A Fractal Analysis of Foreign Exchange Markets.
Long memory in foreign exchange markets is examined for the post-Bretton Woods period using Lo's  modified rescaled range (R/S). Conventional R/S 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 22 countries studied, indicating that 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 nonperiodic cycle ranges from 7 months for Canada and the United Kingdom, to approximately 20 months for Austria, Finland, France, Germany, Ireland, Japan, Malaysia, Netherlands, Sweden, and Switzerland. No support is found for the efficient market hypothesis. Results broadly agree with those pr ovided 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 nonperiodic 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.
Rescaled range (R/S) analysis distinguishes random from nonrandom or deterministic series. The rescaled range is the range divided (rescaled) by the standard deviation. Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and nonrandom components. Conventional statistical techniques lack power to distinguish 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 nonperiodic cyclic long-run dependence as distinguished from short dependence or Markov character and periodic variation [Mandelbrot, 1972a, pp. 259-60]. 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 with pegged currencies. Thus, fractal analysis also indicates the extent to which intervention characterizes the series.
Fractal analysis can also identify ergodic or antipersistent series, for example, 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 suboptimal policy rules that delay intervention, overstate the amount required, or both.
Four techniques are reported in this paper, Hurst's  empirical rule, Mandelbrot and Wallis's  classic, naive R/S, Mandelbrot's [1972a] AR1 R/S, and Lo's  modified R/S. A related technique, Peters's  [V.sub.n], 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 with the classic and AR1 R/S techniques, 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. This 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 7 to 22 months. This finding is far shorter than previously suspected.
The remainder of this 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, and concluding remarks are provided in the sixth section. A complete discussion of the methodology is presented in the 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  shifted his attention from income distribution to speculative prices. Long memory in exchange rates has been studied by Booth et al. , Cheung , Cheung and Lai , Cheung et al. , Peters , Fisher et al. , Barkoulas and Baum [1997c], and Chou and Shih . Andersen and Bollerslev  also found long memory m dollar-deutschemark exchange rates using a generalized autoregressive conditional heteroskedasticity model. Barkoulas and Baum [1997a, 1997b] found long memory m 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 et al. 1998], and derivatives [Fang et al. 1994; Corazza, Malliaris et al. 1997; Barkoulas et al. 1997].
Exchange rate distribution and modeling have received much attention in literature [Cheung et al. 1995; Baum and Barkoulas, 1996; Byers and Peel, 1996; Gazioglu, 1996; Fritsche and Wallace, 1997]. Cheung and Lai  suggest Heiner's  and Kaen and Rosenman's  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 that persistent price movements may suddenly reverse direction, leading to nonperiodic cycles. Program trading introduces the same phenomenon.
A different kind of long memory is suggested by Mussa's  disequilibrium overshooting model, which is based on the contracting approach to introducing monetary nonneutralities into macroeconomic models developed by Fisher , Phelps and Taylor , and Taylor . Mussa's model would be supported by finding antipersistence or ergodicity in exchange rates.
Mandelbrot [1972b, 1974] and Mandelbrot et al.  have developed the multifractal model of asset returns (MMAR). MMAR shares the long memory feature of the FBM model introduced by Mandelbrot and van Ness . The statistical theory necessary to identify empirical regularities and local scaling properties of MMAR processes with local Holder exponents is developed by Calvet et al.  and applied to dollar-deutschemark exchange rates by Fisher et al. .
Mandelbrot's [1972a, 1975, 1977] and Mandelbrot and Wallis's  R/S 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 FBM;
3) random walks; or
4) antipersistent, 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 = [an.sup.H], where R is the average range of all subsamples of size n, S is the average standard deviation for all samples of size n, a is a scaling variable, and n is the size of the subsamples, which is allowed to range from an arbitrarily small value (here, six months) to the largest subsample the data will allow. Putting this expression in logarithms yields log (R/S) = log(a) + Hlog (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 0 to 0.50 for antipersistent series, and is greater than 1 for a persistent or autocorrelated series. Mandelbrot et al.  refer to H as the self-affinity index or scaling exponent.
