1. Academy of Economic Studies - Bucharest Doctoral School of Finance and Banking DOFIN Long Memory in Stock Returns: Research over Markets Supervisor: Professor Dr. Moisă Altăr MSc Student: Silvia Bardoş Bucharest, July 2008

2. Contents Long memory & Motivation Long memory & Motivation Literature review Literature review Steps & data used: Steps & data used: Testing stationarity and long memory Testing stationarity and long memory ADF & KPSS ADF & KPSS Hurst exponent through R/S test & Hurst exponent through wavelet estimator Hurst exponent through R/S test & Hurst exponent through wavelet estimator Determining long memory by estimating fractional differencing parameter Determining long memory by estimating fractional differencing parameter Geweke and Porter-Hudak test & Maximum Likelihood Estimate for an ARFIMA process Geweke and Porter-Hudak test & Maximum Likelihood Estimate for an ARFIMA process Conclusions Conclusions

3. Long Memory & Motivation Long memory has important implications in financial markets because if it is discovered it can be used to construct trading strategies. Long memory has important implications in financial markets because if it is discovered it can be used to construct trading strategies. Long memory or long range dependence means that the information from “today” is not immediately absorbed by the prices in the market and investors react with delay to any such information. Long memory or long range dependence means that the information from “today” is not immediately absorbed by the prices in the market and investors react with delay to any such information.So: A long memory process is a process where a past event has a decaying effect on future events A long memory process is a process where a past event has a decaying effect on future eventsAND Memory is the series property to depend on its own past realizations Memory is the series property to depend on its own past realizations

4. Mathematic view: Long memory processes relates to autocorrelation Mathematic view: Long memory processes relates to autocorrelation If a time series of data exhibits autocorrelation, a value from the data set x s at time t s is correlated with another value x s+z at time t s+z. For a long memory process autocorrelation decays over time and the decay is slower than in a stationary process ( I(0) process ) If a time series of data exhibits autocorrelation, a value from the data set x s at time t s is correlated with another value x s+z at time t s+z. For a long memory process autocorrelation decays over time and the decay is slower than in a stationary process ( I(0) process ) So, if a long memory process exhibits an autocorrelation function that is not consistent for a I(1) process (a process integrated of order 1) nor for an I(0) process (a pure stationary process) we can consider a long memory process as being the layer separating the non-stationary process from the stationary ones – namely a fractionally integrated process. So, if a long memory process exhibits an autocorrelation function that is not consistent for a I(1) process (a process integrated of order 1) nor for an I(0) process (a pure stationary process) we can consider a long memory process as being the layer separating the non-stationary process from the stationary ones – namely a fractionally integrated process. Long Memory & Motivation

5. Literature review Evidence of long memory was first brought up by E. Hurst in 1951 when, testing the behavior of water levels in the Nile river, he observed that the flow of the river was not random, but patterned Evidence of long memory was first brought up by E. Hurst in 1951 when, testing the behavior of water levels in the Nile river, he observed that the flow of the river was not random, but patterned Mandelbrot (1971) was among the first to consider the possibility of long range dependence in asset returns Mandelbrot (1971) was among the first to consider the possibility of long range dependence in asset returns Wright, J. (1999) is detecting evidence of long memory in emerging markets stock returns (Korea, Philippines, Greece, Chile and Colombia) Wright, J. (1999) is detecting evidence of long memory in emerging markets stock returns (Korea, Philippines, Greece, Chile and Colombia) Caporale and Gil-Alana (2002), studying S&P 500 daily returns found results indicating that the degree of dependence remains relatively constant over time, with the order of integration of stock returns fluctuating slightly above or below zero Caporale and Gil-Alana (2002), studying S&P 500 daily returns found results indicating that the degree of dependence remains relatively constant over time, with the order of integration of stock returns fluctuating slightly above or below zero Henry Olan (2002) makes a survey for finding long memory in stock returns from an international perspective. Evidence of long memory is found in the German, Japanese, South Korean and Taiwanese markets against UK, USA, Hong Kong, Singapore and Australia where no sign of long memory appears. Henry Olan (2002) makes a survey for finding long memory in stock returns from an international perspective. Evidence of long memory is found in the German, Japanese, South Korean and Taiwanese markets against UK, USA, Hong Kong, Singapore and Australia where no sign of long memory appears.

