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The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient
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Physica A 336 (2004) 521 – 537
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The Hurst exponent over time: testing the assertion that emerging markets are becoming more e(cient Daniel O. Cajueiroa;∗ , Benjamin M. Tabakb a Universidade
Cat olica de Bras lia-Mestrado em Economia de Empresas, SGAN 916, M odulo B-Asa Norte, DF 70790-160, Brazil b Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900, Brazil Received 25 October 2003
Abstract This paper is concerned with the assertion found in the 3nancial literature that emerging markets are becoming more e(cient over time. To verify whether this assertion is true or not, we propose the calculation of the Hurst exponent over time using a time window with 4 years of data. The data used here comprises the bulk of emerging markets for Latin America and Asia. Our empirical results show that this assertion seems to be true for most countries, but it does not hold for countries such as Brazil, The Philippines and Thailand. Moreover, in order to check whether or not these results depend on the short term memory and the volatility of returns common in such 3nancial asset return data, we 3lter the data by an AR-GARCH procedure and present the Hurst exponents for this 3ltered data. c 2003 Elsevier B.V. All rights reserved. Keywords: Emerging markets; Hurst exponent; GARCH; Long range dependence
1. Introduction The presence of long memory dependence in asset returns has been intriguing academicians as well as 3nancial market professionals for a long time. One of the 3rst to consider the existence of long memory behavior in asset returns was Mandelbrot [1]. Since then, many others have supported Mandelbrot’s results (for details see Ref. [2] and the references therein). In fact, the evidence of long-range memory in 3nancial data causes several drawbacks in modern 3nance: (1) the optimal consumption and portfolio decisions may ∗
Corresponding author.
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.12.031
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become extremely sensitive to the investment horizon [3]; (2) the methods used to price 3nancial derivatives based on martingale models (the most common models, e.g., the Black–Scholes model [4]) are not useful anymore; (3) since the usual tests based on the Capital Asset Pricing Model and Arbitrage Pricing Theory [5] do not take into account long-range dependance, they cannot be applied to series that present such behavior. Moreover, if such long-range persistence is presented in the returns of the 3nancial assets, the random walk hypothesis is not valid anymore and neither does the market e(ciency hypothesis [3]. 1 Therefore, we intend to use the Hurst exponent to test the assertion found in the 3nancial literature that emerging markets are becoming more e(cient over time. It is a very interesting problem since emergent capital markets seem to exhibit some properties which are not present in developed capital markets: (1) while developed capital markets are very e(cient in terms of speed of information, in emerging capital markets investors react to new information slowly; (2) the eBect of capital Cows; (3) the eBect of nonsynchronous trading is expected to be more severe in emerging capital markets than in developed capital markets. The idea behind the assertion is that (1), (2) and (3) have had their inCuence reduced due to the market competition mainly caused by the necessity of diversi3cation of big investors and funds. In general, some works, for instance [7–11], have tried to deal with a simple problem which is to analyze whether a market has or does not have long memory. In special [10,11] compare established markets to emerging markets. They found that the deviations from e(ciency are associated with the degrees of development. Actually, they found the hurst exponent bigger than 0.5 for emerging capital markets. It is appealing that if one wants to study the evolution of a property of a dynamic system over time, it is necessary to have a parameter linked with the observed property which also evolves over time. So, in order to deal with the above problem, we propose the calculation of the Hurst exponent over time using time windows. The concept of time-varying Hurst exponent was introduced in Ref. [12] but it was not discussed in this context and enough in the literature. 2 To the best of our knowledge the only paper which deals with a problem in this context is [13]. 3 Our results are in line with [13] and show that calculating the time-varying Hurst exponent is quite important to access e(ciency and that e(ciency evolves over time. Therefore, this idea is a very relevant one. One can argue that a Hurst exponent cannot be representative for an entire time data series. Besides, a non-trivial problem arises: How should one choose 1 The presence of long-range dependence in asset returns contradicts the weak form of market e(ciency which states that, under the information contained on the set formed by past returns, future returns are unpredictable [6]. 2 We believe that this idea can be useful in a broad class of problems. For instance, there is some evidence that the climate has been changing over time due to the human inCuence. Additionally, it is known that climate variables (pluviometric indexes, temperature etc.) data series usually are characterized by the Hurst exponent bigger than 12 . If the assertion that the climate is becoming more random is true, then it is expected that the Hurst exponent is nearer by 21 today than in the past. 3
We would like to thank Professor Roberto Fernandes Silva Andrade who is now at Physics Department (Universidade Federal da Bahia-Brazil) and who pointed out the existence of this paper when discussing with him our own results.
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Fig. 1. Plots of H and 3ltered H and their histograms for the Brazilian index (Bovespa).
the correct time window in which the Hurst exponent must be calculated? It is clear that there is a trade-oB between the minimum necessary number of points that should be used to calculate the Hurst exponent and the maximum size of the time window which is expected that the Hurst exponent will remain constant (or almost constant). Although 3nding an optimal time-window to calculate the Hurst exponent might be a very important exercise, this problem is beyond the scope of this paper. We choose a 4 year (1008 observations) time-window because it reCects political cycles in most countries and it is su(ciently to give precise estimates. In this work, based on these statements, we have chosen time windows with four years size, but maybe this is not the best size for the time window. On the other hand, to give further support to our results, beside presenting only the plots with the variation of Hurst exponent over time, we also present plots with the histogram of the former results. This additional information may avoid the criticism that the variation of Hurst exponent over time is due to “measurement errors”. Actually, it could be the case, if all the histograms presented here seemed to have a normal distribution.