Because H is the reciprocal of the Mandelbrot-Levy  characteristic exponent, [alpha], estimates of H indicate the probability distribution underlying a time series. H = 1/[alpha] = 1/2 for normally-distributed or Gaussian processes. H = 1 for Cauchy-distributed processes. H = 2 for the Levy 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 that 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 nonperiodic cycle length, the number of observations after which memory of initial conditions is lost, that is, how long it takes for a single outlier's influence to become immeasurably small. If foreign exchange rates are random walks with H = 0.50, then returns are purely random and should lead to investors breaking even over the long run.
If exchange rates are persistent with (0.50 [less than]H [less than] 1.00), then the series are less noisy, exhibiting clearer trends and more persistence the closer H is to 1, and investors should earn positive returns. Neely et al.  found technical trading rules, formalized with a genetic programming algorithm, and provided significant out-of-sample excess returns. Hs close to 1 indicate a high risk of large, abrupt changes, for example, H = 1.00 for the Cauchy distribution.
Finally, if exchange rates are antipersistent, ergodic, or mean-reverting with (0.00 [less than] H [less than] 0.50), then they are more volatile than a random walk. If the highly volatile returns are uncorrelated across different assets, then risk can be minimized by diversification. Ergodicity would support Mussa's  equilibrium overshooting model.
In applying his modified R/S analysis to equity prices, Lo  overturned earlier results based on classical R/S methods finding long memory. In this paper, the Lo technique is applied to exchange rates.
Although earlier attempts at R/S analysis of foreign exchange markets were highly informative (for example, Booth et al. , Cheung , and Peters ), approximately 2 cycle lengths of data are necessary for good estimates of Hurst exponents and average nonperiodic cycle length using classical R/S techniques [Mandelbrot, 1972a; Peters, 1996]. Peters suggested that the cycle length for exchange rates was at least 10 years. More than 25 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 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: program traders, investors who follow deterministic rules, investors who routinely accept exposure approximately one month or longer, and 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 (EMH) 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, then monthly averages are more likely to show short-term dependence than end-of-month observations.
Andersen and Bollerslev [1997, p. 975] used a generalized autoregressive conditional heteroskedasticity specification to model a one-year series of five-minute dollar-deutschemark exchange rates. They 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 . This 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, nonrandom interest rate differentials may introduce systematic bias in the monthly average returns, and any implications for the EMH must be interpreted in this light.  Specifically, nonrandomness in monthly exchange rates cannot disconfirm the EMH 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, [X.sub.t] = In ([P.sub.t]/[P.sub.t-1]), where [X.sub.t] is the logarithmic return on holding the foreign currency at time t, and [P.sub.t] 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  gives an empirical law for use when too few R/S observations are available. This expression, H = [log(R/S)]I[log(n/2)], tends to overstate H if H [greater than] 0.70 and understate H if H [less than] 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 Column 2 of Table 1 and broadly accord with earlier findings using daily data. This suggests that 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.sup.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 unrescaled 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, or the desired steady return which may be withdrawn from the investment each period without depleting the investment.
Mandelbrot and Wallis  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 3 of Table 1. All reported values are close to 2, suggesting highly biased results. H = 2 for the Levy distribution governing tosses of a fair coin. It implies perfect randomness in returns and supports market efficiency.
Mandelbrot and Wallis , Mandelbrot [1972a], and Peters  note the estimate of H may be biased in two major circumstances: nonstationarity of the data or short memory. To overcome either or both sources of bias, they recommend performing RIS analysis on AR1 residuals of the logarithmic returns. This represents a major refinement, removing much, though not all, bias due to short-term dependence.
