- Email: [email protected]

Fractional integration in daily stock market indexes L.A. Gil-Alana T Universidad de Navarra, Faculty of Economics, Edificio Biblioteca, Entrada Este, E-31080 Pamplona, Spain Received 28 April 2004; received in revised form 31 August 2004; accepted 11 February 2005

Abstract I use parametric and semiparametric methods to test for the order of integration in stock market indexes. The results, which are based on the EOE (Amsterdam), DAX (Frankfurt), Hang Seng (Hong Kong), FTSE100 (London), S&P500 (New York), CAC40 (Paris), Singapore All Shares, and the Japanese Nikkei, show that in almost all of the series the unit root hypothesis cannot be rejected. The Hang Seng and the Singapore All Shares seem to be the most nonstationary series with orders of integration higher than one, and the S&P500 is the less nonstationary series, with values smaller than one and showing mean reversion. D 2005 Elsevier Inc. All rights reserved. JEL classification: C22; G14 Keywords: Stock market; Unit roots; Long memory; Mean reversion

1. Introduction An important issue in the empirical analysis of financial time series is whether holding-period returns on a risky asset are serially independent, as required by the efficient market hypothesis in its weak form (i.e., the current stock prices fully reflect all the past stock prices information.). Although a precise formulation of an empirically refutable efficient market hypothesis must be model-specific, historically, most such tests focus on the forecastability of common stock returns.

T Tel.: +34 948 425 625; fax: +34 948 425 626. E-mail address: [email protected] 1058-3300/$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.rfe.2005.02.003

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In this paper I use fractionally integrated techniques to revisit the long memory and mean reversion issues in the stock markets. In particular, I examine if the stock market indexes can be specified in terms of I(d) statistical models, with the possibility that d is a fractional number. For stock indexes, the evidence in favor of long memory may be due to the effect of aggregation. In fact, aggregation is one of the main sources of long memory. The key idea is that aggregation of independent weakly dependent series can produce a strong dependent series: Granger (1980) and Robinson (1978) show that fractional integration can arise as a result of aggregation when: (a) data are aggregated across heterogeneous autoregressive (AR) processes, and (b) data involving heterogeneous dynamic relationships at the individual level are then aggregated to form the time series. Moreover, the existence of long memory in financial asset returns suggests that new theoretical models based on nonlinear pricing models should be elaborated. Mandelbrot (1971) notes that in the presence of long memory, martingale models of asset prices cannot be obtained from arbitrage. In addition, statistical inference concerning asset pricing models based on standard testing procedures may not be appropriate in the context of long memory processes (see, e.g., Barkoulas, Baum, & Travlos, 2000; Yajima, 1985). The paper proceeds as follows. In Section 2 I briefly review the literature on modeling stock market indexes using long memory processes. In Section 3 I define the concept on fractional integration. In Section 4 I describe some of the most frequently used techniques for fractional integration, with special emphasis on a parametric and a semiparametric method that have some distinguishing features compared with other methods. In Section 5, I apply these procedures to several stock market indexes. Section 6 concludes.

2. Historical background Within the paradigm of the efficient market hypothesis, which has been broadly categorized as the brandom walkQ theory of stock prices, the evidence is mixed. For instance, using a variance-ratio test, Lo and MacKinley (1988) and Poterba and Summers (1988), conclude that stock returns exhibit mean reversion. Fama and French (1988), who examine the autocorrelations of one-period returns, also find mean reversion. By contrast, using a generalized form of rescaled range (R\S) statistic, Lo (1991) finds no evidence against the random walk hypothesis. Using annual data and allowing for fractional alternatives, Caporale and Gil-Alana (2002) report that US stock returns are close to being an I(0) series, and point out that their degree of predictability depends on the process followed by the error term. As mentioned in the introduction, in this paper I focus on fractional integration and long memory behavior. The literature in this topic has increased in recent years, but the results are mixed. Some authors find little or no evidence of long memory in stock markets (see, e.g., Hiemstra & Jones, 1997, and the references therein). Barkoulas and Baum (1996), Barkoulas et al. (2000), Cheung and Lai (1995), Crato (1994), Henry (2002), Sadique and Silvapulle (2001), and Tolvi (2003) are among those who find evidence of long memory in monthly, weekly, and daily stock market returns. The study of long range dependence clearly requires a sufficiently long series to justify the application of large sample inference rules based on semiparametric models. However, there is not yet a finite sample theory for rules of parametric inference on long memory. Several recent papers use the Standard and Poor’s (S&P) 500 index of over 17,000 daily observations. Granger and Ding (1995a, 1995b) focus on power transformation or absolute value of the returns (which they use as proxies of volatility). They estimate a long memory process to study persistence in volatility,

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

and establish some stylized facts (temporal and distributional properties). However, in their later study, Granger and Ding (1996) find that the parameters of the long memory model vary considerably from one subseries to the next. Ryden, Terasvirta, and Asbrink (1998) claim that the temporal higher-order dependence observed in return series are better described by a hidden Markov model, and again report that the parameter estimates of the model differ depending on the subseries being considered. However, their model does not account for an important distributional property of absolute returns (i.e., their very slowly decaying autocorrelation function). This model is reproduced by Granger and Terasvirta (1999) in the context of a nonlinear model.