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The R/S method is used here to calculate the Hurst exponent. However, in order to avoid the Lo’s critique [14], i.e., that the R/S method does not distinguish between short-range and long-range dependence 4 , we 3lter the data by means of an AR-GARCH procedure. In fact, the 3ltering procedure based on the AR-GARCH model intends at the same time to 3lter short-range behavior presented in the time series and also to 3lter the volatility of returns. If we calculate the Hurst exponent for volatility non-adjusted returns, the long-range dependence that is found may be due to volatility eBects, which is known to be persistent in 3nancial time series. This could be misleading because it could suggest that there is predictability in the mean. This paper is organized as follows. The methodology used to evaluate the Hurst exponent is introduced in Section 2. In Section 3, the data used in this work is presented. In Section 4, the results are exposed. Finally, Section 5 presents some conclusions of this work. 4
In [14] is also presented a modi3cation of the R/S method which is robust for short-range behavior, however we avoid this method since it was pointed out by [15,16] that this method is conservative for the hypothesis of non-existence of long memory behavior.
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Fig. 3. Plots of H and 3ltered H and their histograms for the Indonesian index (Jakarta Composite).
2. Evaluation of Hurst exponent In this paper, the evaluation of the Hurst exponent is carried out by means of two methodologies. The 3rst one is the usual and most popular methodology in which the R/S analysis [17,18] is applied to the log-return time series to evaluate the Hurst exponent. 5 More explicitly, let X (t) be the price of a stock on a time t and r(t) be the logarithmic return denoted by r(t) = ln(X (t + 1)=X (t)). The R/S statistic is the range of partial sums of deviations of times series from its mean, rescaled by its standard deviation. So, consider a sample of continuously compounded asset returns {r1 ; r2 ; : : : ; r } and let rN denote the sample mean 1= r where is the time span considered. Then the 5 Another often used method to evaluate the Hurst exponent is the well known Detrended Cuctuation analysis (DFA) [19,20]. The use of the R/S may be justi3ed by two main aspects: (a) preliminar tests suggested that the results found by means of these methods were similar; (b) this paper deals with 3nancial time series and this is the most popular method in this case.
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Fig. 4. Plots of H and 3ltered H and their histograms for the Malaysian index (Kuala Lumpur).
R/S statistic is given by 1 (R=S) ≡ max (rt − rN ) − min (rt − rN ) ; 16t6 s 16t6 t=1
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Hurst [17] found that the rescaled range, R/S, for many records in time is very well described by the following empirical relation: (R=S) = (=2)H :
(3)
The second methodology also uses the classical R/S analysis. However, the R/S analysis is now applied to the log-return time series 3ltered by an AR(1)-GARCH(1,1) process;
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for details, see Refs. [21–23]. This process is very common in 3nancial econometrics 6 and its equations are explicitly given by rt = c + rt−1 + t ;
(4)
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where rt is the log-returns of the data series for each t, ht stands for the conditional variance of the residuals for the mean equation for each t and c, , !N and are unknown parameters that need to be estimated. It is relevant to say that prior inspection 6
The Generalized ARCH (GARCH) models belong to the class of the Auto Regressive Conditional Heteroscedasticity (ARCH) models [24]. It was introduced by Bollerslev [21] to give more Cexibility to the ARCH models and also to solve some practical problems presented in the classical ARCH models. The main idea behind these models is that the conditional standard deviations of a data series are a function of their past values.
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of the series suggested that this is a parsimonious representation of these series. We 3t this model for all series and check using Ljung-Box Q-statistics 7 for further correlation but all Q-statistics up to 10th lag were found to be insigni3cant, i.e., the model 3ltered the data for short-range correlation. Therefore, we apply the R/S to the standardized residuals (t) to perform tests for long-range dependence, where (t) = (t)= h(t). These residuals (3ltered returns) are free of the problems inherent to the modi3cation proposed in Ref. [14] as short-range dependencies both in the mean and the conditional variance have been purged. The GARCH process is well de3ned as long as the condition + ¡ 1 is satis3ed. If this condition is not satis3ed the variance process is non-stationary and we would have to 3t other processes for conditional variance such as Integrated GARCH (IGARCH). However, we impose this condition and for all the series this condition is naturally 7
The Ljung and Box Q-statistic test is a test to detect departures from zero autocorrelations in either direction and at all lags. The null hypothesis indicates that there is no serial correlation at the corresponding element of Lags. For details, see appendix.
