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Physica A
journal homepage: www.elsevier.com/locate/physa
Salient features of dependence in daily US stockmarket indicesLuis A. Gil-Alana, Juncal Cunado, Fernando Perez de Gracia ∗
Department of Economics, Universidad de Navarra, Spain
h i g h l i g h t s
• In general, we observe a slight decrease in the degree of dependence across the years in the sample.• The results suggest the presence of one structural break in the series related with the stock market crash in October 1987.• With respect to the volatility processes, there is strong evidence of some degree of stationary long memory.
a r t i c l e i n f o
Article history:Received 14 August 2012Received in revised form 22 December 2012Available online 10 April 2013
Keywords:Long range dependenceVolatilityUS stock marketDay of week effect
a b s t r a c t
This paper deals with the analysis of long range dependence in the US stock market. Wefocus first on the log-values of the Dow Jones Industrial Average, Standard and Poors 500and Nasdaq indices, daily from February, 1971 to February, 2007. The volatility processesare examined based on the squared and the absolute values of the returns series, and thestability of the parameters across time is also investigated in both the level and the volatilityprocesses. A method that permits us to estimate fractional differencing parameters in thecontext of structural breaks is conducted in this paper. Finally, the ‘‘day of the week’’ effectis examined by looking at the order of integration for each day of the week, providing alsoa new modeling approach to describe the dependence in this context.
There is ample financial literature devoted to analyzing, modeling and forecasting the behavior of stock markets. Thefinance theory suggests many stylized facts in daily stock prices. Thus, for example, mean reversion in stock market priceshave been examined in many papers [1–7]. However, the empirical evidence on mean reversion in stock market pricesis still inconclusive. For example, the seminal papers by Fama and French [1] and Poterba and Summers [2] documentedmean reversion in the US stock prices, while other authors such as Lo and MacKinlay [3] detected evidence against meanreversion using weekly US data. Second, the volatility in the stock returns shows an autocorrelation function that decaysslowly. Other studies found that absolute and squared returns have long memory [8–10]. Third, daily stock prices usuallytend to present day of the week effect, along with some other calendar anomalies such as the January effect and the turnof the month effect. The day of the week effect is a relevant stock market anomaly, which is extensively documented in thefinancial literature [11–14]. This anomaly in the stock market has been recently observed in many other countries [15–17].
In this paper, we focus on the above stylized facts. Initially, we examine the long range dependence in three daily USstockmarket indices; then the volatility processes are also examined from a longmemory viewpoint. Finally, the ‘‘day of theweek’’ effect is investigated, proposing a fractionally integrated model where the long run dynamics depend exclusively on
∗ Correspondence to: University of Navarra, School of Economics, E-31080 Pamplona, Spain. Tel.: +34 948 425 625; fax: +34 948 425 626.E-mail address: [email protected] (F.P. de Gracia).
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the day of the week. Long range dependence has been extensively examined in stockmarkets [18–21]. Most of the empiricalevidence using long range dependence is inconclusive. Thus, some authors find little or no evidence of longmemory in stockmarkets [22]. On the other hand, others’ papers find evidence of long memory in monthly, weekly, and daily stock marketreturns [23–27]. Finally, long range cross-correlations among financial indices have been examined by Podobnik et al. [28]and more recently by Wang et al. [29].
The contribution of this work is two-folded: first we provide further evidence of the longmemory properties of the stockmarket indices and their volatility processes along with an analysis of their stability across time. In this context, a recentprocedure to determine fractional integration with structural breaks is implemented. Second, we introduce a new modelalso based on long memory to describe the day-of-the-week effect in financial markets. The rest of the paper is structuredas follows. Section 2 briefly describes the statistical models employed in the paper. Section 3 describes the data; in Section 4we present the main results of the paper, while Section 5 contains some concluding comments and extensions.
