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Empirical evidence of long-range correlations in stock returns
Physica A 287 (2000) 396 – 404
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Empirical evidence of long-range correlations in stock returns Pilar Grau-Carles ∗ Department of Economics, Universidad Rey Juan Carlos, Paseo de los Artilleros s=n, 28032 Madrid, Spain Received 29 April 2000; received in revised form 11 June 2000
Abstract A major issue in nancial economics is the behaviour of stock returns over long horizons. This study provides empirical evidence of the long-range behaviour of various speculative returns. Using di erent techniques such as R=S and modi ed R=S analysis, detrended uctuation analysis (DFA), fractional di erencing test (GPH) and ARFIMA maximum likelihood estimation, we nd little evidence of long memory in returns themselves, by strong evidence of persistence in volatility measured as squared returns or absolute returns. These results allow us to conclude that any stock market model should show no temporal dependence in returns and long-range c 2000 Elsevier Science B.V. All rights reserved. correlation in conditional volatility. Keywords: Long memory processes; R=S analysis; Fractional integration; Detrended uctuation analysis; Periodogram; Stock market prices
1. Introduction The long-range dependence, also known as long memory, is characterized by hyperbolically decaying autocovariance function, by a spectral density that tends to in nity as the frequencies tend to zero and by the self-similarity of aggregated summands. The intensity of these phenomena can be measured either by a parameter d, used as a di erencing parameter in the ARFIMA model, or by the parameter H , that is a scaling parameter. Both parameters are related, in the case of nite variance processes by H = d + 12 , and in the case of in nite variance processes by H = d + 1= [1]. Time ∗
Corresponding author. E-mail address: [email protected] (P. Grau-Carles).
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 7 8 - 2
P. Grau-Carles / Physica A 287 (2000) 396 – 404
397
series with long-range dependence are usually modelled with the ARFIMA (p; d; q). This model is given by (L)(1 − L)d xt = (L)ut ;
ut v i:i:d:(0; 2 ) ;
(1)
where L is the lag operator, d is the fractional di erencing parameter and all the roots of (L) and (L) lie outside the unit circle. For any real number d, the fractional di erence operator (1 − L)d is de ned through a binomial expansion d(d − 1) 2 d(d − 1)(d − 2) 3 (2) L − L + ··· 2! 3! and for −0:5 ¡ d ¡ 0:5 the process is stationary. There is a growing literature in nancial economics that analyses the temporal dependence of stock returns. The random walk hypothesis states that returns are serially random, in other words, that today returns are independent of previous periods stock returns. So the research on, either short, or long-term dependence, has became somehow relevant. For example, the existence of long memory in nancial data would a ect the investment horizon of portfolio decisions. Furthermore, many empirical studies that are based on short-memory statistical techniques would have to be revised. On the other hand, the literature of mean reversion in nancial prices assumes the existence of some mechanism which works over long-time horizons, because the mean-reverting behaviour of stock prices corresponds to the idea that a given change in prices will be followed, in long-time horizons, by changes with the opposite sign. Finally, the bases of the development of ARCH-type family of stochastic models are the ndings of signi cant autocorrelations in volatility measures, such as squared returns or absolute returns. (1 − L)d = 1 − dL +
2. Data We have analyzed the behaviour of the stock market returns using daily data of ve indexes: the Dow Jones (DOW), from 3 January 1927 to 27 September 1999; the Standard & Poor 500 (SP500) from 30 December 1927 to 23 November 1999; the FTSE from 9 September 1993 to 27 September 1999; the NIKKEI from 5 January 1973 to 24 November 1999 and nally the Indice General de la Bolsa de Madrid (IGBM), from 4 November 1985 to 24 September 1999. Each of the data sets has a di erent number of data. The Dow Jones yields 18535 daily entries, the Standard & Poor 500 yields 15646 daily entries, the FTSE 1555 daily entries, the Nikkei 4125 daily entries and the IGBM 3447 daily entries. The rst four indexes correspond to big markets and the fth is a small market in which important anomalies have been found [2]. We have calculated the returns, rt , as the logarithmic di erence in the index. A common nding in much of the empirical literature is that returns themselves contain little serial correlation. However, the absolute returns and their power transformation present long-term correlations. The study of the behaviour of the absolute and square returns has become relevant, one reason is that the investors are in uenced
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not only by the evolution of the sign of the investment, but also by the measure of the absolute return. In other words, the scale becomes more relevant with respect to the evolution of the sign. Also, Granger and Ding [3] proposed the use of the squared return as a measure of risk. Ding et al. [4] investigated the autocorrelation structure of |rt |a where r is stock market return and a is a positive number. They found that the correlation is stronger in the case of a = 1 compared to other values of a. We have looked for long-range correlation on rt ; |rt | and |rt |2 for the whole series of the ve indexes. Four methods have been used, the R=S analysis and the modi ed R=S, the detrended uctuation analysis, the periodogram regression (GPH method) and the ARFIMA estimation by exact maximum likelihood method. Because the last method is computationally burdensome we have used only the last 9000 data for the Dow Jones and the Standard & Poor indexes. The objective is to compare the robustness of the results using di erent methods and di erent markets.