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satis3ed. Hence, we can use the conditional standard deviation to standardize the residuals of the mean equation (4) and obtain residuals which have constant variance. 3. Data The sample employed in this study consists of eleven emerging markets and indices for the United States and Japan, which are included for comparison purposes. We have collected daily closing prices for Argentina (Merval Index), Brazil (Bovespa Index), Chile (IPSA Index), India (Mumbai Sensitive Index), Indonesia (Jakarta Composite Index), Malaysia (Kuala Lumpur Composite Index), Mexico (IPC Index), the Philippines (Philippines Composite Index), South Korea (Korea Composite Index), Taiwan (Taiwan Weighted Index), Thailand (Bangkok SET Index), Japan (Nikkei 225 Index), and the US (S&P 500 Index). The period comprised in this research stems from January 1992 through December 2002. All series were collected from the Bloomberg system. These indices comprise the bulk of emerging markets for Latin America and Asia. The inclusion of indices for the US and Japan is important as comparisons can be made with results from emerging markets.
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Fig. 8. Plots of H and 3ltered H and their histograms for the Philippine index (Philippines Composite).
Additionally, it is interesting to stress that all price indices were deCated by the US dollars and not by inCation. This was done for speci3c reasons. In the 3rst place we have used daily price indices and inCation is measured on a monthly frequency. Therefore, in order to deCate these indices using inCation measures we would have to make some assumptions on the daily inCation behavior, which would certainly add more noise to the data. Furthermore, inCation itself is short memory process and this fact would worsen the problems inherent in the estimation of the Hurst exponents for unadjusted data. Another important reason is that by using dollar-denominated indices, comparisons can be made among these indices. Finally, from an international investor point of view, it makes more sense to analyze these indices and measure e(ciency for these indices using a benchmark currency. 4. Empirical results We perform the estimation of the Hurst exponent for time windows with 1008 observations each, thousands of times. We use the 3rst 1008 observations, calculate
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the Hurst exponent, roll the sample one point forward eliminating the 3rst observation and including the next one, calculating the Hurst exponent for the new time window, and repeat this procedure until the end of the series, in a rolling sample approach. We also estimate an AR(1)-GARCH(1,1) process for the 3rst 1008 observations and calculate the Hurst exponents upon 3ltered returns by using the residuals of the mean equation standardized by the conditional standard deviation as 3ltered returns. Again, we roll the sample and 3t a new AR(1)-GARCH(1,1) with diBerent parameters and calculate the Hurst exponent using standardized residuals. We continue our sampling approach until the end of the series. Therefore, time-varying Hurst exponents were calculated upon both returns and 3ltered returns (3ltering for short-range dependency by the autoregressive term AR(1) and volatility, given by the GARCH(1,1) term). Figs. 1–13 present the empirical results of this paper. In each of them, four plots are presented: the Hurst exponents calculated by means of the methodologies proposed in Section 2 for time windows of four years and their respective histograms. From visual inspection of these 3gures the 3rst issue that it is worth noting is that Hurst exponents seem to be time-varying, as they vary widely over time. This
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suggests that calculating a Hurst exponent for a time series with a 3xed window may be misleading, as most of the literature has done so far. Three possible explanations may be given for time-varying Hurst exponents. In the 3rst place, the capital Cow, which account for a signi3cant proportion of investment in emerging stock markets has increased in the past years, and as these investors enter emerging markets, and have enlarged the base of shareholders providing liquidity or these markets, therefore, the speed of dissemination of information should increase. It is important to notice that in some cases this eBect (capital Cows eBect) can move markets in the opposite direction as capital outCows outperform inCows. Another explanation would be that in some countries the emergence and development of derivative markets in recent years can help investors improve their risk allocations by increasing the speed of information dissemination as well, which should increase market e(ciency. Besides, emerging stock markets have markedly nonsynchronous trading if compared with developed markets. However, this eBect should be reduced as markets become more mature and both trading volume and the number of trades placed in the market tend to increase substantially.
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In order to check whether this time-varying Hurst exponents is due to noise we have plotted the histograms for these parameters and carried out the Jarque-Bera (JB) test of normality (see appendix). From the histograms (see Figs. 1–13) and the JB test (see Table 1), we can infer that these parameters are not normally distributed. In some cases, such as Brazil, Korea and Mexico there seems to be present bimodality (see the AR-GARCH 3ltered Figs. 1, 7 and 5)—clearly bimodality is a clue of “two” Hurst exponents. However, it is worth mentioning that after adjusting for AR-GARCH 3lters the Hurst exponents distribution seems to be nearer to the normal than when compared to parameters calculated using non-adjusted returns. Only for Indonesia (see Fig. 3) there is a signi3cant downward trend for the Hurst exponents (analyzing the non-adjusted Hurst exponent), which suggests a substantial increase in e(ciency in the past years. However, inspection from the adjusted Hurst exponents suggests that the signi3cance of this trend is reduced, which could be due to the fact that this market might be highly volatile and that the adjustments made can make a diBerence when analyzing these 3gures. For Taiwan, the Philippines, Thailand and the US (see Figs. 11, 8, 10 and 13) these parameters possess neither a downward
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Table 1 JB test of H and 3ltered H Countries
JB (H)
p-value (H)
JB (3ltered H)
p-value (3ltered H)
Brazil Chile Indonesia Malaysia Korea Argentina Mexico Philippines India Thailand Taiwan Japan USA
25.70 142.2 230.4 101.4 61.50 7.000 72.50 34.80 125.9 106.7 115.8 48.30 9.900
0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.01
22.88 20.04 234.0 124.0 2.990 6.340 48.99 15.90 130.6 66.59 57.62 48.50 12.22
0.00 0.00 0.00 0.00 0.22 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00
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nor an upward trend. For the remaining countries there seems to be a downward trend in these parameters, which is in line with previous 3ndings in the literature. An interesting 3nding is that the 3gures for the US suggest that this is the most e(cient market (see Fig. 13). The Hurst exponents evolve over time around 0.5 and are very close to 0.5 most of the time. For Japan (see Fig. 12) there is a downward trend as well but at the end of the sample the Hurst exponents Cuctuate around 0.5 as well. This suggests that emerging markets are more ine(cient than developed markets and that there is a trend towards e(ciency in some cases. 5. Conclusions This paper contributes to the literature by showing the importance of studying time-varying Hurst exponents to assess for market e(ciency, instead of relying on single static measures of long-memory dependence. Furthermore, the rolling sample approach used in this paper diBers from most of the work that is found in previous literature. Therefore, our results are not directly comparable to other research.
