices. The present paper applies Lo's adjusted R/S analysis to exchange rates for the first time, providing evidence of long memory in exchange rates, with duration ranging from seven to twenty-two months.  This finding is far shorter than previously suspected.

The remainder of the paper is organized as follows.  A literature review is provided in the second section.  The data are documented in the third section.  Methodology is briefly discussed in the fourth section.  Empirical results are presented in the fifth section.  Concluding remarks are provided in the sixth section.  A complete discussion of the methodology is presented in an appendix.

The search for long memory in capital markets has been a fixture in the literature applying fractal geometry and chaos theory to economics since Mandelbrot [1963] shifted his attention from income distribution to speculative prices.
Long memory in exchange rates has been studied by Booth, Kaen, and Koveos [1982], Cheung [1993], Cheung and Lai [1993], Fung, Lai, and Lo [1995], Peters [1996], Fisher, Calvet, and Mandelbrot [1997], Barkoulas and Baum [1997c], and Chou and Shih [1997]. Andersen and Bollerslev [1997] also found long-memory in dollar/Deutschemark exchange rates using a GARCH model. Barkoulas and Baum [1997a, 1997b] found long memory in eurocurrency returns using spectral regression estimates of fractional differencing parameters.

Fractal analysis has also been applied to equities [Greene and Fielitz, 1977; Lo, 1991; Barkoulas and Baum, 1996; Peters, 1996; Kraemer and Runde, 1997; Barkoulas and Travlos, 1998], interest rates [Duan and Jacobs, 1996; Barkoulas and Baum, 1997a, 1997b], commodities [Barkoulas, Baum, and Oguz, 1998], and derivatives [Fang, Lai, and Lai, 1994; Corazza, Malliaris, and Nardelli, 1997; Barkoulas, Labys, and Onochie, 1997].

Exchange rate distribution and modeling have received much attention in the literature [Cheung, Fung, Lai, and Lo, 1995; Baum and Barkoulas, 1996; Byers and Peel, 1996; Gazioglu, 1996; Fritsche and Wallace, 1997]. Cheung and Lai [1993] suggest Heiner's [1983] and Kaen and Rosenman's [1986] competence-difficulty (C-D) gap hypothesis as a potential source of long memory in asset prices.  This provides a theoretical expectation of long memory.

The C-D gap is a discrepancy between investors' competence to make optimal decisions and the complexity of exogenous risk.  A wide C-D gap leads to investor dependency on deterministic rules, which can lead to persistent price movements in one direction - a crash or speculative bubble.  Due to irregular arrival of new information, Kaen and Rosenman argue persistent price movements may suddenly reverse direction, leading to non-periodic cycles.  Program trading introduces the same phenomenon.

A different kind of long memory is suggested by Mussa's [1984] disequilibrium overshooting model, which is based on the contracting approach to introducing monetary nonneutralities into macroeconomic models developed by Fisher [1977], Phelps and Taylor [1977], and Taylor [1980].  Mussa's model would be supported by finding anti-persistence or ergodicity in exchange rates.

Mandelbrot [1972b, 1974] and Mandelbrot, Fisher, and Calvet [1997] have developed the multifractal model of asset returns (MMAR).  MMAR shares the long-memory feature of the Fractional Brownian Motion (FBM) model introduced by Mandelbrot and van Ness [1968]. The statistical theory necessary to identify empirical regularities and local scaling properties of MMAR processes with local Hölder exponents is developed by Calvet, Fisher, and Mandelbrot [1997] and applied to dollar/Deutschemark exchange rates by Fisher, Calvet, and Mandelbrot [1997].

Mandelbrot's [1972a, 1975, 1977] and Mandelbrot and Wallis's [1969] R/S or rescaled range analysis characterizes time series as one of four types: 1.) dependent or autocorrelated series, 2.) persistent, trend-reinforcing series, also called biased random walks, random walks with drift, or fractional Brownian motion,  3.) random walks, or 4.) anti-persistent, ergodic, or mean-reverting series.