3. Methodology We have begun our investigation with one of the oldest and best-known methods, the R=S analysis. This method, proposed by Mandelbrot and Wallis [5] and based on previous hydrological analysis of Hurst [6], allows the calculation of the self-similarity parameter H . This parameter measures the intensity of long-range dependence in a time series. For a time series {xt }; t = 1; : : : ; n we de ne the range RT ( ) i i X X RT = max (xt − x) min (xt − x) ; (3) 16i6T
t=1
16i6T
t=1
where x is the sample mean of the time series. If the range is rescaled with the sample standard deviation S, the R=S statistics asymptotically follows the relation (R=S)t ˙ Ct H :
(4)
The value of H is generally obtained from the linear regression over a sample of growing temporal horizons (s = t1 ; t2 ; : : : ; T ) ln (R=S)s = ln (C) + H ln (s) :
(5)
An estimated value of H = 12 means that the process has no memory, but H 6= 12 would mean that the process has long memory. More recently, Lo [7] discusses the lack of robustness of the R=S statistic in the presence of short memory or heterocedasticity. Lo suggests the modiÿed rescaled range statistics, which replaces the denominator S, the standard deviation, by a consistent estimator of the square root of the variance of the partial sum of x. ( ), i i X X (xt − x) min (xt − x) sT (q) ; (6) QT = max 16i6T
t=1
16i6T
t=1
P. Grau-Carles / Physica A 287 (2000) 396 – 404
where sT (q) =
T X
i=1
1=2 q T X X 2 (xi − x) 2 =T + j (q) (xi − x)(x i−j − x) T j=1
399
(7)
i=j+1
and also j (q) = 1j=(q + 1) with q ¡ n. Lo derives the limiting distribution of T −1=2 QT under the null of no memory, and shows that the modiÿed rescaled range statistics is robust to short-range dependence. Critical values of the distribution are tabulated by Lo [7, Table II]. In a recent paper, Teverlosky et al. [8] show that Lo test tends to reject the null hypothesis of no long-range dependence when the series is in fact long dependent and that the choice of the truncation lag q is crucial. The second method we have used to measure the long-range dependence is the detrended uctuation analysis (DFA) proposed by Peng et al. [9] and improved in Ref. [10]. The advantage of DFA over Hurst analysis is that it avoids spurious detection of apparent long-range correlation that is an artefact of non-stationarities. The method can0 be summarized as follows. First, the integrate time series y(t 0 ) is obtained, Pt y(t 0 ) = T =1 x(t). Next, the integrate series is divided into non-overlapping intervals, containing m data in each interval. In each interval a least-squared line is tted to the data. The y coordinate of the straight line segments is denoted by ym (t 0 ). Next, the root mean square uctuation of the integrated and detrended time series is calculated v u T u1 X [y(t 0 ) − ym (t 0 )]2 : F(m) = t T 0 t =1
This calculation is repeated over all intervals. A linear relationship on a double log graph of F(m) and the interval size m indicates the presence of a power-law scaling. If there is no correlation or only short correlation F(m) ˙ m1=2 , but if there is long-range power-law correlations then, F(m) ˙ m with 6= 12 . The third method is a semi-parametric procedure to obtain an estimate of the fractional di erencing parameter d. This technique, proposed by Geweke and Porter-Hudak [11], henceforth GPH, is based on the slope of the spectral density function around the angular frequency w = 0. The spectral regression is de ned by w o n + n ; = I; : : : ; v ; (8) ln {I (w )} = a + b ln 4 sin2 2 where I (w ) is the periodogram of the time series at the frequencies w = 2=T with = 1; : : : ; (T − 1)=2, T is the number of observations and v is the number of Fourier frequencies included in the spectral regression. The least-squares estimate of the slope coecient provides an estimation of d. The theoretical error variance is 2 =6 and allows the construction of the t-statistics for the fractional di erencing parameter d. A major issue on the application of this method is the choice of v, Diebold and Rudebusch [12] chose T 1=2 , while Sowell [13] argued that v should be based on the shortest cycle associated with long-run behaviour.
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P. Grau-Carles / Physica A 287 (2000) 396 – 404
The fourth method was developed by Sowell [14] and is a procedure to estimate the stationary ARFIMA models. The estimation requires writing the spectral density function f(w) in terms of the parameter of the model and calculates the autocovariance function at lag k Z 2 1 f(w)eiwk dw : (9)
(k) = 2 0 Then the parameters of the model are estimated by the exact maximum likelihood method. This method uses all the information, long and short term, of the series and allows the calculation of all the parameters of the model. It requires the correct speci cation of the ARMA structure to obtain the nal ARFIMA speci cation. In this investigation, we have estimated all the possible models for p = 0; 1; 2; 3 and q = 0; 1; 2; 3 and we have selected the model following the Akaike Information Citerion (AIC).
4. Results Table 1 gives the results of the traditional R=S analysis of returns, squared returns and absolute returns for all the data sets. For the returns the estimated exponent stays around 0.5, which means that there is no evidence of long-range correlations. However, for the squared returns and absolute returns, there is evidence of long memory with an exponent around 0.7. In all the cases, except for the FTSE index, the intensity of correlation is stronger for absolute returns than for squared returns. Table 2 reports the results of the modi ed R=S analysis for truncation lags q = 0; 5; 10; 25; 50; 100 and 200. It is possible to observe that the modiÿed R=S is sensible to the truncation lag and to the length of the series. In most of the cases, the common nding is that there is no long-range correlation for the returns and there is long memory for the squared returns and absolute returns. Exceptions are the DOW and IGBM returns which reject the null hypothesis for small truncation lags, the NIKKEI which shows statistical signi cance at 5% for returns and the FTSE which does not reject the null hypothesis for the squared returns. Also, it can be observed that in Table 1 Hurst exponent for the returns, absolute returns and squared returns
DOWa DOWb SP500a SP500b FTSE NIKKEI IGBM a The
b The
hole data series. last 9000 data.