6. Steps – Modeling long memory A series x t follows an ARFIMA (p,d,q) process if: where Φ(L), θ(L) are the autoregressive and moving average polynomials, L is the lag, d is the fractional differencing parameter, εt is white noise. For d within (0,0.5), the ARFIMA process is said to exhibit long memory or long range positive dependence For d within (-0.5, 0), the process exhibits intermediate memory or long range negative dependence For d within [0.5, 1) the process is mean reverting and there is no long run impact to future values of the process The process is short memory for d=0 corresponding to a standard ARMA process

7. Testing stationarity Memory is closely related to the order of integration Memory is closely related to the order of integration In the context of non-fractionally integration is equivalent to establish whether the series is I(0) or I(1) and the commonly used tests are ADF and KPSS In the context of non-fractionally integration is equivalent to establish whether the series is I(0) or I(1) and the commonly used tests are ADF and KPSSADF Null hypothesis: H 0 : d = 1 (returns series are containing a unit root) Hassler and Wolter (1994) find that this test of unit root is not consistent against fractional alternatives so the ADF can be inappropriate if we are trying to decide whether a set of data is fractionally integrated or not. KPSS Null hypothesis: H0: d = 0 (return series are stationary) Lee and Schmidt (1996) find that KPSS test can be used to distinguish short memory and long memory stationary processes

8. Testing stationarity KPSS Consider x t ( t = 1, 2, …, N), as the observed return series for which we wish to test stationarity The test decomposes the series into the sum of a random walk, a deterministic trend and a stationary error with the following linear regression model: The KPSS statistics: andis the residual from regressing the series against a constant or a constant and a trend Under the null hypothesis of trend stationary, the residuals e t ( t = 1, 2, …, N ) are from the regression of x on an intercept and time trend. Under the null hypothesis of level stationarity, the residuals e t are from a regression of x on intercept only. Rejection of ADF and KPSS indicates that the process is described by neither I(0) and I(1) processes and that is probable better described by the fractional integrated alternative (d is a non-integer).

9. Estimating long memory using R/S test R/S test Mandelbrot & Wallis (1969) method allows computing parameter H, which measures the intensity of long range dependence in a time series Return time series of length T is divided into n sub-series of length m. For each sub-series m = 1,..., n, we: a) find the mean (E m ) and standard deviation (S m ); b) we subtract the sample mean Z i,m = X i,m − E m for i = 1,..,m; c) produce a time series taking form of W i,m = j,m where i = 1,…,m d) find the range R m = max{W 1,m,…., W n,m } – min{ W 1,m,…., W n,m } e) rescale the range R m by How does this procedure relates to the Hurst exponent?

10. Einstein discovered that the distance covered by a random variable is close related to the square root of time (Brownian motion), where R is the distance covered by the variable, k is a constant and T is the length of the time. Using R/S analysis, Hurst suggested that:, where R/S is the rescaled range, m is the number of observations, k is the constant and H is the Hurst exponent, can be applied to a bigger class of time series (generalized Brownian motion) The Hurst exponent can be than found as: log(R/S)m= log k + H log m H valueReturn time series = 0.5follow a random walk and are independent (0,0.5) are anti-persistent, process covers only a small distance than in the random walk case (0.5,1) are persistent series, process covers a bigger distance than a random walk (long memory) Estimating long memory using R/S test

11. Hurst exponent using wavelet spectral density For computing the Hurst Exponent, the R wavelet estimator uses a discrete wavelet transform then: averages the squares of the coefficients of the transform, performs a linear regression on the logarithm of the average, versus the log of the parameter of the transform The result provides an estimate for the Hurst exponent. Wavelet transform behaves as a microscope that decomposes our return series into components of different frequency so this is why we tend to consider that results obtained for H through the wavelet estimator are being more accurate.