3. Fractional integration I define an I(0) process {u t , t = 0, F1, . . .} as a covariance stationary process with a spectral density function that is positive and finite at the zero frequency. In this context, a given time series {x t , t = 0, F1, . . .} is I(d) if: ð1 LÞd xt ¼ ut ; t ¼ 1; 2; N ; ; xt ¼ 0; tV0;

ð1Þ

where u t is I(0) and L is the lag operator (Lx t = x t1). The polynomial above can be expressed in terms of its binomial expansion, such that for all real d, l X d ðd 1Þ 2 d d L N: ð 1Þ j L j ¼ 1 dL þ ð1 LÞ ¼ j 2 j¼0 The macroeconomic literature stresses the cases of d = 0 and 1. However, d can be any real number. Clearly, if d = 0 in (1), x t = u t , and a bweakly autocorrelatedQ x t is allowed for. However, if d N 0, x t is said to be a long memory process. (I note that if d N 0 the process is also called bstrongly autocorrelatedQ because of the strong association between observations that are widely separated in time.) As d increases beyond 0.5 and through 1, x t can be viewed as becoming bmore nonstationary,Q for example, in the sense that the variance of partial sums increases in magnitude. These processes were initially introduced by Granger (1980, 1981), Granger and Joyeux (1980) and Hosking (1981) (though earlier work by Adenstedt, 1974; Taqqu, 1975 shows an awareness of its representation). They were theoretically justified in terms of aggregation of ARMA processes with randomly varying coefficients by Granger (1980) and Robinson (1978). Chambers (1998), CioczekGeorges and Mandelbrot (1995), and Taqqu, Willinger, and Sherman (1997) also use aggregation to motivate long memory processes, while Parke (1999) uses a closely related discrete time error duration model. More recently, Diebold and Inoue (2001) propose another source of long memory based on regimeswitching models. Empirical applications based on fractional models like (1) are among others Baillie and Bollerslev (1994), Diebold and Rudebusch (1989), Gil-Alana (2000), and Gil-Alana and Robinson (1997). To determine the appropriate degree of integration in a given time series is important from both economic and statistical viewpoints. If d = 0, the series is covariance stationary and possesses bshort

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memoryQ, with the autocorrelations decaying fairly rapidly. If d belongs to the interval (0, 0.5), x t is still covariance stationary. However, the autocorrelations take a much longer time to disappear than in the previous case. If d a [0.5, 1), the series is no longer covariance stationary, but it is still mean reverting, with the effect of the shocks dying away in the long run. Finally, if d z 1, x t is nonstationary and nonmean reverting. Thus, the fractional differencing parameter d plays a crucial role in describing the persistence in the time series behavior: the higher the d, the higher the level of association between the observations.

4. The testing procedures There are many approaches for estimating and testing the fractional differencing parameter d. Earlier studies test the long memory hypothesis in the stock markets using the rescaled–range (R\S) method, which is suggested by Hurst (1951), and is defined as max

1V jbT

RnS ¼

j j X X ðxt x¯ Þ min ðxt x¯ Þ 1VjbT

t¼1

T 1 X ðxt x¯ Þ2 T t¼1

t¼1

! 12

where x¯ is the sample mean of the process x t . The specific estimate of d (Mandelbrot & Wallis, 1968) is given by: logðRnSÞ 1 : dˆ ¼ logT 2 Mandelbrot (1972, 1975), Mandelbrot and Taqqu (1979) and Mandelbrot and Wallis (1969) analyze the properties of this procedure. Beran (1994) explains how to implement the R\S procedure. A problem with this statistic is that the distribution of its test statistic is not well defined and is sensitive to shortterm dependence and heterogeneities of the underlying data generating process. Lo (1991) develops a modified R\S method that addresses these drawbacks of the classical R\S method. Another method widely used in the empirical work is the one proposed by Geweke and Porter-Hudak (GPH, 1983), which is a semiparametric procedure. They use it to obtain an estimate of the fractional differencing parameter d based on the slope of the spectrum around the zero frequency. However, this method has some potential problems. First, it is too sensitive to the choice of the bandwidth parameter numbers, and in the presence of short range dependence such as autoregressive or moving average terms, the GPH estimator is known to be biased in small samples (see, e.g., Agaikloglou, Newbold, & Wohar, 1992). In the context of parametric approaches, Sowell (1992) analyzes the exact maximum likelihood estimates of the parameters of the fractionally ARIMA( p, d, q) model /ð LÞð1 LÞd xt ¼ hð LÞet ;

t ¼ 1; 2; N N ;

ð2Þ

where /(L) and h(L) are polynomials of orders p and q, respectively, with all zeroes of /(L) and h(L) outside the unit circle, and e t is white noise. He uses a recursive procedure

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

that allows quick evaluation of the likelihood function in the time domain, which is given by 1 T=2 1=2 1 jRj exp X TV R XT ; ð2pÞ 2 where X T = (x 1, x 2,. . ., x T)V and X T ~ N(0, R). I use both parametric and semiparametric methods. First, I present a parametric testing procedure developed by Robinson (1994a) that permits me to test I(d) statistical models in raw time series. Then I apply a semiparametric method (Robinson, 1995a). These two methods have several distinguishing features that make them particularly relevant in comparison with other procedures. Thus, they have a standard null limit behavior, unlike other methods for testing in which the limit distribution has to be calculated numerically on a case by case simulation study. Moreover, the limit distribution in Robinson (1994a) is unaffected by the inclusion of deterministic trends or the type of I(0) disturbances used to specify the short run components of the series. Another useful property of these procedures is that they do not require Gaussianity (a condition rarely satisfied in financial time series). Only a moment condition of order two is necessary. 4.1. A parametric testing procedure Robinson (1994a) proposes a Lagrange Multiplier (LM) test of the null hypothesis: H0 : d ¼ d0 :

ð3Þ

in a model given by yt ¼ bVzt þ xt ;

t ¼ 1; 2; N ;

ð4Þ

and (1), for any real value d 0, where y t is the observed time series; b = (b 1, . . ., bk)V is a (k 1) vector of unknown parameters; and z t is a (k 1) vector of deterministic regressors that may include, for example, an intercept, (e.g. z t u 1), or an intercept and a linear time trend (in case of z t = (1,t)V). The test statistic is given by: rˆ ¼