Time series are classified according to the estimated value of the Hurst exponent H, which is defined from the relationship R/S = anH, where R is the average range of all subsamples of size n, S is the average standard deviation  for all samples of size n, a is a scaling variable, and n is the size of the subsamples, which is allowed to range from an arbitrarily small value (here six months) to the largest subsample the data will allow.   Putting this expression in logarithms yields log(R/S) = log(a) + H log(n), which is used to estimate H. H ranges from 1.00 to 0.50 for persistent series, is exactly equal to 0.50 for random walks, ranges from zero to 0.50 for anti-persistent series, and is greater than one for a persistent or autocorrelated series.  Mandelbrot, Fisher, and Calvet [1997] refer to H as the self-affinity index or scaling exponent.

Because H is the reciprocal of the Mandelbrot-LJvy  characteristic exponent ", estimates of H indicate the probability distribution underlying a time series.  H = 1/" = 1/2 for normally-distributed or Gaussian processes.  H = 1 for Cauchy-distributed processes.  H = 2 for the LJvy distribution governing tosses of a fair coin.

In fractal analysis of capital markets, H indicates the relationship between the initial investment R and a constant amount which can be withdrawn, the average return over various samples, providing a steady income without ever totally depleting the portfolio, over all past observations.  Note there is no guarantee against future bankruptcy.

R/S analysis also gives an estimate of the average non-periodic cycle length, the number of observations after which memory of initial conditions is lost, that is, how long it takes for a single outlier's influence to become immeasurably small.  If foreign exchange rates are random walks with H = 0.50, returns are purely random and should lead to investors' breaking even over the long run.

If exchange rates are persistent with (0.50 < H < 1.00), the series are less noisy, exhibiting clearer trends and more persistence the closer H is to one, and investors should earn positive returns. Neely, Weller, and Dittmar [1997] found technical trading rules, formalized with a genetic programming algorithm, provided significant out-of-sample excess returns. H's close to one indicate high risk of large, abrupt changes, e.g., H = 1.00 for the Cauchy distribution.

Finally, if exchange rates are anti-persistent or ergodic or mean-reverting with (0.00 < H < 0.50), they are more volatile than a random walk.  If the highly volatile returns are uncorrelated across different assets, risk can be minimized by diversification. Ergodicity would support Mussa' s [1984] equilibrium overshooting model.

In applying his modified R/S analysis to equity prices, Lo [1991] overturned earlier results based on classical R/S methods finding long memory.  In the present paper, the Lo technique is applied to exchange rates.

Although earlier attempts at R/S analysis of foreign exchange markets were highly informative [e.g., Booth, Kaen, and Koveos 1982, Cheung 1993, Peters 1996], approximately two cycle lengths of data are necessary for good estimates of Hurst exponents and average non-periodic cycle length using classical R/S techniques [Mandelbrot, 1972a; Peters, 1996].  Peters suggested cycle length for exchange rates was at least ten years.  Over twenty-five years have now elapsed since fixed exchange rates were abandoned in 1973.  Since the average cycle length, if it exists, is not known, this time period offers the potential of including a sufficient number of cycles to allow the Hurst exponent and average cycle length to be definitively measured.

The data are monthly average dollar exchange rates from the Federal Reserve Bank of St. Louis economic data (FRED) for a selection of countries.  The sample period is January 1973 to December 1997 - 24 years of data.  For some countries, data is only available for a subsample.  Monthly average exchange rates are of more interest than daily exchange rates for at least four groups of investors: 1.) program traders, 2.) investors who follow deterministic rules, 3.) investors who routinely accept exposure approximately one month or longer, and 4.) currency hedgers.

As long as an investor's average exposure is approximately one month or longer, the average monthly exchange rate better characterizes the asset price than the price on any particular day. In addition, monthly average exchange rates are more relevant for testing the Efficient Market Hypothesis if price adjustments are not instantaneously efficient - a market may be efficient even if imperfect - that is, price adjustments may be efficient on a month-to-month basis, even if not on a day-to-day basis.

Nevertheless, averaged data may have a significantly different marginal distribution than the original data.  Consider the difference between monthly averages and end-of-month observations.  If daily data have short-term dependence or serial correlation over a period of less than 30 days, monthly averages are more likely to show short-term dependence than end-of-month observations.

Andersen and Bollerslev [1997, p. 975], using a GARCH specification to model a one-year series of five-minute dollar-Deutschemark exchange rates, found a slowly mean-reverting fractionally integrated process where short-term volatility could be interpreted as a mixture of many short-run information arrivals with long memory as an intrinsic feature of the data generating process.  Their result supports Mussa [1984]. The present paper's use of monthly average returns removes most short-run volatility, particularly intradaily volatility.