12. The GPH test (1983) Semi-parametric approach to obtain an estimate of the fractional differencing parameter d based on the slope of the spectral density function around the frequency ξ=0 Periodogram (estimator of the spectral density) of x at a frequency ξ I (ξ) = Geweke, J. and S. Porter-Hudak(1983) proposed as an estimate of the OLS estimator of d from the regression:, λ= 1,…..,v the bandwidth v is chosen such that forbut Geweke and Porter-Hudak consider that the power of T has to be within (0.5,0.6). In our test we have considered: V =

13. Maximum likelihood estimates for ARFIMA model In the present paper we have used the MLE implemented based on the approximate maximum likelihood algorithm of Haslett and Raftery (1989) in R. If the estimated d is significantly greater than zero, we consider it an evidence of the presence of long-memory.

14. For testing the existence of long memory we have selected indexes around the world trying to compare return series in mature markets (US, UK, Germany, France, Japan) with emerging markets (Romania, Poland and the BRIC countries) For testing the existence of long memory we have selected indexes around the world trying to compare return series in mature markets (US, UK, Germany, France, Japan) with emerging markets (Romania, Poland and the BRIC countries) For the data series (1997 – 2008) we have first established the length as being 2 (for the wavelet transform performed by the soft) and then we have transformed it in return series through: For the data series (1997 – 2008) we have first established the length as being 2 (for the wavelet transform performed by the soft) and then we have transformed it in return series through: For testing and comparing we have selected mainly, daily returns For testing and comparing we have selected mainly, daily returns Stationarity test were run in Eviews and long memory tests and estimation procedures were run in R Stationarity test were run in Eviews and long memory tests and estimation procedures were run in Rn Data used

15. Is there evidence of long memory in the return time series? S&P 500 daily return series S&P 500 daily return series Null Hypothesis: SP500DAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-47.60859 0.0000 Test critical values:1% level-3.962531 5% level-3.412005 10% level -3.127909 GPH0.450.50.550.750.8 d0.032-0.09060.082143-0.05057-0.03568 tstat sd (d=0)0.258-0.974170.973893-1.25906-1.09493 tstat asd (d=0)0.231-0.826320.935704-1.31442-1.12114 ARFIMA (0,d,0)mleValue d4.583E-05 ADFKPSS R/S Hurst Exponent Diagnostic: 0.4834943 Wavelet estimator for H: 0.4108623 Null Hypothesis: SP500DAY is stationary Exogenous: Constant Bandwidth: 22 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.289248 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

16. FTSE100 daily return series FTSE100 daily return series Null Hypothesis: FTSE100DAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 2 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-29.58352 0.0000 Test critical values:1% level-3.962535 5% level-3.412007 10% level-3.127911 GPH\d=00.450.50.550.750.8 d0.0449066-0.024836-0.01847-0.06293-0.03258 tstat sd0.4092198-0.2552364-0.20357-1.47644-0.91725 tstat asd0.3206718-0.2265169-0.21044-1.63558-1.02361 ARFIMA (0,d,0)mleValue d4.583E-05 ADFKPSS R/S Hurst Exponent Diagnostic: 0.5587119 Wavelet estimator for H: 0.3972144 Null Hypothesis: FTSE100DAY is stationary Exogenous: Constant Bandwidth: 17 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.279658 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

17. Null Hypothesis: BETFIDAYP has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=21) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-29.22895 0.0000 Test critical values:1% level-3.967044 5% level-3.414212 10% level-3.129218 GPH\d=00.450.50.550.750.8 d0.11127650.1880120.268370.1052960.11638 tstat sd0.95316211.8305242.3297162.0236032.708848 tstat asd0.65361441.396422.446052.0694512.717905 ARFIMA (0,d,0)mleValue d0.07864 BET-FI daily return series BET-FI daily return series ADFKPSS R/S Hurst Exponent Diagnostic: 0.6177791 Wavelet estimator for H: 0.6394731 Null Hypothesis: BETFIDAYP is stationary Exogenous: Constant Bandwidth: 7 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.443275 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