rt
|rt |
rt2
0.490 0.556 0.556 0.559 0.480 0.551 0.611
0.772 0.767 0.827 0.825 0.666 0.786 0.716
0.750 0.739 0.715 0.745 0.680 0.721 0.686
P. Grau-Carles / Physica A 287 (2000) 396 – 404
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Table 2 Modi ed rescaled range statistics for the returns, absolute returns and squared returns Vq=0
Vq=5
Vq=10
Vq=25
Vq=50
Vq=100
Vq=200
0.256a
0.580a
0.743b 8.353a 5.332a
0.992 7.026a 4.874a
1.165 5.595a 3.836a
1.316 4.267a 2.962a
1.347 3.220a 2.231a
DOW
rt |rt | rt2
12.722a 7.745a
SP500
rt |rt | rt2
1.321 14.473a 9.480a
1.218 8.317a 6.341a
1.202 6.525a 5.210a
1.235 4.492a 3.748a
1.191 3.325a 2.853a
1.159 2.455a 2.145a
1.210 1.844c 1.670b
FTSE
rt |rt | rt2
0.690 3.488a 1.411
0.954 2.583a 1.044
1.064 2.405a 1.024
1.129 2.109a 1.016
1.164 1.830b 1.019
1.128 1.587 1.038
1.505 1.347 1.054
NIKKEI
rt |rt | rt2
1.991 4.936a 2.471a
2.072c 3.424a 1.880c
2.102a 2.868a 1.682
2.053c 2.171a 1.406
2.039c 1.760b 1.224
1.937c 1.463 1.081
1.757b 1.252 0.982
IGBM
rt |rt | rt2
2.049a 7.131a 5.006a
1.749b 4.776a 3.433a
1.678 3.928a 2.899a
1.514 2.947a 2.324a
1.433 2.378a 1.994c
1.484 1.941a 1.731
1.520 1.619 1.545
9.098a 5.640a
a Statistical
signi cance at 1% level. signi cance at 10% level. c Statistical signi cance at 5% level. b Statistical
Table 3 DFA analysis for the returns, absolute returns and squared returns
DOWa DOWb SP500a SP500b FTSE NIKKEI IGBM a The
b The
rt
|rt |
rt2
0.490 0.497 0.528 0.490 0.393 0.502 0.603
0.627 0.813 0.896 0.825 0.616 0.830 0.835
0.600 0.650 0.835 0.685 0.601 0.724 0.797
hole data series. last 9000 data.
almost all the cases the power of the Lo test is reduced as the truncation parameter is increased. The results of the DFA analysis are shown in Table 3. This technique yields always to smaller exponents than the ones obtained with the traditional R=S analysis for the returns, but not for exponents of the absolute and squared returns. 1 Table 4 reports the spectral regression estimates of d for the data series for the sample size of, v = T 0:5 ; v = T 0:25 ; v = T 0:55 ; v = T 0:575 and v = T 0:6 . Again, most of the 1
The exponent obtained for the absolute returns of the SP500 is smaller than the one obtained in Liu et al. [15] (0:93 ± 0:02) with high frequency data for a three-year period.
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P. Grau-Carles / Physica A 287 (2000) 396 – 404
Table 4 Empirical evidence of the fractional di erencing parameter from periodogram regression d(0:5)
d(0:525)
d(0:55)
DOWa
rt |rt | rt2
0.038 (0.66) 0:528 (9:00)c 0:682 (11:64)c
−0:025 (−0:49) 0:473 (9:22)c 0:551 (10:73)c
0:017 (0:39) 0:503 (11; 18)c 0:474 (10:54)c
DOWb
rt |rt | rt2
−0:447 (−0:62) 0:380 (5:29)c 0:099 (1:38)
−0:005 (−0:08) 0:445 (7:05) c 0:119 (1:89)d
SP500a
rt |rt | rt2
−0:090 (−1:47) 0:773 (12:58)c 0:656 (10:68)c
SP500b
rt |rt | rt2
FTSE
rt |rt | rt2
d(0:575)
d(0:6)
0:031 (0:08) 0:532 (13:44)c 0:470 (11:89)c
0:033 (0:95) 0:547 (15:80)c 0:520 (15:98)c
−0:009 (−0:17) 0:411 (7:37)c 0:109 (1:95)d
0:024 (0:45) 0:370 (7:50)c 0:100 (2:03)e
0:028 (0:65) 0:346 (7:91)c 0:097 (2:20)e
−0:049 (−0:91) 0:777 (14:42)c 0:738 (13:70)c
−0:001 (−0:02) 0:757 (15:99)c 0:747 (15:77)c
−0:039 (−0:95) 0:737 (17:73)c 0:688 (16:55)c
0:076 (2:09)c 0:625 (17:07)c 0:536 (14:63)c
0:006 (0:09) 0:350 (4:87)c 0:155 (2:12)c
0:028 (0:45) 0:413 (7:52)c 0:178 (2:82)c
0:079 (1:42) 0:419 (7:52)c 0:151 (2:71)c
0:073 (1:47) 0:391 (7:92)c 0:125 (2:53)e
0:077 (1:75)d 0:391 (8:95)c 0:116 (2:66)e
−0:195 (−1:63) 0:052 (0:44) −0:005 (−0:03)
−0:071 (−0:67) 0:070 (0:66) 0:004 (0:03)
−0:042 (−0:44) 0:066 (0:68) 0:005 (0:04)
−0:064 (−0:75) 0:066 (0:67) 0:005 (0:04)
−0:025 (−0:32) 0:047 (0:60) 0:004 (0:05)
NIKKEI rt |rt | rt2
0:133 (1:49) 0:438 (4:91)c 0:224 (2:51)c
0:102 (1:31) 0:458 (5:79)c 0:265 (3:34)c
0:032 (0:46) 0:414 (5:86)c 0:243 (3:44)c
0:013 (0:19) 0:437 (6:93)c 0:264 (4:19)c
0:006 (0:11) 0:446 (7:94)c 0:288 (5:13)c
IGBM
0:010 (0:11) 0:409 (4:33)c 0:284 (3:00)c
0:091 (1:09) 0:381 (4:53)c 0:244 (2:89)c
0:104 (1:39) 0:323 (4:33)c 0:206 (2:75)c
0:088 (1:32) 0:341 (5:13)c 0:222 (3:34)c
0:095 (1:59) 0:406 (6:81)c 0:282 (4:74)c
rt |rt | rt2
a The
hole data series. last 9000 data. c Statistical signi cance at the 1% level. d Statistical signi cance at the 10% level. e Statistical signi cance at the 5% level. b The
results are consistent with the ones obtained with the previous tests. It is possible to observe that there is no long-range dependence for returns, but there is strong dependence for absolute and squared returns. However, only for the NIKKEI and IGBM, the results are consistent with the rule H = d + 12 . For the DOW and SP500 H ¡ d + 1=2 for the signi cant cases. Again for the FTSE no evidence of long memory is found neither for returns nor for absolute and squared returns, which agrees with the ndings of Brook eld [16] that using monthly data does not nd signi cant memory for absolute and squared returns. Finally, the results of the ARFIMA estimation 2 are shown in Table 5. With this method, we have obtained that the d parameter is signi cant in all cases except for the squared returns of the FTSE, but for the returns the value of d is very small. The estimation of d agrees with the relation H = d + 12 if we consider the results of the 2
The results are generate using Ox Version 2.20 (see Ref. [17]) and the Ar ma package Version 1.00 [18].