This paper examines the statistical behavior of the average monthly exchange rate, and does not adjust returns for interest rate differentials.  Interest rate differentials may be more easily ignored in daily returns than monthly returns.  Systematic, non-random interest rate differentials may introduce systematic bias in the monthly average returns, and any implications for the Efficient Market Hypothesis must be interpreted in this light.   Specifically, non-randomness in monthly exchange rates cannot disconfirm the Efficient Market Hypothesis unless interest-rate differentials are included in the asset returns.  Randomness in monthly series would still tend to support market efficiency.

The exchange rates are taken as U.S. dollars per foreign currency unit, giving the return to Americans holding foreign currency. The data are converted to logarithmic returns: Xt = ln(Pt/Pt-1), where Xt is the logarithmic return on holding the foreign currency at time t, and Pt  is the exchange rate in dollars per foreign currency unit. Logarithmic returns are more appropriate for R/S analysis than percent price changes because the range in R/S analysis is the cumulative deviation from the average return, and the logarithmic returns sum to the cumulative return.

This section briefly describes the procedures employed to estimate the Hurst exponent and cycle length.  More detailed documentation is presented in the appendix.  R/S analysis examines the behavior of the average range (R) rescaled by the average standard deviation (S), as a function of sample size.

In his pioneering work on the hydrology of the Nile River Valley, Hurst [1951] gives an empirical law for use when too few R/S observations are available.  This expression, H = [log(R/S)]/[log(n/2)], tends to overstate H if H > 0.70 and understate H for H < 0.40.  The empirical rule is extremely information efficient - parsimonious, even - and may be less biased than other conventional R/S measures of H.  Empirical rule estimates of H are reported in the second column of Table 1 and broadly accord with earlier findings using daily data. This suggests the empirical rule is not extremely susceptible to bias due to short-term dependence.

The Hurst exponent H is defined from the relationship R/S = an^H, where R is the average range and S is the average standard deviation, of all samples of observations of size n.  The scaling variable n is allowed to range from an arbitrarily small n = 6 to the largest n permitting the data to be partitioned into two samples.  H defines the average relationship, in past data, between the rescaled range R/S and elapsed time or average sample size n.

The un-rescaled range R is the actual return on foreign currency holdings over a particular sample range. The standard deviation S and time n define the average return, the desired steady return which may be withdrawn from the investment each period without depleting the investment.

Mandelbrot and Wallis [1969] introduced the conventional technique for estimating H by regression.  This original technique is known to be highly biased by short-term dependence. These estimates of H are reported in column three of Table 1.  All reported values are close to two, suggesting highly biased results.  H = 2 for the Lévy distribution governing tosses of a fair coin.  It implies perfect randomness in returns and supports market efficiency.

Mandelbrot and Wallis [1969], Mandelbrot [1972a], and Peters [1994] note the estimate of H may be biased in two major circumstances: nonstationarity of the data and/or short-memory.  To overcome either or both sources of bias, they recommend performing R/S analysis on AR1 residuals of the logarithmic returns.  This represents a major refinement removing much, though not all, bias due to short-term dependence.

H's estimated from AR1 residuals are reported in column four of Table 1.  These values are all smaller than one, suggesting greater plausibility than naively-estimated H's.  At least some bias due to short-term dependence is removed by using AR1 residuals, but it is not clear that AR1 residuals provide completely unbiased estimates of H.  Davies and Harte [1987] show regression estimates of H tend to be biased toward rejection of the null hypothesis of no long memory even for stationary AR1 processes.

Peters [1996] also recommends graphing V_n = (R/S)/(n)1/2 against log(n) to better identify the non-periodic cycle length, but this procedure merely suggested the cycle length is too long to identify with available data.  Inspection of these graphs revealed discontinuities whenever the number of independent samples went down, e.g., from three to two, and because of small-sample properties, the discontinuities became much greater toward the end of the graphs, because that is where the number of sample became smaller.

Lo [1991, pp. 1289-1291] developed the adjusted R/S statistic, Q_n, replacing the denominator of the R/S with the square root of the sum of the sample variance and weighted covariance terms, and which has the property that its statistical behavior is invariant over a general class of short memory processes but deviates for long memory processes.  Lo [1991] and Lo and MacKinlay [1988] used this technique to find little evidence of long memory in stock prices.  Lo's Qn, are reported in column five of Table 1.  Lo's Q-statistic has the advantage that it is robust against short-term dependence, provided Lo's underlying assumptions are satisfied.