T 1=2 ˆ1=2 A aˆ ; rˆ 2

ð5Þ

where T is the sample size and 1 1 1 1 2p TX 2p TX w kj g kj ; sˆ I kj ; rˆ 2 ¼ r2 ðsˆ Þ ¼ g kj ; sˆ I kj ; T T j¼1 j¼1 0 1 !1 TX 1 TX 1 T 1 T 1 X X 2 2 Aˆ ¼ @ w kj w kj eˆ kj V eˆ kj eˆ kj V eˆ kj w kj A; T j¼1 j¼1 j¼1 j¼1

aˆ ¼

kj w kj ¼ log 2sin ; 2

B logg kj ; sˆ ; eˆ kj ¼ Bs

kj ¼

2pj ; T

sˆ ¼ arg min r2 ðsÞ; saT 4

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

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where T* is a compact subset of the R q Euclidean space. I(k j ) is the periodogram of u t evaluated under the null, i.e., !1 T T X X wt wt V wt ð1 LÞd0 yt ; wt ¼ ð1 LÞd0 zt ; uˆ t ¼ ð1 LÞd0 yt bˆ Vwt ; bˆ ¼ t¼1

t¼1

and the function g above is a known function coming from the spectral density function of u t , r2 f k; r2 ; s ¼ g ðk; sÞ; pbkVp: 2p I note that these tests are purely parametric and therefore, they require that I make specific modeling assumptions regarding the short memory specification of u t . Thus, if u t is white noise, g u 1, and if u t is an AR process of form /(L)u t = e t , g = j/(e ik )j2, with r 2 = V(e t ), so that the AR coefficients are a function of s. Based on the null hypothesis H0 (3), Robinson (1994a) establishes that under certain regularity conditions: rˆ Yd N ð0; 1Þ as T Yl;

ð6Þ

and also, the Pitman efficiency of the tests against local departs from the null. This Pitman efficiency theory means that the test is the most efficient when directed against local alternatives. Thus, I am in a classical large sample-testing situation: An approximate one-sided 100a% level test of H0 (3) against the alternative: Ha: d N d 0 (d N d 0) is given by the rule: bReject H0 if rˆ N z a (rˆ b z a )Q, where the probability that a standard normal variate exceeds z a is a. A problem with the parametric procedures is that the model must be correctly specified. Otherwise, the estimates are liable to be inconsistent. (However, I note that the method described just above has nothing to do with the estimation of the fractional differencing parameter. It merely computes diagnostic departures from the null which may be any real number.) In fact, misspecification of the short run components of the process may invalidate the estimation of the long run parameter d. This is the main reason for also using a semiparametric procedure. 4.2. A semiparametric estimation procedure There are several methods for estimating the fractional differencing parameter in a semiparametric way. Examples are the log-periodogram regression estimate (LPE), initially proposed by Geweke and PorterHudak (GPH, 1983) and later modified by Ku¨nsch (1986) and Robinson (1995b); the average periodogram estimate of Robinson (APE, 1994b); and a Gaussian semiparametric estimate (Robinson, 1995a). Robinson’s (1995a) Gaussian semiparametric estimate is basically a local bWhittle estimateQ in the frequency domain, based on a band of frequencies that degenerates to zero. The estimate is implicitly defined by: ! m X PPP 1 ð7Þ logkj ; dˆ ¼ arg min logC ðd Þ 2d d m j¼1 for dað 1=2; 1=2Þ;

PPP

C ðd Þ ¼

m 1 X I kj k2d j ; m j¼1

kj ¼

2pj ; T

m Y0: T

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

where m is a bandwidth parameter number, and I(k j ) is the periodogram of the time series, x t , given by: 2 T 1 X xt eikj t : I kj ¼ 2pT t¼1 Under finiteness of the fourth moment and other mild conditions, Robinson (1995a) proved that: pﬃﬃﬃﬃ m dˆ d0 Y d N ð0; 1=4Þ as T Yl; where d 0 is the true value of d and with the only additional requirement that m Yl slower than T. The exact requirement is that (1/m) + ((m 1+2a (logm)2)/(T 2a ))Y0 as T Yl, where a is determined by the smoothness of the spectral density of the short run component. In case of a stationary and invertible ARMA, a may be set equal to 2 and the condition is (1/m) + ((m 5(logm)2)/(T 4))Y0 as T Yl. Robinson (1995a) showed that m must be smaller than T/2 to avoid aliasing effects. I use the Gaussian Whittle estimate because of its computational simplicity. I note that the Gaussian method requires no additional user-chosen numbers in the estimation (as is the case with the LPE and the APE). Also, the estimate is more efficient than the LPE, and I do not assume Gaussianity to obtain an asymptotic normal distribution.

5. Empirical evidence of long memory in the stock market indexes The time series I analyze correspond to the daily structure of several stock market indexes in Amsterdam (EOE), Frankfurt (DAX), Hong Kong (Hang Seng), London (FTSE100), New York (S&P500), Paris (CAC40), Singapore (Singapore All Shares) and Tokyo (Nikkei) from January 6, 1986 until December 31, 1997, obtained from Franses and Van Dijk (2000). Fig. 1 displays plots of each of the original time series, all of which have a nonstationary appearance. Fig. 2 shows that the first differenced series may be stationary. However, a visual inspection of the correlograms in Fig. 3 shows that there are significant values even at some lags relatively far away from zero. These values may be an indication of fractional integration in the original series with d smaller than or higher than one. Also, the periodograms of the first differenced data in Fig. 4 show that in some of the series there are values close to zero at the smallest frequency, which may indicate that they are now overdifferenced. I note that the periodogram is an estimate of the spectral density function f(k). If a series is overdifferenced, f(0) = 0, so we should expect a similar pattern in the periodogram. Table 1 summarizes the empirical work based on these and other related indexes. Most of the authors use the R\S statistic, the modified R\S, the GPH, and the Gaussian Whittle estimate, along with the Sowell’s (1992) parametric procedure and the Diebold and Rudebusch (1989) method. The latter is a two-step procedure that uses dˆ obtained by GPH, and in the second step, estimates the ARMA parameters from the dˆ -differenced series. The results across this table are mixed. For the Dutch index, d is one or slightly higher. For the German DAX, the values are constrained between 0.97 and 1.11. For the Hang Seng, they range

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48 EOE (Amsterdam)

DAX (Frankfurt)

5000

1200

35

4000 800

3000 2000

400 1000 0 6.Jan.86

0 1.Jul.86

31.Dec.97

Hang Seng (Hong Kong)