P. Grau-Carles / Physica A 287 (2000) 396 – 404
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Table 5 Ar ma estimation for returns, absolute returns and squared returns
DOW
SP500
FTSE
NIKKEI
IGBM
a Statistical
d MA-1 MA-2 d AR-1 AR-2 MA-1 MA-2 d MA-1 MA-2 d AR-1 AR-2 MA-1 MA-2 d AR-1 AR-2 AR-3 MA-1 MA-2
rt
|rt |
rt2
(0; d; 1) −0:042 (−3:69)a 0:148 (9:98)a
(0; d; 1) 0:311 (20:1)a −0:213 (−10:8)a
(0; d; 2) 0:101 (8:41)a −0:037 (−2:78)a 0:401 (37:4)a
(0; d; 1) −0:032 (−2:71)a
(2; d; 2) 0:371 (15:1)a −0:686 (19:2)a 0:236 (6:89)a 0:414 (9:37)a −0:514 (11:9)a
(2; d; 2) 0:140 (11:8)a −0:720 (−8:30)a −0:694 (−20:7)a 0:672 (8:97)a 0:747 (25:2)a
(0; d; 1) −0:124 (−3:47)a −0:148 (−3:21)a
(0; d; 2) 0:111 (3:18)a 0:354 (7:97)a −0:074 (−1:98)a
(0; 0; 1) −0:0002 (−0:0111) 0:917 (97:8)a
(0; d; 2) 0:038 (1:43)a
(2; d; 1) 0:417 (8:67)a 0:434 (8:63)a 0:054 (2:15)a −0:709 (−11:3)a
0:137 (9:13)a
−0:047 (−1:45) −0:086 (−4:28)a (2; d; 2) 0:076 (3:86)a −0:608 (−4:42)a −0:526 (−4:93)a
(0; d; 2) 0:378 (10:8)a
0:729 (5:28)a 0:573 (6:03)a
−0:244 (−6:12)a −0:059 (−2:03)a
(3; d; 2) 0:205 (7:76)a −0:022 (−0:71) −0:042 (−2:06)a (3; d; 2) 0:301 (15:4)a 0:825 (33:5)a −0:811 (−32:3)a −0:167 (−6:85)a −0:991 (−153:)a 0:990 (133)a
signi cance at 5% level.
DFA analysis. With respect to the selected model it can be observed that there is no unanimity. Although it is well known that the AIC tends to overestimate the model order, this fact does not signi cantly a ect the estimation of the long memory term. 5. Conclusions Our investigation on the behaviour of the returns of six stock indexes using nonparametric, semiparametric and parametric statistical tests shows no evidence of long-term dependence, or to a very little degree if we attend to the results of the maximum likelihood estimation. In the case of the absolute and squared returns, long memory is found stronger for absolute returns than for squared returns. We have found important anomalies to this behaviour in the study of the FTSE index. Another remarkable result is that the agreement of the four methods employed decreases if we
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use the whole series of the Dow Jones and the SP500, which can be a consequence of the structural changes that would produce time-varying parameters or regime switches. The pattern of dependence in volatility may not remain constant over time, so any attempt to model the long dependence should reproduce these results. These empirical ndings open a new direction on the research of modelling stock markets. References [1] M.U. Taquu, V. Teverovsky, in: R. Alder, R. Feldman, M.S. Taquu (Eds.), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhauser, Boston, 1998, pp. 177–217. [2] Y. Cheung, K. Lai, J. Int. Money Finance 14 (1995) 597. [3] C.W.J. Granger, Z. Ding, Discussion Paper 93-38. University of California, San Diego, 1993. [4] Z. Ding, C.W.J. Granger, R.F. Engle, J. Empirical Finance 1 (1993) 83. [5] B.B. Mandelbrot, J.R. Wallis, Water Resour. Res. 4 (1968) 967. [6] H.E. Hurst, Trans. Amer. Soc. Civil Eng. 116 (1951) 770. [7] A. Lo, Econometrica 59 (1991) 451–474. [8] V. Teverovsky, M.U. Taquu, W. Willinger, J. Statist. Plann. Inference 80 (1999) 211. [9] C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1684. [10] G.M. Viswanathan, S.V. Buldyrev, S. Havlin, H.E. Stanley, Biophys. J. 72 (1997) 866. [11] J. Geweke, S. Porter-Hudak, J Time Ser. Anal. 4 (1983) 221. [12] J. Hosking, Biometrika 68 (1981) 165. [13] F.B. Sowell, J. Monetary Econ. 29 (1992) 277. [14] F.B. Sowell, J. Econometrics 53 (1992) 165. [15] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Stanley, Phys. Rev. E 60 (1999) 1390. [16] D. Brook eld, Appl. Econ. Lett. 2 (1995) 110. [17] J.A. Doornik, Object-Oriented Matrix Programming Using Ox, 3rd Edition, Timberlake Consultants Press, London, 1999. [18] J.A. Doornik, M. Ooms, A package for estimating, forecasting and simulating Ar ma models, 1999, Oxford: www.nu .ox.ac.uk=Users=Doornik.