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However, our results suggest that Hurst exponents vary over time due to changes in the dynamics of the underlying return time series. This is not explained either by time-varying short-range dependencies nor time-varying volatility as the Hurst exponents are time-varying even after adjusting for short-range dependency and time-varying volatility. This approach can be applied to many other areas such as studies where testing for long memory may be relevant to describe dynamics characteristics. Empirical results for emerging markets suggest that on average there seems to be a downward trend on Hurst exponent for the 1992–2002 time period. Only for Brazil there seems to be an upward trend, which could be explained, at least partially by the reduction in capital inCows for the Brazilian equity market in recent years, which culminated in a low liquidity environment. Finally, developed countries seem to be more e(cient than emerging countries. Acknowledgements We are indebt to Professor JosTe Garcia Vivas Miranda who is now at Geophysics Department (Universidade Federal da Bahia, Brazil) for his useful suggestions. Appendix A A.1. Ljung and Box Q-statistic test The Ljung and Box Q-statistic test is a test to detect departures from zero autocorrelations in either direction and at all lags. The null hypothesis indicates that there is no serial correlation at the corresponding element of Lags. This test was introduced by Box and Pierce [25] in a preliminarily form as follows: m Qm = T 2 (k) ; (A.1) k=1
where T is the sample size, m is the number of autocorrelation lags included in the statistic and 2 (k) is the squared sample autocorrelation at lag k. It is easy to prove that (A.1) is asymptotically distributed as m2 . In order to study small samples, Ljung and Box [26] provided the following 3nite-sample correction which yields a better 3t to the m2 for small sample sizes m 2 (k) Qm = T (T + 2) : (A.2) T −k k=1
Since we want to test the long memory for data 3ltered from short-range dependence, this test plays an important role here. A.2. Jarque–Bera (JB) test of normality The JB test of normality is an asymptotic, or large-sample test. This test 3rst computes the skewness and Kurtosis measures of the OLS residuals and uses the
D.O. Cajueiro, B.M. Tabak / Physica A 336 (2004) 521 – 537
following statistic: 2 S (K − 3)2 ; JB = n + 6 24
537
(A.3)
where n is the sample size, S is the skewness coe(cient and K is the kurtosis coe(cient. For a normally distributed variable, S = 0 and K = 3. Therefore, the JB test of normality is a test of the joint hypothesis that S and K are 0 and 3, respectively. In that case the value of the JB statistics is expected to be zero. Under the null hypothesis that residuals are normally distributed, Jarque and Bera [27] showed that asymptotically the JB statistic given in A.3 follows the chi-square distribution with 2 degrees of freedom. If the computed p value of the JB statistic in an application is su(ciently low, which will happen if the value of the statistic is very diBerent from zero, one can reject the hypothesis that the residuals are normally distributed. But if the p value is reasonably high, which will happen if the value of the statistic is close to zero, we do not reject the normality assumption. References [1] B. Mandelbrot, Rev. Econ. Stat. 53 (1971) 225. [2] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H.E. Stanley, Physica A 279 (2000) 443. [3] B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Springer, New York, 1997. [4] F. Black, M. Scholes, J. Political Econ. 81 (1973) 637. [5] F. Black, M. Jensen, M. Scholes, The capital asset pricing model: some empirical tests, in: M. Jensen (Ed.), Studies in the Theory of Capital Markets, Praeger, New York, 1972. [6] E.F. Fama, J. Am. Stat. Soc. 65 (1970) 1509. [7] J.T. Barkoulas, C.F. Baum, N. Travlos, App. Financial Econ. 10 (2000) 177. [8] O.T. Henry, Appl. Financial Econ. 12 (2002) 725. [9] C.S. Koong, A.K. Tsui, W.S. Chan, Math. Comput. Simulation 43 (1996) 445. [10] M. Beben, A. Orlowski, Euro. Phys. J. B 20 (2001) 527. [11] T. Di Matteo, T. Aste, M.M. Dacorogna, Physica A 324 (2003) 183. [12] S.V. Muniandy, S.C. Lim, R. Murugan, Physica A 301 (2001) 407. [13] R.L. Costa, G.L. Vasconcelos, Long range correlations and nonstationarity in the Brazilian stock market, Physica A, Forthcoming. [14] A.W. Lo, Econometrica 59 (1991) 1279. [15] V. Teverovsky, M.S. Taqqu, W. Willinger, J. Stat. Plann. Inference 80 (1999) 211. [16] W. Willinger, M.S. Taqqu, V. Teverovsky, Finance Stochastics 3 (1999) 1. [17] E. Hurst, Trans. Am. Soc. Civil Eng. 116 (1951) 770. [18] J. Feder, Fractals, Plenum Press, New York, 1988. [19] J.G. Moreira, J.K.L. Silva, S.O. Kamphorst, J. Phys. A 27 (1994) 8079. [20] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1685. [21] B. Bollerslev, J. Econometrics 32 (1986) 307. [22] C. Gourieroux, J. Jasiak, Financial Econometrics, Princeton University Press, Princeton, 2001. [23] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Cambridge University Press, Cambridge, 2000. [24] R. Engle, Econometrica 50 (1982) 987. [25] G. Box, D. Pierce, J. Am. Stat. Soc. 65 (1970) 1509. [26] G. Ljung, G. Box, Biometrika 66 (1978) 67. [27] C.M. Jarque, A.K. Bera, Int. Stat. Rev. 55 (1987) 163.
Physica A 336 (2004) 521 – 537
www.elsevier.com/locate/physa
The Hurst exponent over time: testing the assertion that emerging markets are becoming more e(cient Daniel O. Cajueiroa;∗ , Benjamin M. Tabakb a Universidade