12.Dec.97

FTSE100 (London)

20000

6000

16000 4000

12000 8000

2000

4000 0

6.Jan.86

0 6.Jan.86

31.Dec.97

S&P 500 (New York)

31.Dec.97

CAC40 (Paris) 4000

1200

3000 800 2000 400

0

1000

6.Jan.86

0

31.Dec.97

9.Aug.86

Singapore All Shares (Singapore)

31.Dec.97

Nikkei (Tokyo) 50000

800

40000

600

30000 400 20000 200 0

10000

6.Jan.86

31.Dec.97

0 6.Jan.86

31.Dec.97

Fig. 1. Plots of the original time series.

between 0.95 and 1.23. For the FTSE, d = 1 cannot be rejected with the R\S and the modified R\S. Values of d smaller than one are obtained with the other procedures. For Singapore All Shares, all values are strictly higher than one. For the S&P500, d is constrained between 0.77 and 1.32. For the French CAC, d is between 0.94 and 1.07, and is much higher than one for the Japanese Nikkei. Thus, I can summarize the results across this table by saying that the estimated values of d vary below and above one for all indexes except the Singapore All Shares and the Nikkei. For these two indexes, d is strictly above one, which implies long memory on their corresponding returns.

36

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48 EOE (Amsterdam)

DAX (Frankfurt)

60

400

40 200 20 0

0

-20 -200 -40 -60

7.Jan.86

31.Dec.97

-400

7.Jan.86

Hang Seng (Hong Kong)

31.Dec.97

FTSE100 (London) 200

2000

100

1000

0 0 -100 -1000 -2000

-200 7.Jan.86

31.Dec.97

-300

7.Jan.86

S&P 500 (New York)

31.Dec.97

CAC40 (Paris) 200

60 30

100

0 0 -30 -100

-60 -90 2.Jul.86

12.Dec.97

-200 10.Aug.87

Singapore All Shares (Singapore)

Nikkei (Tokyo)

60

4000

40

2000

20

0

0

-2000

-20

-4000

-40

2.Jul.86

31.Dec.97

31.Dec.97

-6000

2.Jul.86

31.Dec.97

Fig. 2. Plots of the first differenced time series.

5.1. The parametric approach I start by performing the parametric procedure of Robinson (1994a). I denote each of the time series by y t , and employ throughout the model given by (1) and (4), with z t = (1,t)V, t z 1, z t = (0,0)V otherwise. Thus, under the null hypothesis H0 (3): yt ¼ b0 þ b1 t þ xt ; ð1 LÞd0 xt ¼ ut ;

t ¼ 1; 2; N t ¼ 1; 2; N :

ð8Þ ð9Þ

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

37

DAX (Frankfurt)

EOE (Amsterdam) 0,2

0,08 0,04

0,1

0

0

-0,04 -0,1 -0,2

-0,08 1

500

-0,12

1

Hang Seng (Hong Kong)

FTSE100 (London)

0,15

0,12

0,1

0,08

0,05

0,04

0

0

-0,05

-0,04

-0,1

1

500

500

-0,08

1

S&P 500 (New York)

500

CAC40 (Paris) 0,08

0,12 0,08

0,04

0,04 0 0 -0,04

-0,04 -0,08

1

500

-0,08

1

Singapore All Shares (Singapore)

500

Nikkei (Tokyo)

0,2

0,08

0,15

0,04

0,1

0

0,05 -0,04

0

-0,08

-0,05 -0,1

1

500

-0,12

1

500

The large sample standard error under the null hypothesis of no autocorrelation is 1/√T or roughly 0.018

Fig. 3. Correlograms of the first differenced time series.

and treat separately the cases b 0 = b 1 = 0 a priori; b 0 unknown and b 1 = 0 a priori; and b 0 and b 1 unknown, i.e., I consider, respectively, the cases of no regressors in the undifferenced regression (8), an intercept, and an intercept and a linear time trend. I report the test statistic not only for the case of d 0 = 1 (a unit root), but also for d 0 = 0, (0.25), 2, thus including also a test for stationarity (d 0 = 0.5), for I(2) processes (d 0 = 2), as well as other fractionally integrated possibilities. The test statistic reported across Table 2 (and also in Tables 3 and 4) is the one-sided statistic that corresponds to rˆ in (5), so that significantly positive values of rˆ are consistent with orders of integration higher than d 0, but significant negative values are consistent with alternatives of form: d b d 0.

38

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48 EOE (Amsterdam)

DAX (Frankfurt)

40

800

30

600

20

400

10

200

0

1

3126

0

1

Hang Seng (Hong Kong) 30000

800

22500

600

15000

400

7500

200

0

1

3126

0

1

S&P 500 (New York) 800

30

600

20

400

10

200

1

3126

CAC40 (Paris)

40

0

3126

FTSE100 (London)

3126

0

1

Singapore All Shares (Singapore)

2733

Nikkei (Tokyo) 160000

30

120000 20 80000 10 40000 0

1

3126

0

1

3126

Fig. 4. Periodograms of the first differenced time series.

A notable feature that I observe in Table 2, in which u t is taken to be white noise, is the fact that the value of the test statistic monotonically decreases with d 0. This monotonicity is to be expected, in view of the fact that it is a one-sided statistic. I can see, across this table, that the only value of d 0 where H0 (3) cannot be rejected takes place at d 0 = 1. It happens for all series except the FTSE100 and the Singapore All Shares. In these two cases, the unit root null is rejected in favor of higher orders of integration and, although d 0 = 1.25 is also rejected, this time it is rejected against smaller values of d 0, suggesting that the order of integration for these series should be constrained between these two values. In general, the results seem to be robust to the different specifications for z t in (4), implying that the intercept and the time trend may be unimportant when modeling these series.