2. The statistical models
The statistical models employed across the paper are all based on different versions of fractionally integrated models.This allows a greater degree of flexibility than the standard approaches based on the stationary I(0) or nonstationary I(1)models since the number of differences required to get I(0) series may non-necessarily be an integer number but a realvalue. Following the work of Granger [30], Granger and Joyeux [31] and Hosking [32], a rapidly growing body of literaturehas emerged on fractionally integrated ARFIMA processes. Robinson [33,34], Beran [35], Baillie [36] and more recently Gil-Alana and Hualde [37] present surveys on this topic. A process xt is said to be integrated of order d if,
(1 − L)dxt = ut , t = 1, 2, . . . , (1)
with xt = 0, t ≤ 0, where ut is an I(0) process, defined as a covariance stationary process with spectral density functionthat is positive and finite, and L is the backward shift operator (Lxt = xt−1). In the event that d is not an integer, the series xtrequires fractional differencing in order to obtain a stationary (possibly) ARMA series. ARIMA(p, d, q) models in which d isa positive integer are special cases of the general process in (1). If d > 0 in (1), xt is said to long memory, so-named becauseof the strong association between observations widely separated in time.
For stock indices, the evidence in favor of long memory may be due to the effect of aggregation. In fact, that is one of themain sources of longmemory. The key idea is that, aggregation of independentweakly dependent series can produce a strongdependent series. Robinson [38] and Granger [30] showed that fractional integration can arise as a result of aggregationwhen data are aggregated across heterogeneous autoregressive (AR) processes; data involving heterogeneous dynamicrelationships at the individual level are then aggregated to form the time series. Moreover, the existence of long memoryin financial asset returns suggests that new theoretical models based on nonlinear pricing models should be elaborated.1Mandelbrot [40] notes that in the presence of long memory, martingale models of asset prices cannot be obtained fromarbitrage. In addition, statistical inference concerning asset pricing models based on standard testing procedures may notbe appropriate in the context of long memory processes [41].
Throughout the paperwe focus onRobinson’s [42] parametric approach,which does not require preliminary differencing;it allows us to test any real value d in (1) encompassing stationary and nonstationary hypotheses. We use the followingmodel:
yt = β ′zt + xt , (2)
where yt is the time series we observe, β is a (k × 1) vector of unknown parameters; zt is a (k × 1) vector of deterministiccomponents, and xt is given by (1), testing the null hypothesis:
Ho : d = do, (3)
for any real value do. Thus, the null hypothesized model is:
and a trend-stationary representation is obtained if zt = (1, t)T and do = 0; a unit root model if do = 1; and fractional I(d)models if do is a fractional value. Another advantage of this approach is that the limit distribution is standard normal, andthis limit behavior holds independently of the deterministic regressors used for zt in (2) and the type of weak dependence(e.g. ARMA) processes used for the I(0) disturbance term ut . In the final part of the article, Eq. (1) will be replaced by:
(1 − L5)dyt = ut , t = 1, 2, . . . , (4)
to take into account ‘‘day of the week’’ long memory effects. The functional form of this procedure can be found inRobinson [42] and in any of its numerous empirical applications [45]. Another approach that will be employed in the paper,due to Gil-Alana [44], will permit us to estimatemodels like (1) and (2) in the context of structural breaks where the numberof breaks and the break dates are endogenously determined by the model.
1 Peters [39] defined the ‘‘Fractional Market Hypothesis’’ for modeling long-term dependence features in financial time series.
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3. Data description
We used daily data for the Dow Jones Industrial Average (DJIA), Standard and Poors 500 (S&P) and Nasdaq indicesfrom February 5, 1971 to February 7, 2007 yielding 9395 observations. All data are obtained from the Bloomberg web site(http://www.bloomberg.com) and they are all in natural logs. We focus on this period not taking into account the data after2007 to avoid the effects that the subprime crisis might have had on the stock market structures.
Fig. 1 displays the plots of the three US stock markets time series indices: DJIA, S&P and Nasdaq. Apparently the threeseries displays a very similar shape over the sample period and shows a clear upward trend. Summary statistics on the threeUS stock markets indices (in logs) and their corresponding returns are shown in Table 1. The Nasdaq index presents thehighest volatility over the sample period, while S&P and the DJIA are more stable markets. When we analyze the log stockmarket prices in first differences (i.e., returns), all series present a similar mean and volatility behavior across the sampleperiod. The skewness and kurtosis coefficients reveal departures from normality in the data, confirmed by the Jarque–Berastatistic. Nevertheless, the versions of the Robinson’s [42] method employed in this article are robust against non-Gaussiandisturbances.