Cat olica de Bras lia-Mestrado em Economia de Empresas, SGAN 916, M odulo B-Asa Norte, DF 70790-160, Brazil b Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900, Brazil Received 25 October 2003
Abstract This paper is concerned with the assertion found in the 3nancial literature that emerging markets are becoming more e(cient over time. To verify whether this assertion is true or not, we propose the calculation of the Hurst exponent over time using a time window with 4 years of data. The data used here comprises the bulk of emerging markets for Latin America and Asia. Our empirical results show that this assertion seems to be true for most countries, but it does not hold for countries such as Brazil, The Philippines and Thailand. Moreover, in order to check whether or not these results depend on the short term memory and the volatility of returns common in such 3nancial asset return data, we 3lter the data by an AR-GARCH procedure and present the Hurst exponents for this 3ltered data. c 2003 Elsevier B.V. All rights reserved. Keywords: Emerging markets; Hurst exponent; GARCH; Long range dependence
1. Introduction The presence of long memory dependence in asset returns has been intriguing academicians as well as 3nancial market professionals for a long time. One of the 3rst to consider the existence of long memory behavior in asset returns was Mandelbrot [1]. Since then, many others have supported Mandelbrot’s results (for details see Ref. [2] and the references therein). In fact, the evidence of long-range memory in 3nancial data causes several drawbacks in modern 3nance: (1) the optimal consumption and portfolio decisions may ∗
Corresponding author.
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.12.031
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become extremely sensitive to the investment horizon [3]; (2) the methods used to price 3nancial derivatives based on martingale models (the most common models, e.g., the Black–Scholes model [4]) are not useful anymore; (3) since the usual tests based on the Capital Asset Pricing Model and Arbitrage Pricing Theory [5] do not take into account long-range dependance, they cannot be applied to series that present such behavior. Moreover, if such long-range persistence is presented in the returns of the 3nancial assets, the random walk hypothesis is not valid anymore and neither does the market e(ciency hypothesis [3]. 1 Therefore, we intend to use the Hurst exponent to test the assertion found in the 3nancial literature that emerging markets are becoming more e(cient over time. It is a very interesting problem since emergent capital markets seem to exhibit some properties which are not present in developed capital markets: (1) while developed capital markets are very e(cient in terms of speed of information, in emerging capital markets investors react to new information slowly; (2) the eBect of capital Cows; (3) the eBect of nonsynchronous trading is expected to be more severe in emerging capital markets than in developed capital markets. The idea behind the assertion is that (1), (2) and (3) have had their inCuence reduced due to the market competition mainly caused by the necessity of diversi3cation of big investors and funds. In general, some works, for instance [7–11], have tried to deal with a simple problem which is to analyze whether a market has or does not have long memory. In special [10,11] compare established markets to emerging markets. They found that the deviations from e(ciency are associated with the degrees of development. Actually, they found the hurst exponent bigger than 0.5 for emerging capital markets. It is appealing that if one wants to study the evolution of a property of a dynamic system over time, it is necessary to have a parameter linked with the observed property which also evolves over time. So, in order to deal with the above problem, we propose the calculation of the Hurst exponent over time using time windows. The concept of time-varying Hurst exponent was introduced in Ref. [12] but it was not discussed in this context and enough in the literature. 2 To the best of our knowledge the only paper which deals with a problem in this context is [13]. 3 Our results are in line with [13] and show that calculating the time-varying Hurst exponent is quite important to access e(ciency and that e(ciency evolves over time. Therefore, this idea is a very relevant one. One can argue that a Hurst exponent cannot be representative for an entire time data series. Besides, a non-trivial problem arises: How should one choose 1 The presence of long-range dependence in asset returns contradicts the weak form of market e(ciency which states that, under the information contained on the set formed by past returns, future returns are unpredictable [6]. 2 We believe that this idea can be useful in a broad class of problems. For instance, there is some evidence that the climate has been changing over time due to the human inCuence. Additionally, it is known that climate variables (pluviometric indexes, temperature etc.) data series usually are characterized by the Hurst exponent bigger than 12 . If the assertion that the climate is becoming more random is true, then it is expected that the Hurst exponent is nearer by 21 today than in the past. 3
We would like to thank Professor Roberto Fernandes Silva Andrade who is now at Physics Department (Universidade Federal da Bahia-Brazil) and who pointed out the existence of this paper when discussing with him our own results.