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

39

Table 1 Empirical work based on stock market prices Country (index)

Authors

Data frequency

Time period

Methods

Results

The Netherlands (EOE)

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Jacobsen (1996)

Monthly

Dec 52–Dec 90

Mod. R\S: d = 1 GPH: d = 1.06, 1.31 d =1

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Lux (1996)

Daily

Jan 88–Oct 95

Jacobsen (1996)

Monthly

Dec 52–Aug 90

Tolvi (2003)

Monthly

Jan 60–Oct 99

Lipka and Los (2003)

Daily

Nov 90–Oct 00

Henry (2002)

Monthly

Jan 82–Sep 98

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Jacobsen (1996)

Monthly

Dec 52–Dec 90

Henry (2002)

Monthly

Jan 82–Sep 98

Tolvi (2003)

Monthly

Jan 60–Sep 99

Lipka and Los (2003)

Daily

Apr 84–Oct 00

Henry (2002)

Monthly

Jan 82–Sep 98

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Jacobsen (1996)

Monthly

Dec 52–Dec 90

Hiemstra and Jones (1997) Lobato and Savin (1997) Henry (2002)

Daily Daily Monthly

Jul 62–Dec 91 Jul 62–Dec 94 Jan 82–Sep 98

Tolvi (2003)

Monthly

Jan 60–Sep 99

Mod. R\S GPH R\S Mod. R\S Mod. R\S GPH R\S Mod RS GPH R\S Mod. R\S Sowell, S S + Outliers R\S Mod. R\S GPH QMLE D&R Mod. R\S GPH Mod. R\S GPH R\S Mod. R\S GPH QMLE D&R Sowell, S S + Outliers R\S Mod. R\S GPH QMLE D&R Mod. R\S GPH Mod. R\S GPH R\S Mod. R\S Mod R\S QMLE GPH QMLE D&R Sowell, S S + Outliers

Germany (DAX)

Hong Kong (Hang Seng)

U.K. (FTSE)

Singapore (All Shares)

U.S. (S&P500)

Mod. R\S: d = 1 GPH: d = 1.08, 1.10 R\S Mod R\S: d = 1 GPH d = 1.11, 0.97 d =1 S: d = 1.048 S + Outliers: d = 1.043 R\S: d = 1.01 Mod R\S: d = 1.03 GPH: d = 1.18, 1.23 QMLE: d = 0.95 D&R: d = 0.98 Mod. R\S: d = 1 GPH: d = 0.99, 1.01 Mod. R\S: d = 1 GPH: d = 0.90, 0.96 d =1 GPH: d = 1.05, 0.80 QMLE: d = 0.97 D&R: d = 0.76 S: d = 0.947 S + Outliers: d = 1.013 R\S: d = 1.01 Mod R\S: d = 1.03 GPH: d = 1.03, 0.91 QMLE: d = 1.01 D&R: d = 1.06 Mod. R\S: d = 1 GPH: d = 1.03, 1.09 Mod. R\S: d = 1 GPH: d = 1, 0.862 d =1 d =1 d =1 GPH: d = 1.32, 0.77 QMLE: d = 1.10 D&R: d = 0.98 S: d = 0.953 S + Outliers: d = 0.988 (continued on next page)

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

Table 1 (continued) Country (index)

Authors

Data frequency

Time period

Methods

Results

France (CAC)

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Jacobsen (1996)

Monthly

Dec 52–Dec 90

Mod. R\S: d = 1 GPH: d = 1.03, 1.07 d =1

Tolvi (2003)

Monthly

Jan 60–Oct 99

Lipka and Los (2003)

Daily

Apr 84–Oct 00

Cheung and Lai (1995)

Monthly

Jan 70–Aug 92

Jacobsen (1996)

Monthly

Dec 52–Dec 90

Henry (2002)

Monthly

Jan 82–Sep 98

Mod. R\S GPH R\S Mod. R\S Sowell, S S + Outliers R\S Mod. R\S Mod. R\S GPH R\S Mod. R\S GPH QMLE D&R

Japan (Nikkei)

S: d = 0.95 S Outliers: d = 0.955 R\S: d = 1.01 Mod R\S: d = 0.94 Mod. R\S: d = 1 GPH: d = 1.46, 1.41 d =1 GPH: d = 1.48, 1.40 QMLE: d = 1.04 D&R: d = 1.02

Cheung and Lai (1995) use for all countries Morgan Stanley Capital International (MSCI) indices. Jacobsen (1996) employs a CBS-Index of the Dutch Stock Market. Hiemstra and Jones (1997) use New York and American Stock Exchanges obtained from the CRSP Stock Files.

Table 2 Testing H0 (3) in (1) and (4) in the stock market indices with white noise disturbances Series

zt

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

EOE

– 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1,

237.51 237.51 249.67 233.60 233.60 238.09 251.16 251.16 229.00 236.60 236.60 227.18 235.95 235.95 245.75 198.64 198.64 194.31 249.53 249.53 205.05 250.95 250.99 236.46

236.23 225.57 232.21 223.99 216.24 215.34 230.36 229.73 178.80 221.06 210.72 192.89 235.55 217.87 227.69 163.78 175.55 168.07 173.11 216.38 182.10 229.97 219.43 212.09

148.58 170.93 156.79 130.88 147.09 133.66 118.65 135.72 104.18 118.87 130.68 111.00 144.59 155.01 137.62 93.42 105.36 98.14 102.49 131.08 123.69 144.05 138.99 141.19

36.88 43.40 40.20 35.73 35.91 34.04 33.67 35.82 34.61 35.34 36.93 35.61 33.95 33.44 30.48 29.48 29.73 29.20 37.82 47.96 48.11 37.25 37.59 37.81

0.21 0.68 0.68 0.48 1.31 1.31 1.28 1.34 1.34 2.01 3.66 3.66 0.09 1.47 1.47 1.13 0.44 0.44 4.31 8.16 8.16 0.23 0.35 0.35

11.63 11.93 11.93 11.78 12.41 12.41 11.44 11.15 11.50 10.47 9.12 9.12 11.23 11.56 11.55 10.18 10.34 10.34 9.02 7.15 7.15 11.62 11.70 11.70

17.12 16.94 16.94 17.37 17.40 17.40 17.28 17.26 17.26 16.36 15.33 15.33 16.75 16.54 16.54 15.61 15.52 15.52 15.32 14.14 14.14 16.84 16.65 16.65

20.22 19.88 19.88 20.44 20.22 20.22 20.41 20.35 20.35 19.67 18.87 18.87 19.95 19.56 19.55 18.68 18.51 18.51 19.01 18.05 18.05 19.83 19.47 19.47

22.16 21.83 21.83 22.35 22.03 22.03 22.30 22.22 22.22 21.74 21.13 21.13 21.96 21.56 21.54 20.60 20.44 20.44 21.30 20.52 20.52 21.75 21.32 21.32

DAX

Hang Seng

FTSE100

S&P500

CAC40

Singapore All Shares

Nikkei

t)V

t)V

t)V

t)V

t)V

t)V

t)V

t)V

In bold: The non-rejection values of the null hypothesis at the 5% significance level.