Statistically significant Q-statistics indicate rejection of the null hypothesis of no long-term dependence or long memory.   It is important to note [Corazza, Malliaris, and Nardelli, 1997, p. 455]  rejection of the null does not necessarily imply long memory but more precisely that the underlying stochastic process does not simultaneously satisfy all the conditions specified by Lo [1991, p. 1282]. The largest lag order of statistically significant Q-statistics gives the average non-periodic cycle length, beyond which most long memory (memory of initial conditions) is lost.

The Lo analysis identifies non-periodic cycles ranging from 7 to 22 months for various countries.  Contrast this result with the classical R/S technique which is unable to identify cycle length in over twenty-five years of data.

Strong evidence of short-term dependence is found with as few as seven months of data for Canada and the U.K., and as many as twenty-two for Austria and Sweden.   This suggests a high level of independence between the U.S. dollar and the freely-floating Canadian dollar and the U.K. pound, and a relatively high level of dependence of the value of most other foreign currencies on the dollar, over average periods of up to 22 months.

Lo provided a procedure, known as "Rolling Lo," [see Cheung and Lai, 1993, pp. 190-193,] which performs the Lo analysis using every possible starting point in the data set.  This technique is not undertaken here, but is of special interest because it would be robust to monetary policy regime shifts.

 Table 1 presents a comparison of different R/S methodologies.  Hurst's empirical rule, gives H's reported in column two, that broadly accord with earlier findings with daily data, such as Peters [1996].  Peters found H = 0.64 for Japan and Germany, H = 0.61 for the U.K., and H = 0.50 for Singapore.  From the monthly data, India, Sri Lanka, and Malaysia have H's approximately equal to 0.50.  It is not surprising that India and Sri Lanka have similar H's, even in the absence of a pegging arrangement.  Sri Lanka administers a managed float with limited flexibility.

Peters found H = 0.50 for Singapore, a country which pegs to the dollar.  The finding that India and Sri Lanka, countries which do not peg to the dollar, have H's near 0.50, supports the weak form of the Efficient Market Hypothesis, but only for those two countries, not the remaining 22. However, caution is warranted in interpreting these negative results: systematic bias may have been introduced into the average monthly returns by the presence of systematic, non-random interest rate differentials.

The empirical rule tends to overstate H if H > 0.70 and understate H for H < 0.40.  It is computed here using the largest R/S for comparison only.  The empirical law is an approximation to be used when too few observations are available to allow computation of more than one R/S value.  Here, many R/S's can be computed and averaged, for various sample sizes.  Even the largest R/S is the average over two samples. Virtually all H's computed with the empirical rule were in the region of downward bias, suggesting the true H is closer to 0.50 if the assumption of serial independence is valid.

H's computed by Mandelbrot and Wallis's [1969] conventional technique are reported in column three with standard errors in parentheses.  These H's are estimated by regression of log (n) on log (R/S).  All H's are greater than one, and virtually all are very close to two.  This result contrasts markedly with earlier published findings.

Interpreted literally, H's near two indicate data generating processes approximating the LJvy distribution governing tosses of a fair coin.  This result appears to support strong market efficiency, but is apparently entirely due to bias caused by short memory.

This unexpected outcome may be due to serial correlation distinct from long-term memory.  A relatively short order of autocorrelation may have been introduced by taking the average of thirty daily exchange rates.  This is plausible if the daily data is autocorrelated approximately ten days or longer, which might make the monthly averages autocorrelated of order one or, at most, two months.

Mandelbrot and Wallis [1969], Mandelbrot [1972a], and Peters [1996] recommend H be estimated with AR1 residuals to avoid this problem with serial correlation.  H's computed on the AR1 residuals are reported in column four with standard errors in parenthesis.  These H's may still be biased, but at least some bias due to nonstationarity or short-memory has been removed.  Most H's are in the neighborhood of 0.60 with standard errors in the neighborhood of 0.010. Malaysia has the highest H at 0.758.