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the correct time window in which the Hurst exponent must be calculated? It is clear that there is a trade-oB between the minimum necessary number of points that should be used to calculate the Hurst exponent and the maximum size of the time window which is expected that the Hurst exponent will remain constant (or almost constant). Although 3nding an optimal time-window to calculate the Hurst exponent might be a very important exercise, this problem is beyond the scope of this paper. We choose a 4 year (1008 observations) time-window because it reCects political cycles in most countries and it is su(ciently to give precise estimates. In this work, based on these statements, we have chosen time windows with four years size, but maybe this is not the best size for the time window. On the other hand, to give further support to our results, beside presenting only the plots with the variation of Hurst exponent over time, we also present plots with the histogram of the former results. This additional information may avoid the criticism that the variation of Hurst exponent over time is due to “measurement errors”. Actually, it could be the case, if all the histograms presented here seemed to have a normal distribution.
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The R/S method is used here to calculate the Hurst exponent. However, in order to avoid the Lo’s critique [14], i.e., that the R/S method does not distinguish between short-range and long-range dependence 4 , we 3lter the data by means of an AR-GARCH procedure. In fact, the 3ltering procedure based on the AR-GARCH model intends at the same time to 3lter short-range behavior presented in the time series and also to 3lter the volatility of returns. If we calculate the Hurst exponent for volatility non-adjusted returns, the long-range dependence that is found may be due to volatility eBects, which is known to be persistent in 3nancial time series. This could be misleading because it could suggest that there is predictability in the mean. This paper is organized as follows. The methodology used to evaluate the Hurst exponent is introduced in Section 2. In Section 3, the data used in this work is presented. In Section 4, the results are exposed. Finally, Section 5 presents some conclusions of this work. 4
In [14] is also presented a modi3cation of the R/S method which is robust for short-range behavior, however we avoid this method since it was pointed out by [15,16] that this method is conservative for the hypothesis of non-existence of long memory behavior.
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2. Evaluation of Hurst exponent In this paper, the evaluation of the Hurst exponent is carried out by means of two methodologies. The 3rst one is the usual and most popular methodology in which the R/S analysis [17,18] is applied to the log-return time series to evaluate the Hurst exponent. 5 More explicitly, let X (t) be the price of a stock on a time t and r(t) be the logarithmic return denoted by r(t) = ln(X (t + 1)=X (t)). The R/S statistic is the range of partial sums of deviations of times series from its mean, rescaled by its standard deviation. So, consider a sample of continuously compounded asset returns {r1 ; r2 ; : : : ; r } and let rN denote the sample mean 1= r where is the time span considered. Then the 5 Another often used method to evaluate the Hurst exponent is the well known Detrended Cuctuation analysis (DFA) [19,20]. The use of the R/S may be justi3ed by two main aspects: (a) preliminar tests suggested that the results found by means of these methods were similar; (b) this paper deals with 3nancial time series and this is the most popular method in this case.
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R/S statistic is given by 1 (R=S) ≡ max (rt − rN ) − min (rt − rN ) ; 16t6 s 16t6 t=1
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Hurst [17] found that the rescaled range, R/S, for many records in time is very well described by the following empirical relation: (R=S) = (=2)H :
(3)
The second methodology also uses the classical R/S analysis. However, the R/S analysis is now applied to the log-return time series 3ltered by an AR(1)-GARCH(1,1) process;
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for details, see Refs. [21–23]. This process is very common in 3nancial econometrics 6 and its equations are explicitly given by rt = c + rt−1 + t ;
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The Generalized ARCH (GARCH) models belong to the class of the Auto Regressive Conditional Heteroscedasticity (ARCH) models [24]. It was introduced by Bollerslev [21] to give more Cexibility to the ARCH models and also to solve some practical problems presented in the classical ARCH models. The main idea behind these models is that the conditional standard deviations of a data series are a function of their past values.