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

41

Table 3 Testing H0 (3) in (1) and (4) in the stock market indices with AR(1) disturbances Series

zt

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

EOE

– 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1,

0.62 0.62 0.98 1.30 1.30 1.95 0.38 0.38 1.53 0.15 0.15 2.77 0.82 0.82 1.84 0.93 0.93 0.06 4.49 4.49 7.04 0.20 0.20 0.19

12.92 12.96 14.13 13.73 13.68 14.36 11.57 11.69 12.98 10.40 10.51 12.52 11.05 11.29 13.67 12.89 12.69 12.32 10.27 10.15 10.17 12.76 12.55 11.42

20.17 8.09 7.44 20.87 2.94 1.73 2.16 0.76 1.64 20.95 4.38 2.05 12.79 0.54 0.70 25.91 5.64 0.22 23.69 10.98 5.82 18.00 1.23 7.31

0.95 25.39 26.41 1.37 20.71 21.62 12.89 16.32 16.46 5.84 12.58 13.17 4.65 17.80 18.57 10.88 12.58 13.20 13.96 14.25 14.38 5.94 20.43 19.24

0.50 1.03 1.03 1.74 0.72 0.72 1.89 2.02 2.02 0.70 0.32 0.32 1.00 3.10 3.10 0.63 0.66 0.67 1.61 1.59 1.59 0.94 1.50 1.50

4.73 9.31 9.31 4.25 9.30 9.31 6.82 7.29 7.29 4.26 7.59 7.59 6.14 9.63 9.63 2.47 7.57 7.57 4.01 6.85 6.85 7.17 10.34 10.34

9.59 13.19 13.19 9.59 13.78 13.78 12.45 12.86 12.86 9.26 12.07 12.06 10.52 13.23 13.23 7.38 11.55 11.55 8.19 11.56 11.56 11.50 14.54 14.54

13.33 15.74 15.74 13.52 16.62 16.62 16.17 16.45 16.45 13.02 15.09 15.09 13.97 15.71 15.71 11.33 14.22 14.22 11.87 14.60 14.60 14.66 17.06 17.06

16.11 17.68 17.68 16.36 18.63 18.63 18.70 18.91 18.90 15.83 17.26 17.27 16.50 17.56 17.53 14.24 16.16 16.16 14.76 16.77 16.77 16.99 18.79 18.88

DAX

Hang Seng

FTSE100

S&P500

CAC40

Singapore All Shares

Nikkei

t)V

t)V

t)V

t)V

t)V

t)V

t)V

t)V

In bold: The non-rejection values of the null hypothesis at the 5% significance level.

However, even bearing in mind the monotonicity obtained in the previous results, the significance of these results might be due in large part to the unaccounted-for I(0) autocorrelation in u t . Thus, I also fit other models, taking into account a weakly autocorrelated structure on the disturbances. First, I impose an AR(1) process for u t . Table 3 presents the results. The first noticeable thing here is that there is a lack of monotonicity in the value of rˆ with respect to d 0, especially for small values of d 0. This lack of monotonicity might be explained in terms of model misspecification, as is argued, for example, in Gil-Alana and Robinson (1997). However, it may also be due to the fact that the AR coefficients are Yule-Walker estimates. Therefore, although they are smaller than one in absolute value, the coefficients can be arbitrarily close to one. A problem may occur, in that these coefficients may be capturing the order of integration of the series by means, for example, of a coefficient of 0.99 when I use AR(1) disturbances. I observe that if d 0 is equal to or higher than 0.75, monotonicity is always achieved and the non-rejection values take place when d 0 = 1. Moreover, H0 (3) cannot be rejected with d 0 = 0.75 for the EOE and the DAX if z t = 0. Also, the unit root null is rejected against higher values of d for the Hang Seng, and against smaller orders of integration for the S&P500. AR modeling of I(0) processes is conventional, but there are many other types of I(0) processes, including some that are outside the stationary and invertible ARMA class. One that seems especially

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

Table 4 Testing H0 (3) in (1) and (4) in the stock market indices with Bloomfield (1) disturbances Series

zt

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

EOE

– 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1, – 1 (1,

149.74 149.73 159.06 146.68 146.67 150.13 158.89 158.87 X140.4 149.65 149.64 142.39 146.95 146.93 157.46 123.86 123.91 119.98 160.61 160.76 124.45 160.69 160.65 150.07

141.94 136.98 139.88 130.89 126.94 125.92 135.45 134.93 96.59 129.78 125.70 108.71 145.45 130.41 136.60 90.46 100.98 93.68 93.91 128.95 103.31 134.71 128.73 121.82

83.61 94.67 87.29 73.25 78.31 71.23 60.74 72.49 52.90 63.04 68.65 54.51 81.10 85.65 75.52 47.69 54.48 50.37 49.32 67.47 60.45 77.90 72.75 75.38

23.05 27.98 24.46 21.54 21.80 20.88 19.04 20.24 18.98 18.33 17.70 16.33 20.17 18.80 16.67 15.85 16.31 15.72 18.50 22.30 22.48 21.72 21.43 21.67

0.34 1.10 1.10 1.53 0.75 0.75 2.08 2.17 2.17 0.32 0.13 0.13 0.55 3.03 3.02 0.46 0.59 0.59 0.98 2.28 2.28 1.06 1.91 1.91