www.elsevier.com/locate/physa
Empirical evidence of long-range correlations in stock returns Pilar Grau-Carles ∗ Department of Economics, Universidad Rey Juan Carlos, Paseo de los Artilleros s=n, 28032 Madrid, Spain Received 29 April 2000; received in revised form 11 June 2000
Abstract A major issue in nancial economics is the behaviour of stock returns over long horizons. This study provides empirical evidence of the long-range behaviour of various speculative returns. Using di erent techniques such as R=S and modi ed R=S analysis, detrended uctuation analysis (DFA), fractional di erencing test (GPH) and ARFIMA maximum likelihood estimation, we nd little evidence of long memory in returns themselves, by strong evidence of persistence in volatility measured as squared returns or absolute returns. These results allow us to conclude that any stock market model should show no temporal dependence in returns and long-range c 2000 Elsevier Science B.V. All rights reserved. correlation in conditional volatility. Keywords: Long memory processes; R=S analysis; Fractional integration; Detrended uctuation analysis; Periodogram; Stock market prices
1. Introduction The long-range dependence, also known as long memory, is characterized by hyperbolically decaying autocovariance function, by a spectral density that tends to in nity as the frequencies tend to zero and by the self-similarity of aggregated summands. The intensity of these phenomena can be measured either by a parameter d, used as a di erencing parameter in the ARFIMA model, or by the parameter H , that is a scaling parameter. Both parameters are related, in the case of nite variance processes by H = d + 12 , and in the case of in nite variance processes by H = d + 1= [1]. Time ∗
Corresponding author. E-mail address: [email protected] (P. Grau-Carles).
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 7 8 - 2
P. Grau-Carles / Physica A 287 (2000) 396 – 404
397
series with long-range dependence are usually modelled with the ARFIMA (p; d; q). This model is given by (L)(1 − L)d xt = (L)ut ;
ut v i:i:d:(0; 2 ) ;
(1)
where L is the lag operator, d is the fractional di erencing parameter and all the roots of (L) and (L) lie outside the unit circle. For any real number d, the fractional di erence operator (1 − L)d is de ned through a binomial expansion d(d − 1) 2 d(d − 1)(d − 2) 3 (2) L − L + ··· 2! 3! and for −0:5 ¡ d ¡ 0:5 the process is stationary. There is a growing literature in nancial economics that analyses the temporal dependence of stock returns. The random walk hypothesis states that returns are serially random, in other words, that today returns are independent of previous periods stock returns. So the research on, either short, or long-term dependence, has became somehow relevant. For example, the existence of long memory in nancial data would a ect the investment horizon of portfolio decisions. Furthermore, many empirical studies that are based on short-memory statistical techniques would have to be revised. On the other hand, the literature of mean reversion in nancial prices assumes the existence of some mechanism which works over long-time horizons, because the mean-reverting behaviour of stock prices corresponds to the idea that a given change in prices will be followed, in long-time horizons, by changes with the opposite sign. Finally, the bases of the development of ARCH-type family of stochastic models are the ndings of signi cant autocorrelations in volatility measures, such as squared returns or absolute returns. (1 − L)d = 1 − dL +
2. Data We have analyzed the behaviour of the stock market returns using daily data of ve indexes: the Dow Jones (DOW), from 3 January 1927 to 27 September 1999; the Standard & Poor 500 (SP500) from 30 December 1927 to 23 November 1999; the FTSE from 9 September 1993 to 27 September 1999; the NIKKEI from 5 January 1973 to 24 November 1999 and nally the Indice General de la Bolsa de Madrid (IGBM), from 4 November 1985 to 24 September 1999. Each of the data sets has a di erent number of data. The Dow Jones yields 18535 daily entries, the Standard & Poor 500 yields 15646 daily entries, the FTSE 1555 daily entries, the Nikkei 4125 daily entries and the IGBM 3447 daily entries. The rst four indexes correspond to big markets and the fth is a small market in which important anomalies have been found [2]. We have calculated the returns, rt , as the logarithmic di erence in the index. A common nding in much of the empirical literature is that returns themselves contain little serial correlation. However, the absolute returns and their power transformation present long-term correlations. The study of the behaviour of the absolute and square returns has become relevant, one reason is that the investors are in uenced
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not only by the evolution of the sign of the investment, but also by the measure of the absolute return. In other words, the scale becomes more relevant with respect to the evolution of the sign. Also, Granger and Ding [3] proposed the use of the squared return as a measure of risk. Ding et al. [4] investigated the autocorrelation structure of |rt |a where r is stock market return and a is a positive number. They found that the correlation is stronger in the case of a = 1 compared to other values of a. We have looked for long-range correlation on rt ; |rt | and |rt |2 for the whole series of the ve indexes. Four methods have been used, the R=S analysis and the modi ed R=S, the detrended uctuation analysis, the periodogram regression (GPH method) and the ARFIMA estimation by exact maximum likelihood method. Because the last method is computationally burdensome we have used only the last 9000 data for the Dow Jones and the Standard & Poor indexes. The objective is to compare the robustness of the results using di erent methods and di erent markets.
