A Fractal Analysis of Foreign Exchange Markets
Mulligan, Robert F., International Advances in Economic Research
ROBERT F. MULLIGAN [*]
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. …