7.14 8.94 8.92 6.67 8.27 8.26 5.79 6.05 6.05 7.12 7.50 7.50 7.59 9.41 9.40 6.20 7.36 7.36 6.61 6.70 6.71 8.34 9.52 9.53

10.85 12.13 12.13 10.81 11.72 11.72 9.98 10.19 10.19 10.97 11.14 11.14 10.97 12.31 12.31 9.86 10.45 10.45 10.92 10.87 10.87 12.11 12.77 12.77

13.24 14.07 14.07 13.00 13.83 13.83 12.23 12.75 12.75 13.09 13.48 13.48 13.35 14.33 14.32 12.28 12.81 12.81 13.33 13.35 13.35 14.19 14.68 14.68

14.77 15.05 15.06 14.66 15.20 15.20 14.00 14.08 14.07 14.80 14.71 14.72 14.89 15.41 115.33 13.89 14.14 14.14 14.61 14.97 14.98 15.58 16.24 16.24

DAX

Hang Seng

FTSE100

S&P500

CAC40

Singapore All Shares

Nikkei

t)V

t)V

t)V

t)V

t)V

t)V

t)V

t)V

In bold: The non-rejection values of the null hypothesis at the 5% significance level.

relevant and convenient in the context of the present tests is that proposed by Bloomfield (1973), in which the spectral density function is given by: ! k X r2 sr cosðkrÞ ; ð10Þ f ðk; sÞ ¼ exp 2 2p r¼1 where k is now the number of parameters required to describe the short run dynamics of the series. Bloomfield (1973) shows that the logarithm of an estimated spectral density function of an ARMA( p, q) process is often a well-behaved function and can thus be approximated by a truncated Fourier series. He shows that (10) approximates it well if p and q have small values, which usually happens in economics. Like the stationary AR( p) model, the Bloomfield (1973) model has exponentially decaying autocorrelations, so I can use a model like this for u t in (9). Formulae for Newton-type iteration for estimating the s r involve no matrix inversion, and updating formulae with k is also straightforward. I can replace Aˆ below (5) by the population quantity l X l¼kþ1

l 2 ¼

k X p2 l 2 ; 6 l¼1

that is constant for s r, which is different from what happens in the AR case.

L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

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Table 4 presents the results based on the Bloomfield (1973) exponential model (with k = 1). I also use other values of k and obtain results that are similar to those reported in this table. Monotonicity is achieved for all series and all values of do, and like the previous tables, the unit root null is the only case where H0 (3) cannot be rejected. Thus, for the EOE, the DAX, the FTSE100, and the CAC40, the unit root hypothesis cannot be rejected independently of the inclusion or not of an intercept and/or a linear time trend. For the S&P500, the Nikkei, and the Singapore All Shares, this hypothesis cannot be rejected if z t = 0. However, if I include deterministic trends, it is rejected in favor of smaller values of d for the S&P500 and the Nikkei, and in favor of d N 1 for the Singapore All Shares. Finally, for the Hang Seng, the unit root null is rejected in all cases in favor of higher orders of integration. In the light of the results reported across these tables, I can conclude that there is some evidence in favor of unit roots in the stock market indexes, although in some cases I observe fractional degrees of integration with d slightly smaller or higher than one. To be more precise on the appropriate order of integration of each of the series, I recompute the Robinson’s (1994a) tests, but this time using values of d 0 = 0, (0.01), 2. Table 5 reports for each time series, each type of regressor, and each type of disturbances, the confidence intervals of those values of d 0 where H0 (3) cannot be rejected at the 5% significance level. The table also reports the value of d 0 (denoted by d*0) that produces the lowest statistics in absolute value across d 0. Table 5 Confidence intervals and values of do which produces the lowest statistics across d o Series

ut

EOE

White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield White noise AR Bloomfield

DAX

Hang Seng

FTSE100

S&P500

CAC40

Singapore All Shares

Nikkei

No regressor

An intercept

A linear time trend

Conf. Int.

d*o

Conf. Int.

d*o

Conf. Int.

d*o

[0.99–1.02] [0.94–1.12] [0.98–1.04] [0.99–1.02] [1.01–1.14] [1.00–1.07] [1.00–1.04] [1.01–1.08] [1.01–1.09] [1.01–1.05] [0.99–1.07] [0.98–1.05} [0.98–1.02] [0.89–1.02] [0.96–1.02] [1.00–1.04] [1.01–1.21] [0.98–1.05] [1.04–1.08] [1.07–1.10] [0.99–1.06] [0.98–1.02] [0.92–1.02] [0.96–1.01]

1.00 1.03 1.01 1.01 1.08 1.03 1.02 1.05 1.05 1.03 1.03 1.01 1.00 0.96 0.98 1.02 1.11 1.01 1.06 1.08 1.02 1.00 0.97 0.98

[0.98–1.01] [0.96–1.01] [0.96–1.01] [0.97–1.00] [0.96–1.01] [0.96–1.01] [1.00–1.04] [1.01–1.08] [1.02–1.08] [1.03–1.08] [0.96–1.03] [0.96–1.03] [0.96–1.00] [0.91–0.96] [0.91–0.96] [0.99–1.03] [0.95–1.02] [0.95–1.02] [1.09–1.13] [1.00–1.07] [1.02–1.08] [0.98–1.01] [0.95–1.00] [0.95–0.99]

0.99 0.98 0.98 0.98 0.99 0.98 1.02 1.04 1.05 1.05 0.99 1.00 0.98 0.93 0.93 1.01 0.98 0.99 1.11 1.04 1.05 1.00 0.97 0.97

[0.98–1.01] [0.96–1.01] [0.96–1.01] [0.97–1.00] [0.96–1.01] [0.96–1.01] [1.00–1.04] [1.01–1.08] [1.02–1.08] [1.03–1.08] [0.96–1.03] [0.96–1.02] [0.96–1.00] [0.91–0.96] [0.91–0.96] [0.99–1.03] [0.95–1.02] [0.95–1.03] [1.09–1.13] [1.00–1.07] [1.02–1.08] [0.98–1.01] [0.95–1.00] [0.95–0.99]

0.99 0.98 0.98 0.98 0.99 0.98 1.02 1.04 1.05 1.05 0.99 1.00 0.98 0.94 0.93 1.01 0.98 0.98 1.11 1.04 1.05 1.00 0.97 0.97

In bold, the values of d*o which are strictly below 1.