3. Methodology We have begun our investigation with one of the oldest and best-known methods, the R=S analysis. This method, proposed by Mandelbrot and Wallis [5] and based on previous hydrological analysis of Hurst [6], allows the calculation of the self-similarity parameter H . This parameter measures the intensity of long-range dependence in a time series. For a time series {xt }; t = 1; : : : ; n we de ne the range RT ( ) i i X X RT = max (xt − x) min (xt − x) ; (3) 16i6T
t=1
16i6T
t=1
where x is the sample mean of the time series. If the range is rescaled with the sample standard deviation S, the R=S statistics asymptotically follows the relation (R=S)t ˙ Ct H :
(4)
The value of H is generally obtained from the linear regression over a sample of growing temporal horizons (s = t1 ; t2 ; : : : ; T ) ln (R=S)s = ln (C) + H ln (s) :
(5)
An estimated value of H = 12 means that the process has no memory, but H 6= 12 would mean that the process has long memory. More recently, Lo [7] discusses the lack of robustness of the R=S statistic in the presence of short memory or heterocedasticity. Lo suggests the modiÿed rescaled range statistics, which replaces the denominator S, the standard deviation, by a consistent estimator of the square root of the variance of the partial sum of x. ( ), i i X X (xt − x) min (xt − x) sT (q) ; (6) QT = max 16i6T
t=1
16i6T
t=1
P. Grau-Carles / Physica A 287 (2000) 396 – 404
where sT (q) =
T X
i=1
1=2 q T X X 2 (xi − x) 2 =T + j (q) (xi − x)(x i−j − x) T j=1
399
(7)
i=j+1
and also j (q) = 1j=(q + 1) with q ¡ n. Lo derives the limiting distribution of T −1=2 QT under the null of no memory, and shows that the modiÿed rescaled range statistics is robust to short-range dependence. Critical values of the distribution are tabulated by Lo [7, Table II]. In a recent paper, Teverlosky et al. [8] show that Lo test tends to reject the null hypothesis of no long-range dependence when the series is in fact long dependent and that the choice of the truncation lag q is crucial. The second method we have used to measure the long-range dependence is the detrended uctuation analysis (DFA) proposed by Peng et al. [9] and improved in Ref. [10]. The advantage of DFA over Hurst analysis is that it avoids spurious detection of apparent long-range correlation that is an artefact of non-stationarities. The method can0 be summarized as follows. First, the integrate time series y(t 0 ) is obtained, Pt y(t 0 ) = T =1 x(t). Next, the integrate series is divided into non-overlapping intervals, containing m data in each interval. In each interval a least-squared line is tted to the data. The y coordinate of the straight line segments is denoted by ym (t 0 ). Next, the root mean square uctuation of the integrated and detrended time series is calculated v u T u1 X [y(t 0 ) − ym (t 0 )]2 : F(m) = t T 0 t =1
This calculation is repeated over all intervals. A linear relationship on a double log graph of F(m) and the interval size m indicates the presence of a power-law scaling. If there is no correlation or only short correlation F(m) ˙ m1=2 , but if there is long-range power-law correlations then, F(m) ˙ m with 6= 12 . The third method is a semi-parametric procedure to obtain an estimate of the fractional di erencing parameter d. This technique, proposed by Geweke and Porter-Hudak [11], henceforth GPH, is based on the slope of the spectral density function around the angular frequency w = 0. The spectral regression is de ned by w o n + n ; = I; : : : ; v ; (8) ln {I (w )} = a + b ln 4 sin2 2 where I (w ) is the periodogram of the time series at the frequencies w = 2=T with = 1; : : : ; (T − 1)=2, T is the number of observations and v is the number of Fourier frequencies included in the spectral regression. The least-squares estimate of the slope coecient provides an estimation of d. The theoretical error variance is 2 =6 and allows the construction of the t-statistics for the fractional di erencing parameter d. A major issue on the application of this method is the choice of v, Diebold and Rudebusch [12] chose T 1=2 , while Sowell [13] argued that v should be based on the shortest cycle associated with long-run behaviour.
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P. Grau-Carles / Physica A 287 (2000) 396 – 404
The fourth method was developed by Sowell [14] and is a procedure to estimate the stationary ARFIMA models. The estimation requires writing the spectral density function f(w) in terms of the parameter of the model and calculates the autocovariance function at lag k Z 2 1 f(w)eiwk dw : (9)
(k) = 2 0 Then the parameters of the model are estimated by the exact maximum likelihood method. This method uses all the information, long and short term, of the series and allows the calculation of all the parameters of the model. It requires the correct speci cation of the ARMA structure to obtain the nal ARFIMA speci cation. In this investigation, we have estimated all the possible models for p = 0; 1; 2; 3 and q = 0; 1; 2; 3 and we have selected the model following the Akaike Information Citerion (AIC).
4. Results Table 1 gives the results of the traditional R=S analysis of returns, squared returns and absolute returns for all the data sets. For the returns the estimated exponent stays around 0.5, which means that there is no evidence of long-range correlations. However, for the squared returns and absolute returns, there is evidence of long memory with an exponent around 0.7. In all the cases, except for the FTSE index, the intensity of correlation is stronger for absolute returns than for squared returns. Table 2 reports the results of the modi ed R=S analysis for truncation lags q = 0; 5; 10; 25; 50; 100 and 200. It is possible to observe that the modiÿed R=S is sensible to the truncation lag and to the length of the series. In most of the cases, the common nding is that there is no long-range correlation for the returns and there is long memory for the squared returns and absolute returns. Exceptions are the DOW and IGBM returns which reject the null hypothesis for small truncation lags, the NIKKEI which shows statistical signi cance at 5% for returns and the FTSE which does not reject the null hypothesis for the squared returns. Also, it can be observed that in Table 1 Hurst exponent for the returns, absolute returns and squared returns
DOWa DOWb SP500a SP500b FTSE NIKKEI IGBM a The
b The
hole data series. last 9000 data.