Hs estimated from AR1 residuals are reported in Column 4 of Table 1. These values are all smaller than 1, suggesting greater plausibility than naively estimated Hs. 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  show that regression estimates of H tend to be biased toward rejection of the null hypothesis of no long memory even for stationary AR1 processes.
Peters  also recommends graphing [V.sub.n] = (R/S)/[(n).sub.1/2] against log(n) to better identify the nonperiodic 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, for example, from 3 to 2, and because of small-sample properties, with the discontinuities becoming much greater toward the end of the graphs because that is where the number of samples became smaller.
Lo [1991, pp. 1289-91] developed the adjusted R/S statistic, [Q.sub.n], replacing the denominator of the R/S with the square root of the sum of the sample variance and weighted covariance terms. It also has the property that its statistical behavior is invariant over a general class of short memory processes but deviates for long memory processes. Lo  and Lo and MacKinlay  used this technique to find little evidence of long memory in stock prices. Lo's [Q.sub.n] figures are reported in Column 5 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 longterm dependence or long memory. It is important to note that 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] [Corazza et al., 1997, p. 455]. The largest lag order of statistically significant Q-statistics gives the average nonperiodic cycle length, beyond which most long memory (memory of initial conditions) is lost.
The Lo analysis identifies nonperiodic 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 more than 25 years of data.
Strong evidence of short-term dependence is found with as few as 7 months of data for Canada and the United Kingdom, and as many as 22 for Austria and Sweden. This suggests a high level of independence between the U.S. dollar and the free-floating Canadian dollar and the United Kingdom pound. It also suggests 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. 1903]). It performs the Lo analysis using every possible starting point in the data set. This technique is not undertaken here but is especially interesting 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 Hs, reported in Column 2, that broadly accord with earlier findings with daily data such as Peters . Peters found H = 0.64 for Japan and Germany, H = 0.61 for the United Kingdom, and H = 0.50 for Singapore. From the monthly data, India, Sri Lanka, and Malaysia have Hs approximately equal to 0.50. It is not surprising that India and Sri Lanka have similar Hs 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 Hs near 0.50 supports the weak form of the EMH, but only for those two countries. 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, nonrandom interest rate differentials.
The empirical rule tends to overstate H if H [greater than] 0.70 and understate H if H [less than] 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/Ss can be computed and averaged for various sample sizes. Even the largest R/S is the average over two samples. Virtually all Hs 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.
Hs computed by Mandelbrot and Wallis's  conventional technique are reported in Column 3. These Hs are estimated by regression of log(n) on log(R/S). All Hs are greater than 1, and virtually all are very close to 2. This result contrasts markedly with earlier published findings.
Interpreted literally, Hs near 2 indicate data generating processes, approximating the Levy 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 30 daily exchange rates. This is plausible if the daily data is autocorrelated approximately 10 days or longer, which might make the monthly averages autocorrelated of order 1 or, at most, 2 months.
Mandelbrot and Wallis , Mandelbrot [1972a], and Peters  recommend that H be estimated with AR1 residuals to avoid this problem with serial correlation. Hs computed on the AR1 residuals are reported in Column 4. These Hs may still be biased, but at least some bias due to nonstationarity or short memory has been removed. Most Hs 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 that U.S. and 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 that Canadians practice currency substitution with the U.S. dollar.
In their study of eurocurrency deposit rates, Barkoulas and Baum [1997a, p. 363] found the Euro-Canadian dollar rate was the only rate that their long-memory model was unable to predict better than a linear forecasting model. The present result is consistent with this finding.
Hs different from 0.50 apparently demonstrate that exchange rates are not random walks, shedding some doubt on weak market efficiency and indicating that 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, systematic bias due to interest rate differentials, or both.
Neither the conventional or AR1 R/S analysis indicated the average nonperiodic cycle length. Failure to detect a nonperiodic cycle in 24 years of data suggests that the average cycle length, if it exists, is 12 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 5 along with indications of 5 and 10 percent significance. The Lo analysis demonstrates that at least some bias in the estimates of H is due to short-term dependence, not interest rate differentials.
