Long memory features evolve in the time-varying process in Asia-Pacific foreign exchange markets

Available online at www.sciencedirect.com ScienceDirect Procedia Economics and Finance 14 ( 2014 ) 286 294 International Conference on Applied Economics (ICOAE) 2014 Long memory features evolve in the time-varying process in Asia-Pacific foreign exchange markets Sang Hoon Kang a, Ron Mclver b, Sung-Yong Park c, Seong-Min Yoon c, * a Department of Business Adminstration, Pusan National Univerity, Busan, Korea b School of Commerce, University of South Australia, Adelaide, Australia c Department of Economics, Pusan National University, Busan, Korea Abstract We investigated the presence of, and changes in, long memory features in the returns and volatility dynamics of six Asia- Pacific foreign exchange markets (Australian dollar, Japanese yen, Korean won, New Zealand dollar, Singaporean dollar, and Taiwan dollar) using time-varying Hurst exponents. In particular, instead of relying on a single static measure of long memory, we explored time-varying long memory features over time to assess changes in market efficiency by analyzing the returns and volatility of the markets. Furthermore, considering a time-varying rolling approach, we estimated values of the Hurst exponent for time windows with 1,000 observations (about 4 years of data) in each window. The estimation results indicated that both the returns and the volatility series possessed strong long memory features and that the degree of the long memory features changed over time. Additionally, the Hurst exponent showed an upward trend during the 1997 Asian currency crisis and 2008 global financial crisis, indicating that exchange rate markets became inefficient and predictable during the financial crisis. 2014 The Authors. Published by by Elsevier Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and/or peer-review under responsibility of the Organising Committee of ICOAE 2014. Selection and/or peer-review under responsibility of the Organizing Committee of ICOAE 2014 Keywords: Hurst exponent, time-varying long memory, efficient market hypothesis, Asia-Pacific foreign exchange markets ; * Corresponding author. Department of Economics, Pusan National University, Jangjeon 2-dong, Geumjeong-gu, Busan, 609-735, Republic of Korea, Tel.: +82-51-510-2557; fax: +82-51-581-3143. E-mail address: smyoon@pusan.ac.kr (S.-M. Yoon). 2212-5671 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and/or peer-review under responsibility of the Organizing Committee of ICOAE 2014 doi:10.1016/s2212-5671(14)00714-x

Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 287 1. Introduction The efficient market hypothesis (EHM) requires that the arrival of new information be promptly arbitraged away; that is, the dynamics of asset returns are unpredictable and follow a random-walk process (Fama, 1970). Thus, according to the theory, it is impossible to consistently outperform the market by using any information. The weak-form EHM suggests that excess returns cannot be earned in the long run by using any form of technical analysis based on historical prices or volume. In turn, this means that prices exhibit no serial dependency and so future price movements are determined entirely by information not contained in the price series, referring to a random-walk process. However, it is commonly observed that asset returns do not follow a truly random-walk process but exhibit strong dependence or long memory features (Lo and MacKinlay, 1997; Lo, 1991; Peters, 1991, 1994). Long memory means that the statistical dependence of asset returns decays at a very slow mean-reverting hyperbolic rate. Such a phenomenon raises questions about the appropriateness of asset pricing models based on EMH (Barkoulas, Baum and Travlos, 2000; Sadique and Silvapulle, 2001; Henry, 2002). Lo (1991) pointed out that the presence of long memory causes several drawbacks: (1) optimal consumption/savings and portfolio decisions may become extremely sensitive to the investment horizon, (2) derivative pricing models based on Martingale methods, such as the Black-Scholes model, are no longer accurate, and (3) traditional tests of the capital asset pricing model (CAPM) and arbitrage pricing theory (APT) are no longer valid. Moreover, evidence of long memory directly contradicts the validity of weak-form efficiency because its presence is closely connected with price predictability. The presence of long memory is an important topic in both theoretical and empirical research because its presence is closely connected with the predictability in first and/or second moments of the price distribution. Despite extensive research on long memory in international equity markets, the issue remains unresolved, and recent empirical studies have re-examined the presence or absence of long memory in returns and volatility (Kiliç, 2004; Vougas, 2004; Christodoulou-Volos and Siokis, 2006; Assaf, 2008; McMillan and Thupayagale, 2008; Cajueiro, Gogas and Tabak, 2009; Kasman, Turgutlu and Ayhan, 2009; Kasman, Kasman and Torun, 2009; McMillan and Ruiz, 2009; Wang and Wu, 2012; Hull and McGroarty, 2014). The primary aim of the present study was to examine whether Asia-Pacific exchange rate markets are efficient over time. The extensive literature mentioned above has examined long memory features in international equity markets. However, few studies have focused on foreign exchange rate markets in the Asia-Pacific region. Additionally, this study considers the Hurst exponent, which, as a single static measure of long memory over long time periods, can be used to classify the degree of market efficiency. However, it has been found that the presence of long memory evolves over time (Cajueiro and Tabak, 2004, 2005). In this regard, the present study takes a rolling sample approach to calculate time-varying Hurst exponents. The rolling sample approach is a useful tool for measuring the degree of market efficiency over time. The rest of this paper is organized as follows. Section 2 describes the characteristics of the sample data set. Section 3 explains the methodology of the rescaled range analysis. Section 4 provides the estimation results of the Hurst exponent over time. The final section provides conclusions. 2. Data For the analysis, we used sample data consisting of the daily nominal spot exchange rates (against the US dollar) for six Asia-Pacific countries: Australian dollar (AUD), Japanese yen (YEN), Korean won (KRW), New Zealand dollar (NZD), Singaporean dollar (SGD), and Taiwan dollar (TWD). All samples were drawn from the database of the Federal Reserve Bank of St. Louis; the sample period was from January 2, 1990 to June 28, 2013. The exchange rates were converted into a nominal percentage return series for all exchange rate series, i.e., rt ln( yt / y t-1) 100 for t 1,2,..., T, in which r t is the return and y t is exchange rate at t time. Figure 1 shows the dynamics of the six exchange rate returns.

288 Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 Table 1. Descriptive statistics and unit root tests for the six exchange rate returns Panel A: Descriptive statistics AUD YEN KRW NZD SGD TWD Mean 0.003-0.007 0.009 0.005-0.007 0.002 Maximum 7.703 3.342 13.64 5.932 2.762 4.082 Minimum -8.212-5.630-19.75-6.178-4.144-3.423 Std. Dev. 0.758 0.695 0.845 0.761 0.353 0.318 Skewness -0.618-0.454-0.944-0.361-0.428 0.864 Kurtosis 16.44 7.369 112.2 9.376 15.45 35.21 J-B test 44878*** 4904*** 2920195*** 10141*** 38374*** 250103*** Panel B: Unit root tests ADF -79.38*** -76.34*** -16.58*** -76.33*** -79.99*** -57.45*** PP -79.38*** -76.34*** -68.09*** -76.35*** -79.94*** -79.84*** KPSS 0.138 0.084 0.086 0.105 0.l71 0.205 Notes: The J-B (Jarque-Bera) test corresponded to the statistic for the null hypothesis of normality in the sample return distributions. Mackinnon s 1% critical value was -3.435 for the ADF and PP tests. The critical value for the KPSS test was 0.739 at the 1% significance level. *** indicates rejection of the null hypothesis at the 1% significance level.

Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 289 Fig. 1. Dynamics of the six exchange rate returns over the sample period Table 1 shows the descriptive statistics and the results of the unit root test for the six exchange rate returns. In Panel A of Table 1, we observe the following. (1) The means of all sample returns were positive except for the YEN and SGD. (2) The KRW had the highest standard deviation value, and the TWD had the lowest. (3) Based on the values of skewness, kurtosis, and Jarque-Bera tests, the distributions of all return series were not normally distributed or leptokurtic, meaning that none of the exchange rate returns followed a true randomwalk process. Panel B of Table 1 shows the results of three unit root tests: the augmented Dickey-Fuller (ADF), Phillips- Perron (PP), and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) tests. The ADF and PP test statistics were large and negative, rejecting the null hypothesis of a unit root; the KPSS test statistic did not reject the null hypothesis of stationarity at the 1% level of significance. Thus, all exchange rate returns were stationary processes. 3. Time-varying Hurst exponent approach Hurst (1951) developed the re-scaled range statistic ( R/ S statistic). He observed many natural time series that followed a biased random walk or a special pattern, and measured the trend using an exponent, now referred to as the Hurst exponent. The R/ S statistic is built on a range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. Consider a sample return series { r1, r2, r3,..., r n } and let r 1 n n denote the sample mean r, where t 1 t n is the time length. Then, the R/ S statistic is given by: n t t 1 ( R/ S) n max rn rn min rk rn S 1 t n 1 t n n k 1 k 1 where S n is the standard deviation, as follows: 1 2 2 1 Sn rt r n n t, (1). (2) Hurst found that the R/ S statistic was proportional to the time length n with the Hurst exponent ( H ): ( R/ S) n c ( n) H, (3) where c is a constant. Taking logarithms yields log( R/ S) n logc Hlog( n). (4)

290 Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 By performing a least-squares regression with log( R/ S ) n as the dependent variable and log( n ) as the independent one, we find the slope of the regression, which is the estimate of the Hurst exponent ( H ). The R/ S statistic can classify a time series into a random or non-random process according to the estimated value of the Hurst exponent. For example, if the value of H for a time series is 0.5, this indicates that the time series follows a random walk process: that is, an independent process. For a time series with a long memory process, H lies between 0.5 and 1, indicating that what happens now has an impact in the future. This property of a time series provides the predictability of a time series because it shows a trend. If H lies between 0 and 0.5, it indicates that the time series possesses an anti-persistent process or negative autocorrelation (Jin and Frechette, 2004). Furthermore, considering the time-varying rolling approach proposed by Cajueiro and Tabak (2004, 2005), we estimated the values of the Hurst exponent for time windows with 1,000 observations (about 4 years of data) each. In the first step, we calculated the Hurst exponent of the first time window. Then, we rolled the sample one point forward by eliminating the first observation and adding the next one, and calculating the Hurst exponent for the new time window. This procedure was repeated until the end of the exchange rate returns. To estimate the exchange rate volatility, we used the residuals of the AR(1)-GARCH(1,1) model as a proxy for the volatility of exchange rate returns. We calculated the time-varying Hurst exponent of the volatility again using the same rolling sample approach explained above; we repeated the calculation almost 5,000 times. 4. Empirical results To assess the degree of market efficiency over time, we estimated the Hurst exponent for sub-sample windows of the 1,000 observations using the rolling approach. Specifically, the initial Hurst exponent covered the period from January 2, 1990 to December 22, 1993, whereas the finishing Hurst exponent was from December 23, 1993 to June 28, 2013. As a result, we obtained almost 5,000 Hurst exponent values for the returns and volatility of exchange rates. Table 2 shows the descriptive statistics of the estimated Hurst exponents for the returns (Panel A) and volatility (Panel B) of the six exchange rates. As indicated by the values of skewness, kurtosis, and Jarque- Bera test statistics, the probability distributions of these Hurst exponents were also not normal. The medians of the Hurst exponents were in the range of 0.544-0.621 for the returns and 0.934-0.807 for the volatility. These results indicate that both the returns and volatility of the exchange rate exhibited long memory properties and suggest the invalidity of the EMH. Figures 2 and 3 show graphs of the time-varying Hurst exponents for the returns and volatility of the six exchange rates, respectively; the graphs were estimated using rolling windows of the 1,000 observations. In terms of the exchange rate returns, all graphs show similar patterns. From October 1998, the patterns of the time-varying Hurst exponents exhibited an upward trend because of the Asian currency crisis in 1997. A substantial downward trend was observed from the end of 2004; the Hurst exponents were close to 0.5 and some were even lower. These results suggest a substantial increase in market efficiency in the decade after the Asian currency crisis. Table 2. Descriptive statistics of time-varying Hurst exponents AUD YEN KRW NZD SGD TWD Panel A: Time-varying Hurst exponent for returns Mean 0.566 0.543 0.589 0.583 0.558 0.619 Median 0.576 0.544 0.600 0.584 0.562 0.621 Maximum 0.666 0.726 0.717 0.674 0.707 0.753

Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 291 Minimum 0.435 0.402 0.411 0.424 0.386 0.494 Std. Dev. 0.046 0.066 0.063 0.046 0.078 0.049 Skewness -0.604 0.511-0.616-0.271-0.175 0.197 Kurtosis 2.973 3.073 2.536 2.729 1.921 2.693 J-B test 299.05*** 214.52*** 351.59*** 127.88*** 263.43*** 49.49*** Panel B: Time-varying Hurst exponent for volatility Mean 0.920 0.914 0.835 0.911 0.865 0.794 Median 0.927 0.934 0.837 0.924 0.877 0.807 Maximum 1.009 1.002 0.937 1.001 0.988 0.981 Minimum 0.775 0.806 0.695 0.745 0.653 0.552 Std. Dev. 0.045 0.051 0.043 0.062 0.095 0.093 Skewness -0.749-0.508-0.445-0.629-0.601-0.282 Kurtosis 3.715 1.922 2.967 2.383 2.253 2.564 J-B test 564.12*** 499.36*** 161.09*** 401.47*** 409.85*** 101.33*** Note: *** indicates rejection of the null hypothesis at the 1% significance level. In terms of volatility, the values of the time-varying Hurst exponents were much greater than 0.5, and the degree of the long memory property was affected by the exogenous shocks, corresponding to the financial crisis. For example, since October 2005, the values of the Hurst exponents increased because of the U.S. subprime mortgage crisis and the subsequent global financial crisis from 2007 to 2009. These findings confirm that the degree of the long memory property changed over time. Our results differ from those of Cajueiro and Tabak (2004, 2005), who asserted that equity markets were becoming more efficient over time. This difference may be due to our use of more recent and volatile data in the present study. In this regard, we conclude that the degree of the long memory features changed over time in the returns and volatility of the six Asia-Pacific exchange rates.

292 Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 Fig. 2. Time-varying Hurst exponents for the six exchange rate returns Note: The estimation of Hurst exponents was repeated almost 5,000 times using the rolling sample approach.

Sang Hoon Kang et al. / Procedia Economics and Finance 14 ( 2014 ) 286 294 293 Fig. 3. Time-varying Hurst exponent for volatility Note: See Fig. 2. 5. Conclusions In this paper, we re-examined the degree of market efficiency in six Asia-Pacific exchange rate markets. In particular, instead of relying on a single static measure of long memory, we explored the time-varying long memory features over time to assess changes in market efficiency by analyzing the returns and volatility of the markets. To investigate the degree of market efficiency, we considered the time-varying Hurst exponent by taking a rolling sample approach with a time window of 1,000 observations. The rolling sample approach is a useful tool for evaluating the degree of market efficiency over time. The estimation results for the time-varying Hurst exponent indicated that both the dynamics of returns and volatility of the markets possessed strong long memory features and that the degree of the long memory features changed over time. Additionally, the Hurst exponent showed an upward trend during the 1997 Asian currency crisis and 2008 global financial crisis, indicating that the exchange rate markets became more inefficient and predictable in recent years during the financial crisis. References Assaf, A., 2008, Long Memory in International Equity Markets: Revisited, Applied Financial Economics Letters 4, 433 437. Barkoulas, J. T., Baum, C. F., Travlos, N., 2000, Long Memory in the Greek Stock Market, Applied Financial Economics 10, 177 184. Cajueiro, D. O., Tabak, B. M., 2004, The Hurst Exponent over Time: Testing the Assertion that Emerging Markets are Becoming More Efficient, Physica A 336, 521 537. Cajueiro, D. O., Tabak, B. M., 2005, Testing for Time-Varying Long-Range Dependence in Volatility for Emerging Markets, Physica A 346, 577 588.

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