rt
|rt |
rt2
0.490 0.556 0.556 0.559 0.480 0.551 0.611
0.772 0.767 0.827 0.825 0.666 0.786 0.716
0.750 0.739 0.715 0.745 0.680 0.721 0.686
P. Grau-Carles / Physica A 287 (2000) 396 – 404
401
Table 2 Modi ed rescaled range statistics for the returns, absolute returns and squared returns Vq=0
Vq=5
Vq=10
Vq=25
Vq=50
Vq=100
Vq=200
0.256a
0.580a
0.743b 8.353a 5.332a
0.992 7.026a 4.874a
1.165 5.595a 3.836a
1.316 4.267a 2.962a
1.347 3.220a 2.231a
DOW
rt |rt | rt2
12.722a 7.745a
SP500
rt |rt | rt2
1.321 14.473a 9.480a
1.218 8.317a 6.341a
1.202 6.525a 5.210a
1.235 4.492a 3.748a
1.191 3.325a 2.853a
1.159 2.455a 2.145a
1.210 1.844c 1.670b
FTSE
rt |rt | rt2
0.690 3.488a 1.411
0.954 2.583a 1.044
1.064 2.405a 1.024
1.129 2.109a 1.016
1.164 1.830b 1.019
1.128 1.587 1.038
1.505 1.347 1.054
NIKKEI
rt |rt | rt2
1.991 4.936a 2.471a
2.072c 3.424a 1.880c
2.102a 2.868a 1.682
2.053c 2.171a 1.406
2.039c 1.760b 1.224
1.937c 1.463 1.081
1.757b 1.252 0.982
IGBM
rt |rt | rt2
2.049a 7.131a 5.006a
1.749b 4.776a 3.433a
1.678 3.928a 2.899a
1.514 2.947a 2.324a
1.433 2.378a 1.994c
1.484 1.941a 1.731
1.520 1.619 1.545
9.098a 5.640a
a Statistical
signi cance at 1% level. signi cance at 10% level. c Statistical signi cance at 5% level. b Statistical
Table 3 DFA analysis for the returns, absolute returns and squared returns
DOWa DOWb SP500a SP500b FTSE NIKKEI IGBM a The
b The
rt
|rt |
rt2
0.490 0.497 0.528 0.490 0.393 0.502 0.603
0.627 0.813 0.896 0.825 0.616 0.830 0.835
0.600 0.650 0.835 0.685 0.601 0.724 0.797
hole data series. last 9000 data.
almost all the cases the power of the Lo test is reduced as the truncation parameter is increased. The results of the DFA analysis are shown in Table 3. This technique yields always to smaller exponents than the ones obtained with the traditional R=S analysis for the returns, but not for exponents of the absolute and squared returns. 1 Table 4 reports the spectral regression estimates of d for the data series for the sample size of, v = T 0:5 ; v = T 0:25 ; v = T 0:55 ; v = T 0:575 and v = T 0:6 . Again, most of the 1
The exponent obtained for the absolute returns of the SP500 is smaller than the one obtained in Liu et al. [15] (0:93 ± 0:02) with high frequency data for a three-year period.
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P. Grau-Carles / Physica A 287 (2000) 396 – 404
Table 4 Empirical evidence of the fractional di erencing parameter from periodogram regression d(0:5)
d(0:525)
d(0:55)
DOWa
rt |rt | rt2
0.038 (0.66) 0:528 (9:00)c 0:682 (11:64)c
−0:025 (−0:49) 0:473 (9:22)c 0:551 (10:73)c
0:017 (0:39) 0:503 (11; 18)c 0:474 (10:54)c
DOWb
rt |rt | rt2
−0:447 (−0:62) 0:380 (5:29)c 0:099 (1:38)
−0:005 (−0:08) 0:445 (7:05) c 0:119 (1:89)d
SP500a
rt |rt | rt2
−0:090 (−1:47) 0:773 (12:58)c 0:656 (10:68)c
SP500b
rt |rt | rt2
FTSE
rt |rt | rt2
d(0:575)
d(0:6)
0:031 (0:08) 0:532 (13:44)c 0:470 (11:89)c
0:033 (0:95) 0:547 (15:80)c 0:520 (15:98)c
−0:009 (−0:17) 0:411 (7:37)c 0:109 (1:95)d
0:024 (0:45) 0:370 (7:50)c 0:100 (2:03)e
0:028 (0:65) 0:346 (7:91)c 0:097 (2:20)e
−0:049 (−0:91) 0:777 (14:42)c 0:738 (13:70)c
−0:001 (−0:02) 0:757 (15:99)c 0:747 (15:77)c
−0:039 (−0:95) 0:737 (17:73)c 0:688 (16:55)c
0:076 (2:09)c 0:625 (17:07)c 0:536 (14:63)c
0:006 (0:09) 0:350 (4:87)c 0:155 (2:12)c
0:028 (0:45) 0:413 (7:52)c 0:178 (2:82)c
0:079 (1:42) 0:419 (7:52)c 0:151 (2:71)c
0:073 (1:47) 0:391 (7:92)c 0:125 (2:53)e
0:077 (1:75)d 0:391 (8:95)c 0:116 (2:66)e
−0:195 (−1:63) 0:052 (0:44) −0:005 (−0:03)
−0:071 (−0:67) 0:070 (0:66) 0:004 (0:03)
−0:042 (−0:44) 0:066 (0:68) 0:005 (0:04)
−0:064 (−0:75) 0:066 (0:67) 0:005 (0:04)
−0:025 (−0:32) 0:047 (0:60) 0:004 (0:05)
NIKKEI rt |rt | rt2
0:133 (1:49) 0:438 (4:91)c 0:224 (2:51)c
0:102 (1:31) 0:458 (5:79)c 0:265 (3:34)c
0:032 (0:46) 0:414 (5:86)c 0:243 (3:44)c
0:013 (0:19) 0:437 (6:93)c 0:264 (4:19)c
0:006 (0:11) 0:446 (7:94)c 0:288 (5:13)c
IGBM
0:010 (0:11) 0:409 (4:33)c 0:284 (3:00)c
0:091 (1:09) 0:381 (4:53)c 0:244 (2:89)c
0:104 (1:39) 0:323 (4:33)c 0:206 (2:75)c
0:088 (1:32) 0:341 (5:13)c 0:222 (3:34)c
0:095 (1:59) 0:406 (6:81)c 0:282 (4:74)c
rt |rt | rt2
a The
hole data series. last 9000 data. c Statistical signi cance at the 1% level. d Statistical signi cance at the 10% level. e Statistical signi cance at the 5% level. b The
results are consistent with the ones obtained with the previous tests. It is possible to observe that there is no long-range dependence for returns, but there is strong dependence for absolute and squared returns. However, only for the NIKKEI and IGBM, the results are consistent with the rule H = d + 12 . For the DOW and SP500 H ¡ d + 1=2 for the signi cant cases. Again for the FTSE no evidence of long memory is found neither for returns nor for absolute and squared returns, which agrees with the ndings of Brook eld [16] that using monthly data does not nd signi cant memory for absolute and squared returns. Finally, the results of the ARFIMA estimation 2 are shown in Table 5. With this method, we have obtained that the d parameter is signi cant in all cases except for the squared returns of the FTSE, but for the returns the value of d is very small. The estimation of d agrees with the relation H = d + 12 if we consider the results of the 2
The results are generate using Ox Version 2.20 (see Ref. [17]) and the Ar ma package Version 1.00 [18].