Canada is the only country with an H not statistically different from 0.50, indicating a random walk.  This is surprising in light of the fact that Canada does not peg to the U.S. dollar and may stem from the high level of integration between the two economies.  This result may indicate U.S.-Canadian interest-rate differentials are both random and small enough to remove them as a source of systematic bias.

In addition, the flow of trade between the U.S. and Canada is the largest in the world and U.S. trading in the Canadian dollar, and Canadian trading in the U.S. dollar are much less dominated by speculative trading than other currencies.  The foreign exchange market should, however, be more efficient and more random, the higher the percentage of speculative trades. Another unique feature of the Canadian economy is the extent Canadians practice currency substitution with the U.S. dollar.

In their study of eurocurrency deposit rates, Barkoulas and Baum [1997, p. 363] found the Euro-Canadian dollar rate was the only rate their long-memory model was unable to predict better than a linear forecasting model.  The present result is consistent with this finding.

H's different from 0.50 apparently demonstrate exchange rates are not random walks, shedding some doubt on weak market efficiency and indicating technical analysis of exchange rates can provide systematic returns.  Nevertheless, this finding may be due to short-term dependence still present after taking AR1 residuals, or systematic bias due to interest-rate differentials, or both.

Neither the conventional or AR1 R/S analysis indicated the average non-periodic cycle length.  Failure to detect a non-periodic cycle in twenty-four years of data suggests the average cycle length, if it exists, is twelve years or longer, however, short-term dependence may mask the truly interesting long memory.
The Lo analysis detects long memory distinct from short-term dependence, including serial correlation and Markovian dependence, which biases conventional estimates of the Hurst exponent.  Q-statistics and the order of long-term dependence are reported in column five, along with asterisks indicating ten and five percent significance. The Lo analysis demonstrates that at least some bias in the estimates of H is due to short-term dependence, and not to interest-rate differentials.

The null hypothesis tested by Lo's Qn is no long memory, and is rejected at the ten percent significance level for all samples.  The null hypothesis is rejected at lower significance levels for larger samples.  Using the one percent significance level, the average cycle length is the largest n for which Q is not significant at the one percent level.  Q_n is reported for different n's in increments of five, and the largest n significant at the five percent level, but not the one percent level.  This largest n reported for each country is the average non-periodic cycle length.

The Lo analysis indicates definite average non-periodic cycle lengths of seven months for Canada and the U.K.; eight for Belgium; twelve for Australia, Denmark, the E.U., and South Africa; fourteen for Spain; fifteen for New Zealand; sixteen for India, Italy, Norway, and Portugal; seventeen for Sri Lanka; nineteen for Finland, France, Germany, Ireland, Japan, and Malaysia; twenty-one for the Netherlands; and  twenty-two for Austria and Sweden.

Average non-periodic cycle should not be interpreted as a precise measurement.  The short Canadian cycle suggests a high level of integration with the U.S. economy.  The U.K. cycle is shortest of all E.U. countries suggesting a high level of independence for the U.K. from the rest of Europe, but not from the U.S.  Countries with highly integrated economies should have similar cycle lengths - e.g. India (16 months) and Sri Lanka (17 months); Australia (12 months) and New Zealand (15 months); Japan and Malaysia (both 19 months.)  Lo's modified R/S analysis says nothing about whether economies with the same cycle length run in phase, however.

Lo's analysis is robust to variance non-stationarity, but not to mean non-stationarity.  Logarithmic returns are mean stationary by construction.  In addition, all the log return series are subjectively mean stationary by inspection.

Currencies pegged to the dollar should have Hurst exponents approximately equal to 0.50, indicating the exchange rate changes in a purely random, normally-distributed manner.  Currencies not pegged should display time persistence with H > 0.50, unless market efficiency imposes randomness and normality anyway. Currencies with unbroken free float against the dollar since 1973 should provide the best estimates of H.

This study finds no support for market efficiency except for Canada, and the evidence for EMH in Canada must be qualified by the high likelihood of biased estimates of H from AR1 R/S.   Using Lo's modified R/S, this paper finds much shorter average non-periodic cycles than previously suspected - seven months for Canada and the U.K. The longest cycle length found was twenty-two months for Austria and Sweden.  Long memory and non-periodic cycles were found for all twenty-two countries studied.

Countries with regime shifts, such as Argentina and Mexico, and particularly the former communist countries, present special difficulties of interpretation.  Though their exchange rates may have different H's and cycle lengths for different policy regimes, it may be impossible to measure H when the duration of the regime is too small compared to the cycle length.