18. BOVESPA daily return series BOVESPA daily return series ADFKPSS Null Hypothesis: BOVESPADAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-44.17009 0.0000 Test critical values:1% level-3.962531 5% level-3.412005 10% level-3.127909 GPH\d=00.450.50.550.750.8 d0.0554950.1242070.138253-0.0504179-0.03275 tstat sd0.47031721.1993461.650381-1.505468-1.09286 tstat asd0.39628221.1328311.574859-1.310459-1.02899 ARFIMA (0,d,0)mleValue d0.0003773 R/S Hurst Exponent Diagnostic: 0.5681442 Wavelet estimator for H: 0.5579485 Null Hypothesis: BOVESPADAY is stationary Exogenous: Constant Bandwidth: 16 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.414950 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

19. RTS daily return series ADFKPSS Null Hypothesis: RTSDAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-42.96666 0.0000 Test critical values:1% level-3.962531 5% level-3.412005 10% level-3.127909 GPH\d=00.450.50.550.750.8 d-0.1386964-0.11365-0.05167-0.00301980.028219 tstat sd-1.185368-1.29636-0.67683-0.08284270.909179 tstat asd-0.990412-1.03654-0.58861-0.07848920.886579 ARFIMA (0,d,0)mle\d=0Value d0.03032 R/S Hurst Exponent Diagnostic: 0.543887 Wavelet estimator for H: 0.531688 Null Hypothesis: RTSDAY is stationary Exogenous: Constant Bandwidth: 1 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.083395 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

20. SENSEX daily return series ADFKPSS Null Hypothesis: SENSEXDAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-41.41383 0.0000 Test critical values:1% level-3.962531 5% level-3.412005 10% level-3.127909 GPH\d=00.450.50.550.750.8 d0.07590820.0097280.031772-0.01299930.013071 tstat sd0.55964460.0789110.34705-0.32476420.402457 tstat asd0.54204980.0887280.361916-0.33787650.410681 ARFIMA (0,d,0)mle\d=0Value d0.04682 R/S Hurst Exponent Diagnostic: 0.568345 Wavelet estimator for H: 0.525448 Null Hypothesis: SENSEXDAY is stationary Exogenous: Constant Bandwidth: 10 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.509953 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

21. Hang Seng daily return series ADFKPSS Null Hypothesis: HANGSENGDAY has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=25) t-Statistic Prob.* Augmented Dickey-Fuller test statistic-45.92523 0.0000 Test critical values:1% level-3.962531 5% level-3.412005 10% level -3.127909 GPH\d=00.450.50.550.750.8 d0.1315890.0513350.0390960.02789820.019198 tstat sd0.8962070.4694990.482140.72543830.60606 tstat asd0.9396620.4682030.4453470.72512950.603159 ARFIMA (0,d,0)mleValue d4.583E-05 R/S Hurst Exponent Diagnostic: 0.528084 Wavelet estimator for H: 0.495059 Null Hypothesis: HANGSENGDAY is stationary Exogenous: Constant Bandwidth: 5 (Newey-West using Bartlett kernel) LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.499391 Asymptotic critical values*:1% level 0.739000 5% level 0.463000 10% level 0.347000

22. Index dailyH value via R/S S&P 5000.48 FTSE1000.5587 CAC400.4738 DAX0.5189 NIKKEI 2250.5045 WIG0.593 BET0.4232 BET C0.6306 BET FI0.6187 BOVESPA0.5681 RTS0.5439 SENSEX0.5683 HANG SENG0.5281 Index dailyH value via Wavelet estimator S&P 5000.4109 FTSE1000.3072 CAC400.4161 DAX0.4957 NIKKEI 2250.4927 WIG0.4786 BET0.5337 BET C0.5894 BET FI0.6395 BOVESPA0.5579 RTS0.5317 SENSEX0.5254 HANG SENG0.4951 Comparison between indices - Hurst BRIC

23. Conclusions Using a range of test and estimation procedures we have investigated whether stock returns exhibit long memory Our results come to increase a bit the idea that emerging markets have a weak form of long memory as resulted in case of Russia and India or a stronger form like discovered in case of Romania (BET-FI), China and Brazil. Mature markets, in which we include US & UK among Germany, France show mixed evidence We have tested for long memory the return series for BRIC countries indicesWhy? Because it is important to see is there is some kind of correlation between distant observations in these markets as emerging markets are of great interest to potential investors first taking into account their returns and second because they can be used in case of portfolio diversification as emerging market countries have low correlation with mature markets.