P. Grau-Carles / Physica A 287 (2000) 396 – 404
403
Table 5 Ar ma estimation for returns, absolute returns and squared returns
DOW
SP500
FTSE
NIKKEI
IGBM
a Statistical
d MA-1 MA-2 d AR-1 AR-2 MA-1 MA-2 d MA-1 MA-2 d AR-1 AR-2 MA-1 MA-2 d AR-1 AR-2 AR-3 MA-1 MA-2
rt
|rt |
rt2
(0; d; 1) −0:042 (−3:69)a 0:148 (9:98)a
(0; d; 1) 0:311 (20:1)a −0:213 (−10:8)a
(0; d; 2) 0:101 (8:41)a −0:037 (−2:78)a 0:401 (37:4)a
(0; d; 1) −0:032 (−2:71)a
(2; d; 2) 0:371 (15:1)a −0:686 (19:2)a 0:236 (6:89)a 0:414 (9:37)a −0:514 (11:9)a
(2; d; 2) 0:140 (11:8)a −0:720 (−8:30)a −0:694 (−20:7)a 0:672 (8:97)a 0:747 (25:2)a
(0; d; 1) −0:124 (−3:47)a −0:148 (−3:21)a
(0; d; 2) 0:111 (3:18)a 0:354 (7:97)a −0:074 (−1:98)a
(0; 0; 1) −0:0002 (−0:0111) 0:917 (97:8)a
(0; d; 2) 0:038 (1:43)a
(2; d; 1) 0:417 (8:67)a 0:434 (8:63)a 0:054 (2:15)a −0:709 (−11:3)a
0:137 (9:13)a
−0:047 (−1:45) −0:086 (−4:28)a (2; d; 2) 0:076 (3:86)a −0:608 (−4:42)a −0:526 (−4:93)a
(0; d; 2) 0:378 (10:8)a
0:729 (5:28)a 0:573 (6:03)a
−0:244 (−6:12)a −0:059 (−2:03)a
(3; d; 2) 0:205 (7:76)a −0:022 (−0:71) −0:042 (−2:06)a (3; d; 2) 0:301 (15:4)a 0:825 (33:5)a −0:811 (−32:3)a −0:167 (−6:85)a −0:991 (−153:)a 0:990 (133)a
signi cance at 5% level.
DFA analysis. With respect to the selected model it can be observed that there is no unanimity. Although it is well known that the AIC tends to overestimate the model order, this fact does not signi cantly a ect the estimation of the long memory term. 5. Conclusions Our investigation on the behaviour of the returns of six stock indexes using nonparametric, semiparametric and parametric statistical tests shows no evidence of long-term dependence, or to a very little degree if we attend to the results of the maximum likelihood estimation. In the case of the absolute and squared returns, long memory is found stronger for absolute returns than for squared returns. We have found important anomalies to this behaviour in the study of the FTSE index. Another remarkable result is that the agreement of the four methods employed decreases if we
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P. Grau-Carles / Physica A 287 (2000) 396 – 404
use the whole series of the Dow Jones and the SP500, which can be a consequence of the structural changes that would produce time-varying parameters or regime switches. The pattern of dependence in volatility may not remain constant over time, so any attempt to model the long dependence should reproduce these results. These empirical ndings open a new direction on the research of modelling stock markets. References [1] M.U. Taquu, V. Teverovsky, in: R. Alder, R. Feldman, M.S. Taquu (Eds.), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhauser, Boston, 1998, pp. 177–217. [2] Y. Cheung, K. Lai, J. Int. Money Finance 14 (1995) 597. [3] C.W.J. Granger, Z. Ding, Discussion Paper 93-38. University of California, San Diego, 1993. [4] Z. Ding, C.W.J. Granger, R.F. Engle, J. Empirical Finance 1 (1993) 83. [5] B.B. Mandelbrot, J.R. Wallis, Water Resour. Res. 4 (1968) 967. [6] H.E. Hurst, Trans. Amer. Soc. Civil Eng. 116 (1951) 770. [7] A. Lo, Econometrica 59 (1991) 451–474. [8] V. Teverovsky, M.U. Taquu, W. Willinger, J. Statist. Plann. Inference 80 (1999) 211. [9] C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1684. [10] G.M. Viswanathan, S.V. Buldyrev, S. Havlin, H.E. Stanley, Biophys. J. 72 (1997) 866. [11] J. Geweke, S. Porter-Hudak, J Time Ser. Anal. 4 (1983) 221. [12] J. Hosking, Biometrika 68 (1981) 165. [13] F.B. Sowell, J. Monetary Econ. 29 (1992) 277. [14] F.B. Sowell, J. Econometrics 53 (1992) 165. [15] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Stanley, Phys. Rev. E 60 (1999) 1390. [16] D. Brook eld, Appl. Econ. Lett. 2 (1995) 110. [17] J.A. Doornik, Object-Oriented Matrix Programming Using Ox, 3rd Edition, Timberlake Consultants Press, London, 1999. [18] J.A. Doornik, M. Ooms, A package for estimating, forecasting and simulating Ar ma models, 1999, Oxford: www.nu .ox.ac.uk=Users=Doornik.