The United States' monetary policy can clearly be divided into two regimes: pre- and post-Volker, breaking in June 1979.  This imposes at least one regime shift on the dollar exchange rate for any other country, in addition to shifts due to foreign monetary policy.  The "Rolling Lo" technique may be used to sort out regime shifts, and Hurst's empirical law allows an approximate measure of H for short data series.

No support is found for weak market efficiency, except for Canada, and only with the short-term- dependency-biased H estimated from AR1 residuals.  This paper finds no support for Mussa's [1984] equilibrium overshooting model, but this could easily be due to the use of monthly data.

The results presented here show conventional estimates of H are biased by short-term memory violating the assumption of serial independence of subsamples, perhaps severely.

Long memory has been clearly demonstrated for foreign exchange markets for all twenty-two countries studied..  Average non-periodic cycle lengths are found to vary greatly across countries, and to be much shorter than conventional R/S analysis suggested - always less than two years. Unfortunately, no definitive indication of the character of this long memory has been found.  Additional work needs to be done to fully characterize the short and long memory processes present in exchange rates.

In fractal analysis of capital markets, H indicates the relationship between the amount of an initial investment R and a constant amount which can be withdrawn or reinvested, the average yield over various samples, providing a steady income without depleting the portfolio, over all past observations.  Note there is no guarantee against future bankruptcy.

The first step in R/S analysis consists of constructing the logarithmic return on an asset:  Xt = ln(Pt/Pt-1), where Xt is the logarithmic return on the asset at time t, and Pt  is the price of the asset at time t.  In this context, Pt is the average exchange rate for any particular month, and Xt is the logarithmic average return for holding a currency from one month to the next.  Logarithmic returns are more appropriate for R/S analysis than percent changes in prices because the range used in R/S analysis is the cumulative deviation from the average return, and the logarithmic returns sum to the cumulative return.

Next, the R/S time series is constructed for all sample periods ranging from an arbitrary minimum to the largest sample size allowing the data to be partitioned into at least two subsamples.  The minimum sample size here is six months.

The data is initially partitioned into as many sequential six-month subsamples as possible.  In the absence of short-term dependence, (e.g., serial correlation,) each subsample is independent.  For each six-month subsample, the range (R) and the standard deviation (S) are calculated to form the rescaled range or R/S.  The range is rescaled by dividing by the sample standard deviation.  An R/S is computed for each six-month subsample, and the average is taken as the observation of R/S for n = 6.

The procedure is repeated for n = 7, and so on, until n equals one-half the number of observations of the logarithmic return time series.  This procedure provides a time series of average R/S's.   The Hurst exponent H is defined from the relationship R/S = anH.  H defines the average relationship, in past data, between the rescaled range R/S and elapsed time.

The un-rescaled range R is the actual return on foreign currency holdings.  The standard deviation S and time n define the average return, the desired steady return which may be withdrawn from the investment in each period without depleting the investment.
To estimate H, the logarithm of R/S is graphed on the vertical axis, against the logarithm of the number of observations on the horizontal axis.  The slope of the linear part of the graph gives an estimate of the Hurst exponent.  The extent of the linear part, measured by the number of observations, gives the average non-periodic cycle length.

Mandelbrot and Wallis [1969], Mandelbrot [1972a], and Peters [1994] note the estimate of H may be biased in two major circumstances: nonstationarity of the data and/or short-memory.  To overcome either or both of these sources of bias, they recommended graphing the logarithm of the AR1 residuals of the original regression on the vertical axis against the logarithm of n on the horizontal axis.  H's estimated on AR1 residuals are less biased, but often do not have all short-term dependency bias removed.

Lo [1991, pp. 1289-1291] developed an adjusted R/S statistic, Qn, which replaces the denominator of the R/S with the square root of the sum of the sample variance and weighted covariance terms, and which has the property that its statistical behavior is invariant over a general class of short memory processes but deviates for long memory processes.  Lo [1991] and Lo and MacKinlay [1988] used this technique to find little evidence of long memory in stock prices.  Lo's modified R/S analysis is presented in this paper, with Q-statistics and their significance levels presented in Table 1.  Q tests the null hypothesis of no long memory, and is robust against short-term dependence, including serial-correlation, ARIMA, and Markov processes.

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