The null hypothesis tested by Lo's [Q.sub.n] is no long memory and is rejected at the 10 percent significance level for all samples. The null hypothesis is rejected at lower significance levels for larger samples. Using the 1 percent significance level, the average cycle length is the largest n for which Q is not significant at the 1 percent level. [Q.sub.n] is reported for different ns in increments of five, and the largest n is significant at the 5 percent level but not the 1 percent level. This largest n reported for each country is the average nonperiodic cycle length.
The Lo analysis indicates definite average nonperiodic cycle lengths of 7 months for Canada and the United Kingdom, 8 for Belgium, 12 for Australia, Denmark, the European Union, and South Africa, 14 for Spain, 15 for New Zealand, 16 for India, Italy, Norway, and Portugal, 17 for Sri Lanka, 19 for Finland, France, Germany, Ireland, Japan, and Malaysia, 21 for the Netherlands, and 22 for Austria and Sweden.
"Average nonperiodic 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 United Kingdom cycle is the shortest of all the European Union countries, suggesting a high level of independence for the United Kingdom from the rest of Europe but not from the U.S. Countries with highly integrated economies should have similar cycle lengths, for example, India (16 months) and Sri Lanka (17 months), Australia (12 months) and New Zealand (15 months), and 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 nonstationarity but not to mean nonstationarity. 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 Hs 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 [greater than] 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 nonperiodic cycles than previously suspected, 7 months for Canada and the United Kingdom. The longest cycle length found was 22 months for Austria and Sweden. Long memory and nonperiodic cycles were found for all 22 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 Hs and cycle lengths for different policy regimes, it may be impossible to measure H when the duration of the regime is too small as compared with the cycle length.
The U.S. 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  equilibrium overshooting model, but this could easily be due to using 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 22 countries studied. Average nonperiodic cycle lengths are found to vary greatly across countries and are much shorter than conventional R/S analysis suggested, always less than 2 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 that are present in exchange rates.
(*.) Western Carolina University--U.S.A. This paper was presented at the Forty-Sixth International Atlantic Economic Conference, Boston, MA, October 8-11, 1998. The author is profoundly indebted to the session discussant, Takashi Kamihigashi, and to Nicholas Apergis for many helpful comments, to colleague Patrick Allen Hays who provided FORTRAN programs to estimate H and perform the Lo analysis and who provided immeasurably invaluable advice and support, to student Mark Douglas Wells, Jr. who assisted as part of the undergraduate honors project in the Honors College of Western Carolina University, and to an anonymous referee whose comments greatly improved the paper. Responsibility for any shortcomings belong to the author.
(1.) The Mandelbrot-Levy family of probability distributions is also referred to as Levy-stable, L-stable, stable, stable-Paretian, and Pareto-Levy. Samuelson  popularized the term Mandelbrot-Levy, but Mandelbrot avoids this expression, perhaps out of modesty, and the other terms remain current.
(2.) Federal Reserve Economic Data (or FRED(R)) can be found on the internet at http://www.stls.frb.org/fred/data/exchange.html.
(3.) The author is particularly indebted to Professor Takashi Kamihigashi for this insight.