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of the series suggested that this is a parsimonious representation of these series. We 3t this model for all series and check using Ljung-Box Q-statistics 7 for further correlation but all Q-statistics up to 10th lag were found to be insigni3cant, i.e., the model 3ltered the data for short-range correlation. Therefore, we apply the R/S to the standardized residuals (t) to perform tests for long-range dependence, where (t) = (t)= h(t). These residuals (3ltered returns) are free of the problems inherent to the modi3cation proposed in Ref. [14] as short-range dependencies both in the mean and the conditional variance have been purged. The GARCH process is well de3ned as long as the condition + ¡ 1 is satis3ed. If this condition is not satis3ed the variance process is non-stationary and we would have to 3t other processes for conditional variance such as Integrated GARCH (IGARCH). However, we impose this condition and for all the series this condition is naturally 7
The Ljung and Box Q-statistic test is a test to detect departures from zero autocorrelations in either direction and at all lags. The null hypothesis indicates that there is no serial correlation at the corresponding element of Lags. For details, see appendix.
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satis3ed. Hence, we can use the conditional standard deviation to standardize the residuals of the mean equation (4) and obtain residuals which have constant variance. 3. Data The sample employed in this study consists of eleven emerging markets and indices for the United States and Japan, which are included for comparison purposes. We have collected daily closing prices for Argentina (Merval Index), Brazil (Bovespa Index), Chile (IPSA Index), India (Mumbai Sensitive Index), Indonesia (Jakarta Composite Index), Malaysia (Kuala Lumpur Composite Index), Mexico (IPC Index), the Philippines (Philippines Composite Index), South Korea (Korea Composite Index), Taiwan (Taiwan Weighted Index), Thailand (Bangkok SET Index), Japan (Nikkei 225 Index), and the US (S&P 500 Index). The period comprised in this research stems from January 1992 through December 2002. All series were collected from the Bloomberg system. These indices comprise the bulk of emerging markets for Latin America and Asia. The inclusion of indices for the US and Japan is important as comparisons can be made with results from emerging markets.
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Additionally, it is interesting to stress that all price indices were deCated by the US dollars and not by inCation. This was done for speci3c reasons. In the 3rst place we have used daily price indices and inCation is measured on a monthly frequency. Therefore, in order to deCate these indices using inCation measures we would have to make some assumptions on the daily inCation behavior, which would certainly add more noise to the data. Furthermore, inCation itself is short memory process and this fact would worsen the problems inherent in the estimation of the Hurst exponents for unadjusted data. Another important reason is that by using dollar-denominated indices, comparisons can be made among these indices. Finally, from an international investor point of view, it makes more sense to analyze these indices and measure e(ciency for these indices using a benchmark currency. 4. Empirical results We perform the estimation of the Hurst exponent for time windows with 1008 observations each, thousands of times. We use the 3rst 1008 observations, calculate
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the Hurst exponent, roll the sample one point forward eliminating the 3rst observation and including the next one, calculating the Hurst exponent for the new time window, and repeat this procedure until the end of the series, in a rolling sample approach. We also estimate an AR(1)-GARCH(1,1) process for the 3rst 1008 observations and calculate the Hurst exponents upon 3ltered returns by using the residuals of the mean equation standardized by the conditional standard deviation as 3ltered returns. Again, we roll the sample and 3t a new AR(1)-GARCH(1,1) with diBerent parameters and calculate the Hurst exponent using standardized residuals. We continue our sampling approach until the end of the series. Therefore, time-varying Hurst exponents were calculated upon both returns and 3ltered returns (3ltering for short-range dependency by the autoregressive term AR(1) and volatility, given by the GARCH(1,1) term). Figs. 1–13 present the empirical results of this paper. In each of them, four plots are presented: the Hurst exponents calculated by means of the methodologies proposed in Section 2 for time windows of four years and their respective histograms. From visual inspection of these 3gures the 3rst issue that it is worth noting is that Hurst exponents seem to be time-varying, as they vary widely over time. This
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Fig. 10. Plots of H and 3ltered H and their histograms for the Thailand index (Bangkok set).
suggests that calculating a Hurst exponent for a time series with a 3xed window may be misleading, as most of the literature has done so far. Three possible explanations may be given for time-varying Hurst exponents. In the 3rst place, the capital Cow, which account for a signi3cant proportion of investment in emerging stock markets has increased in the past years, and as these investors enter emerging markets, and have enlarged the base of shareholders providing liquidity or these markets, therefore, the speed of dissemination of information should increase. It is important to notice that in some cases this eBect (capital Cows eBect) can move markets in the opposite direction as capital outCows outperform inCows. Another explanation would be that in some countries the emergence and development of derivative markets in recent years can help investors improve their risk allocations by increasing the speed of information dissemination as well, which should increase market e(ciency. Besides, emerging stock markets have markedly nonsynchronous trading if compared with developed markets. However, this eBect should be reduced as markets become more mature and both trading volume and the number of trades placed in the market tend to increase substantially.
D.O. Cajueiro, B.M. Tabak / Physica A 336 (2004) 521 – 537 Hurst exponent over time
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Fig. 11. Plots of H and 3ltered H and their histograms for the Taiwanese index (Taiwan Weighted).