The values in brackets refer to the 95% confidence intervals.
4. Salient features of dependence in US stock market indices
This section is divided into various sub-sections. In the first one we examine the degree of dependence of the three seriesby estimating their orders of integration. Here we employ Robinson’s [42] parametric approach along with an estimate of dbased on the Whittle function in the frequency domain [43]. The second sub-section is devoted to the volatility processes,and the sameprocedures as in the previous sub-section are applied here to the squared and the absolute values of the returns,which are used as proxies for the volatility. In both subsections we also examine the stability of the fractional differencingparameters across time. The third sub-section deals with the ‘‘day of the week’’ effect.
4.1. Initial results on dependence
Table 2 summarizes the results of Robinson’s [42] parametric approach in (1) and (2) for the three log-series, assumingthat ut in (1) is white noise; since we must also specify the deterministic components of zt in (2), we consider the threestandard cases, i.e., no deterministic terms (i.e., zt = 0), an intercept (zt ≡ 1), and an intercept with a linear time trend(zt = (1, t)T ). The inclusion of a linear time trend may appear unrealistic in the case of financial time series. However,it should be noted that in the context of fractional (or integer) differences, the time trend disappears in the long run. Forexample, suppose that ut in (1) is white noise. Then, testing Ho (3) in (1) and (2) with zt ≡ 1 and d = 1, the series becomes,for t > 1, a pure randomwalk process, and a randomwalkwith an intercept if zt = (1, t)T . Table 2 shows the test results; thenumbers in bold are the approximatedmaximum likelihood estimates of d obtainedwith theWhittle function. The estimateswere obtained based on the first differenced data, adding then 1 to the resulting values to obtain the proper estimates ofd in the log-series. Table 2 also shows the 95% confidence bands for the non-rejection values of do using Robinson’s [42]parametric approach. Here, we test Ho (3) in the model given by (1) and (2), using do-values from 0 to 2 in 0.001 increments,i.e., do = 0, 0.001, 0.002, . . . , 1.998, 1.999 and 2.
The first thing we observe in Table 2 is that the estimated values of d are slightly above 1 in the three series and thishappens for the three types of models employed, being higher for the Nasdaq than for the other two indices. Moreover,the confidence intervals for the non-rejection values of do include the unit root (and also some values of d below 1) in allcases with the exception of the Nasdaq with an intercept and/or a linear trend. The fact that in most of the cases we cannotreject the unit root null hypothesis suggests that the markets may be efficient according to this simple specification. Thisis consistent with many others’ empirical works. However, these significant results might be due in large part to the un-accounted-for I(0) autocorrelation in ut . Thus, we also computed results using autoregressions for the disturbance term.Table 3 displays the results of Robinson’s [42] statistic assuming that ut in (1) is AR(1), and we take do-values in (3) equal to0, 0.10, . . . , 1.40 and 1.50. The limit distribution of the test statistic is unaffected by the inclusion of (weakly) autocorrelatedterms, and thus it is still standard N(0, 1). A significant feature of these results is that the value of the test statistic does notmonotonically decrease with do. Such monotonicity is a characteristic that should be expected in these results given correctspecification and adequate sample size since the test statistic is one-sided. Thus, for example, if Ho (3) is rejected with
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Table 3Values of the test statistic [42] with AR(1) disturbances.
The values in brackets refer to the 95% confidence intervals.
do = 0.20 against the alternative d > 0.20, the test statistic is significantly positive and above the critical value, and a moresignificant result in this direction should be expected when do = 0.10 or 0 is tested.