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

I observe here that for the Hang Seng and the Singapore All Shares, all intervals exclude the unit root case, d*0, which oscillates between 1.02 and 1.08 for the former series and between 1.02 and 1.11 for the latter. This result is in line with the results reported in previous studies on these two indexes (Cheung & Lai, 1995; Henry, 2002; see Table 1). For the FTSE100, most of the values are also higher than one, although d*0 is equal to 0.99 for AR(1) u t with an intercept and a linear time trend. For the remaining series, most of the values are around one, although they are sometimes above and sometimes below. However, if I permit deterministic trends most of them are below one, which implies that a small degree of mean reversion may take place for these

Fig. 5. QMLE of Robinson (1995a) based on the first differenced data for a range of values J = 500, 2500.

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45

series. These values are 0.98 and 0.99 for the EOE, the DAX, and the CAC40. They are slightly smaller (0.97 and 0.98) for the Nikkei and range between 0.93 and 0.98 for the S&P500 (Fig. 5). 5.2. The semiparametric approach Next, I perform the semiparametric procedure described in Section 4.2. Fig. 4 reports the results based on the Whittle semiparametric estimate of Robinson (1995a), i.e., where dˆ is given by (7) for a range of values of m from 500 to 1500. Since the time series are clearly nonstationary, I perform the analysis based on the first differenced data, adding one to the estimated values of d to obtain the proper orders of integration of the series. Fig. 4 also shows the 95% confidence intervals corresponding to the I(0) null hypothesis. I observe that for two of the series, the Hang Seng and the Singapore All Shares (and in some cases the FTSE), the estimated values of d are almost all above the I(0) interval. This finding implies that the orders of integration of the original series are above one. This finding is consistent with the results obtained with Robinson’s (1994a) parametric procedure that these series do not present mean reverting behavior. For the EOE, the CAC40, the DAX, and the Nikkei, the values oscillate around one. The S&P500 presents values which are in all cases smaller than one, suggesting that mean reversion takes place in this series. This result is again consistent with the parametric results obtained just above. My results presented are also in line with other previous works about the long memory property in the stock markets. For example, Cheung and Lai (1995) and Crato (1994) find evidence of d = 1 in several international stock market indexes. Barkoulas et al. (2000) report values of d higher than one for the Greek stock market. Tolvi (2003) finds evidence of mean reversion (i.e. d b 1) for the Finish data. Moreover, the fact that I obtain the highest evidence of mean reversion for the S&P500 is consistent with other works in the US market (Lo & MacKinley, 1988; Poterba & Summers, 1988). However, at the same time, given that the unit root cannot statistically be rejected, my results do not contradict the results reported in other articles, that are based on cointegration between stock prices and dividends. This is also of considerable importance for the validity of the market efficiency hypothesis of the stock market.

6. Conclusion In this paper I use fractionally integrated techniques to examine the stochastic behavior of several stock market indexes. By using these techniques, I capture the dynamic behavior of the series in a much more flexible way than can those methods based on traditional I(0) and I(1) approaches. I use a parametric testing procedure originally developed by Robinson (1994a) and a semiparametric Whittle estimation method (Robinson, 1995a). I use these procedures because of the distinguishing features that make them particular relevant in comparison with other methods. Robinson’s (1994a) tests allow me to consider roots with integer and fractional orders of integration with no effect on the standard null limit distribution. This limit behavior is also unaffected by the inclusion of deterministic trends and different types of I(0) disturbances. In addition, the tests are most efficient when directed against the appropriate (fractional) alternatives. The reason for using the Whittle semiparametric method of Robinson (1995a) is based on its computational simplicity. Also, this method requires a single bandwidth parameter, unlike other procedures that require a trimming number. A FORTRAN code with the programs is available from the author on request.

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L.A. Gil-Alana / Review of Financial Economics 15 (2006) 28–48

I apply these methods to the daily structure of several stock market indexes, the EOE (Amsterdam), the DAX (Frankfurt), the Hang Seng (Hong Kong), the FTSE100 (London), the S&P500 (New York), the CAC40 (Paris), the Singapore All Shares, and the Nikkei (Tokyo). Franses and Van Dijk (2000) analyze these series in a non-linear fashion. However, there has not yet been a long-memory study of these series that investigates if mean reversion occurs. Moreover, the disparities in capitalization, sophistication, and market microstructure across the eight countries reduce concerns about sample or market-specific results. When I try to summarize the conclusions for the individual series, I find that the order of integration of the Singapore All Shares and the Hang Seng is much higher than one. Thus, for these series I find conclusive evidence against mean reversion, but long memory on the returns. (I note that the order of integration is invariant to the log-transformation of the series.). Moreover, my findings imply that long-range dependence may be a source of long-horizon predictability. Relative transaction costs are greater for trading strategies based on short-term predictability than for those strategies based on long-term predictability. Thus, the long-horizon strategy may represent an unexploited profit opportunity. On the other hand, the S&P500 appears to be the less nonstationary series, with d below one and thus, showing mean-reverting behavior. For the remaining series, (the EOE, the DAX, the CAC40, the FTSE, and the Japanese Nikkei), the values oscillate around the unit root. Clearly, to determine if a small degree of mean reversion occurs, a more careful study of these series should be performed. The fact that some of these series share a common degree of integration provides some support of fractional cointegration, an issue that has not yet been widely studied in the literature to date.

Acknowledgement The author gratefully acknowledges financial support from the Minsterio de Ciencia y Tecnologia (SEC2002-01839, Spain). Comments of an anonymous referee are gratefully acknowledged.

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