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Estimates of H for Various Countries H H H Country Empirical Rule Naive R/S AR1 Residuals (2) (3) (4) Australia 0.431 2.080 0.546 (73.01 - 98.01) (0.019) (0.012) Austria 0.750 1.889 0.625 (73.01 - 98.01) (0.034) (0.010) Belgium 0.344 2.023 0.623 (73.01 - 98.01) (0.017) (0.010) Canada 0.347 2.016 0.505 (73.01 - 98.01) (0.015) (0.005) Denmark 0.347 2.187 0.621 (73.01 - 98.01) (0.034) (0.009) European Union 0.380 2.137 0.679 (80.01 - 98.01) (0.025) (0.009) Finland 0.396 2.072 0.585 (73.01 - 98.01) (0.021) (0.007) France 0.356 1.992 0.663 (73.01 - 98.01) (0.015) (0.014) Germany 0.373 2.138 0.609 (73.01 - 98.01) (0.020) (0.012) [Q.sub.n] Country Modified R/S (5) Australia [Q.sub.5] = 2.628 [**] (73.01 - 98.01) [Q.sub.10] = 1.957 [*] [Q.sub.12] = 1.807 [*] Austria [Q.sub.5] = 3.357 [**] (73.01 - 98.01) [Q.sub.10] = 2.501 [**] [Q.sub.15] = 2.091 [*] [Q.sub.20] = 1.840 [*] [Q.sub.22] = 1.764 [*] Belgium [Q.sub.5] = 2.189 [**] (73.01 - 98.01) [Q.sub.8] = 1.807 [*] Canada [Q.sub.5] = 2.112 [**] (73.01 - 98.01) [Q.sub.7] = 1.844 [*] Denmark [Q.sub.5] = 2.531 [**] (73.01 - 98.01) [Q.sub.10] = 1.902 [**] [Q.sub.12] = 1.762 [*] European Union [Q.sub.5] = 2.658 [**] (80.01 - 98.01) [Q.sub.10] = 1.980 [*] [Q.sub.12] = 1.765 [*] Finland [Q.sub.5] = 3.045 [**] (73.01 - 98.01) [Q.sub.10] = 2.294 [**] [Q.sub.15] = 1.944 [*] [Q.sub.19] = 1.771 [*] France [Q.sub.5] = 3.244 [**] (73.01 - 98.01) [Q.sub.10] = 2.414 [**] [Q.sub.15] = 2.018 [*] [Q.sub.19] = 1.749 [*] Germany [Q.sub.5] = 3.108 [**] (73.01 - 98.01) [Q.sub.10] = 2.318 [**] [Q.sub.15] = 1.940 [*] [Q.sub.19] = 1.766 [*] India 0.528 2.134 0.584 [Q.sub.5] = 2.924 [**] (75.01 - 98.01) (0.015) (0.008) [Q.sub.10] = 2.179 [**] [Q.sub.15] = 1.825 [*] [Q.sub.16] = 1.774 [*] Ireland 0.374 2.043 0.611 [Q.sub.5] = 3.104 [**] (73.01 - 98.01) (0.015) (0.011) [Q.sub.10] = 2.326 [**] [Q.sub.15] = 1.958 [*] [Q.sub.19] = 1.773 [*] Italy 0.378 2.133 0.611 [Q.sub.5] = 2.845 [**] (73.01 - 98.01) (0.020 (0.011) [Q.sub.10] = 2.133 [**] [Q.sub.15] = 1.800 [*] [Q.sub.16] = 1.747 [*] Japan 0.366 1.980 0.565 [Q.sub.5] = 3.094 [**] (73.01 - 98.01) (0.020) (0.009) [Q.sub.10] = 2.318 [**] [Q.sub.15] = 1.951 [*] [Q.sub.19] = 1.766 [*] Malaysia 0.477 2.041 0.758 [Q.sub.5] = 3.145 [**] (73.01 - 98.01) (0.026) (0.013) [Q.sub.10] = 2.351 [**] [Q.sub.15] = 1.972 [*] [Q.sub.19] = 1.782 [*] Netherlands 0.343 2.089 0.608 [Q.sub.5] = 3.294 [**] (73.01 - 98.01) (0.017) (0.015) [Q.sub.10] = 2.451 [**] [Q.sub.15] = 2.048 [*] [Q.sub.20] = 1.802 [*] [Q.sub.21] = 1.764 [*] New Zealand 0.436 2.015 0.562 [Q.sub.5] = 2.830 [**] (73.01 - 98.01) (0.016) (0.007) [Q.sub.10] = 2.115 [**] [Q.sub.15] = 1.775 [*] Norway 0.355 1.960 0.611 [Q.sub.5] = 2.906 [**] (73.01 - 98.01) (0.017) (0.013) [Q.sub.10] = 2.181 [**] [Q.sub.15] = 1.837 [*] [Q.sub.16] = 1.788 [*] Portugal 0.430 2.022 0.614 [Q.sub.5] = 2.829 [**] (75.01 - 98.01) (0.016) (0.009) [Q.sub.10] = 2.127 [**] [Q.sub.15] = 1.795 [*] [Q.sub.16] = 1.747 [*] South Africa 0.462 1.988 0.640 [Q.sub.5] = 2.474 [**] (73.01 - 98.01) (0.028) (0.008) [Q.sub.10] = 1.884 [*] [Q.sub.12] = 1.753 [*] Sri Lanka 0.553 2.073 0.520 [Q.sub.5] = 2.993 [**] (75.01 - 98.01) (0.022) (0.009) [Q.sub.10] = 2.240 [**] [Q.sub.15] = 1.882 [*] [Q.sub.17] = 1.784 [*] Spain 0.430 2.006 0.601 [Q.sub.5] = 2.725 [**] (75.01 - 98.01) (0.017) (0.010) [Q.sub.10] = 2.050 [*] [Q.sub.14] = 1.783 [*] Sweden 0.404 2.033 0.561 [Q.sub.5] = 3.334 [**] (73.01 - 98.01) (0.019) (0.014) [Q.sub.10] = 2.485 [**] [Q.sub.15] = 2.079 [*] [Q.sub.20] = 1.834 [*] [Q.sub.22] = 1.727 [*] Switzerland 0.348 2.029 0.666 [Q.sub.5] = 3.111 [**] (73.01 - 98.01) (0.021) (0.015) [Q.sub.10] = 2.324 [**] [Q.sub.15] = 1.949 [*] [Q.sub.19] = 1.761 [*] United Kingdom 0.412 2.074 0.649 [Q.sub.5] = 1.992 [*] (73.01 - 98.01) (0.014) (0.010) [Q.sub.7] = 1.751 [*] Notes: (*.) and (**.)denotes 5 and 10 percent significance, respectively Standard errors are in parentheses.
The R/S Methodology and its Refinements
In fractal analysis of capital markets, H indicates the relationship between the amount of an initial investment R and a constant amount that 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,[X.sub.t] = ln([P.sub.t]/[P.sub.t-1]), where [X.sub.t] is the logarithmic return on the asset at time t, and [P.sub.t] is the price of the asset at time t. In this context, [P.sub.t] is the average exchange rate for any particular month, and [X.sub.t] 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 (for example, serial correlation), each subsample is independent. For each six-month subsample, the range (R) and the standard deviation (S) are calculated to form the 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/Ss. The H is defined from the relationship R/S = [an.sup.H]. H defines the average relationship in past data between the R/S and elapsed time.
The unrescaled range R is the actual return on foreign currency holdings. The standard deviation S and time n define the average return, or 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 nonperiodic cycle length.
Mandelbrot and Walls , Mandelbrot [1972a], and Peters  note the estimate of H may be biased in two major circumstances: nonstationarity of the data 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. Hs estimated on AR1 residuals are less biased, but often do not have all short-term dependency bias removed.
Lo [1991, pp. 1289-91] developed an adjusted R/S statistic, [Q.sub.n]. This replaces the denominator of the R/S with the square root of the sum of the sample variance and weighted covariance terms and has the property that its statistical behavior is invariant over a general class of short memory processes but deviates for long memory processes. Lo  and Lo and MacKinlay  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, autoregressive integrated moving average, and Markov processes.
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|Author:||MULLIGAN, ROBERT F.|
|Publication:||International Advances in Economic Research|
|Date:||Feb 1, 2000|
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