In order to check whether this time-varying Hurst exponents is due to noise we have plotted the histograms for these parameters and carried out the Jarque-Bera (JB) test of normality (see appendix). From the histograms (see Figs. 1–13) and the JB test (see Table 1), we can infer that these parameters are not normally distributed. In some cases, such as Brazil, Korea and Mexico there seems to be present bimodality (see the AR-GARCH 3ltered Figs. 1, 7 and 5)—clearly bimodality is a clue of “two” Hurst exponents. However, it is worth mentioning that after adjusting for AR-GARCH 3lters the Hurst exponents distribution seems to be nearer to the normal than when compared to parameters calculated using non-adjusted returns. Only for Indonesia (see Fig. 3) there is a signi3cant downward trend for the Hurst exponents (analyzing the non-adjusted Hurst exponent), which suggests a substantial increase in e(ciency in the past years. However, inspection from the adjusted Hurst exponents suggests that the signi3cance of this trend is reduced, which could be due to the fact that this market might be highly volatile and that the adjustments made can make a diBerence when analyzing these 3gures. For Taiwan, the Philippines, Thailand and the US (see Figs. 11, 8, 10 and 13) these parameters possess neither a downward
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Fig. 12. Plots of H and 3ltered H and their histograms for the Japanese index (Nikkei 225).
Table 1 JB test of H and 3ltered H Countries
JB (H)
p-value (H)
JB (3ltered H)
p-value (3ltered H)
Brazil Chile Indonesia Malaysia Korea Argentina Mexico Philippines India Thailand Taiwan Japan USA
25.70 142.2 230.4 101.4 61.50 7.000 72.50 34.80 125.9 106.7 115.8 48.30 9.900
0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.01
22.88 20.04 234.0 124.0 2.990 6.340 48.99 15.90 130.6 66.59 57.62 48.50 12.22
0.00 0.00 0.00 0.00 0.22 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00
D.O. Cajueiro, B.M. Tabak / Physica A 336 (2004) 521 – 537 Histogram X Normal distribution
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Fig. 13. Plots of H and 3ltered H and their histograms for the North-American index (S&P 500).
nor an upward trend. For the remaining countries there seems to be a downward trend in these parameters, which is in line with previous 3ndings in the literature. An interesting 3nding is that the 3gures for the US suggest that this is the most e(cient market (see Fig. 13). The Hurst exponents evolve over time around 0.5 and are very close to 0.5 most of the time. For Japan (see Fig. 12) there is a downward trend as well but at the end of the sample the Hurst exponents Cuctuate around 0.5 as well. This suggests that emerging markets are more ine(cient than developed markets and that there is a trend towards e(ciency in some cases. 5. Conclusions This paper contributes to the literature by showing the importance of studying time-varying Hurst exponents to assess for market e(ciency, instead of relying on single static measures of long-memory dependence. Furthermore, the rolling sample approach used in this paper diBers from most of the work that is found in previous literature. Therefore, our results are not directly comparable to other research.
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However, our results suggest that Hurst exponents vary over time due to changes in the dynamics of the underlying return time series. This is not explained either by time-varying short-range dependencies nor time-varying volatility as the Hurst exponents are time-varying even after adjusting for short-range dependency and time-varying volatility. This approach can be applied to many other areas such as studies where testing for long memory may be relevant to describe dynamics characteristics. Empirical results for emerging markets suggest that on average there seems to be a downward trend on Hurst exponent for the 1992–2002 time period. Only for Brazil there seems to be an upward trend, which could be explained, at least partially by the reduction in capital inCows for the Brazilian equity market in recent years, which culminated in a low liquidity environment. Finally, developed countries seem to be more e(cient than emerging countries. Acknowledgements We are indebt to Professor JosTe Garcia Vivas Miranda who is now at Geophysics Department (Universidade Federal da Bahia, Brazil) for his useful suggestions. Appendix A A.1. Ljung and Box Q-statistic test The Ljung and Box Q-statistic test is a test to detect departures from zero autocorrelations in either direction and at all lags. The null hypothesis indicates that there is no serial correlation at the corresponding element of Lags. This test was introduced by Box and Pierce [25] in a preliminarily form as follows: m Qm = T 2 (k) ; (A.1) k=1
where T is the sample size, m is the number of autocorrelation lags included in the statistic and 2 (k) is the squared sample autocorrelation at lag k. It is easy to prove that (A.1) is asymptotically distributed as m2 . In order to study small samples, Ljung and Box [26] provided the following 3nite-sample correction which yields a better 3t to the m2 for small sample sizes m 2 (k) Qm = T (T + 2) : (A.2) T −k k=1
Since we want to test the long memory for data 3ltered from short-range dependence, this test plays an important role here. A.2. Jarque–Bera (JB) test of normality The JB test of normality is an asymptotic, or large-sample test. This test 3rst computes the skewness and Kurtosis measures of the OLS residuals and uses the
D.O. Cajueiro, B.M. Tabak / Physica A 336 (2004) 521 – 537
following statistic: 2 S (K − 3)2 ; JB = n + 6 24
537
(A.3)
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