Weobserve in Table 3 thatmonotonicity is not satisfied if d is small for anymodel and any series. This lack ofmonotonicitymay be explained in terms of model specification as is argued for example in Gil-Alana and Robinson [45], though it mayalso be a consequence of the competition between the fractional differencing parameter and the AR coefficient in describingthe nonstationarity. Thus, we observe that if d = 0, Ho cannot be rejected in most of the cases, and, though not reported,the associated AR coefficients were then very close to 1 in all cases. We finally observe that only if d > 1 (in the case ofno deterministic terms) or if d > 0.60 (with an intercept and/or a time trend) is monotonicity satisfied. Table 4 displaysthe Whittle estimates of d and the 95% confidence bands for those regions of d where monotonicity is achieved. Here, weobserve that if no deterministic terms are included, the unit root null hypothesis is rejected for the three series in favor ofhigher orders of integration; however, if an intercept and/or a linear trend are included, the values of d are strictly smallerthan 1 for the DJIA and the S&P, implying a small degree of mean reversion for these two series. On the other hand, for theNasdaq, d is strictly above 1 in all cases. The rejection of the unit root in the case of the Nasdaq index in favor of higher ordersof integration indicates that, according to this specification, shocks are here permanent, and moreover, that stock marketreturns are long memory, implying some degree of predictability in its behavior. Higher AR orders were also employed andthe results were completely in line with those reported here. Note that, given the fractional nature of the d-differencingpolynomial in (1) the process can itself be expressed in terms of an AR(∞) process, and thus, the contribution of the shortrun AR(k) dynamics only affects the first k terms.
The following two figures display the estimates of d using samples of one complete year of observations. In doing so wewant to investigate if the order of integration of the series has changed across the years. Fig. 2 refers to the case of whitenoise disturbances while Fig. 3 corresponds to the AR(1) case in both cases with zt ≡ 1. We focus here on the case withan intercept (zt ≡ 1) since it seems the most realistic model for the series analyzed in the paper. Note that the estimatedvalues of d are very similar for the three cases of no regressors, an intercept, and an intercept with a linear trend, and,though not reported, the coefficients associated with the time trends were found to be insignificantly different from zero atconventional statistical levels in virtually all cases.
Starting with the white noise case (in Fig. 2) we observe that for the DJIA and the S&P, the values decrease slowly acrossthe sample from values above 1 at the beginning of the sample to values below 1 at the end of the sample. For the Nasdaq, thedecrease is more pronounced. In fact, for the first half of the sample, the values are strictly above 1 and only for the last tenyears there are estimates below unity. A similar pattern is observed if AR(1) disturbances are considered, though here theconfidence bands include the unit root in all cases, also for the values corresponding to the Nasdaq index. In general, a slight
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Fig. 2. Estimates of d based on 1-year samples with white noise disturbances.
reduction is observed in the orders of integration of the series across the sample, in some cases, below unity, implying thatthe market is becoming more inefficient as time goes by, especially for the DJIA and the S&P indices. On the other hand, forthe Nasdaq index, the values tend towards unity from values above 1, implying that this market is becoming more efficientin recent years.
In view of the preceding results, it might be of interest to examine the potential presence of structural breaks in thedata. Note that, the time period of data employed in this work spans from 1971 to 2007. This period consists of many majorevents which may have caused (multiple) structural breaks in the series. Diebold and Inoue [46] have shown that regimeswitching and long memory are intimately linked and that regime switching can also be employed as a means of bridgingthe gap between stationary ARMA models and nonstationary infinite variance unit root processes. Similarly, Granger andHyung [47] show that occasional breaks in the data generate slow decaying autocorrelations and other properties of I(d)processes. Gil-Alana [44] proposes amethod for estimating fractional integration and structural breaks with the dates of thebreaks being endogenously determined by the model. Applying it in our data, it is found a single break in each of the threeseries. Therefore, the model considered is:
yt = α1 + xt; (1 − L)d1xt = ut , t = 1, 2, . . . Tb,yt = α2 + xt; (1 − L)d2xt = ut , t = Tb + 1, . . . T ,
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Fig. 3. Estimates of d based on 1-year samples with AR(1) disturbances.
Table 5aEstimates in a model with a single break and white noise disturbances.
Series Break date First sub-sample Second sub-sampled1 α1 d2 α2
DJIA Oct 5, 1987 1.066* 6.7755 1.037* 7.5981(153.44) (184.35)
S&P Oct 5, 1987 1.041* 4.5703 1.053* 5.5293(103.07) (137.02)
Nasdaq Oct 5, 1987 1.123 4.6115 1.088* 5.7769(86.82) (85.27)
* Means that the unit root null (i.e. d = 1) cannot be rejected at the 5% level.The values in parenthesis are t-values.
where the α’s are the coefficients corresponding to the intercepts; d1 and d2 may be real values, ut is I(0), and Tb is the timeof the break date. The results for the two cases of white noise and AR(1) disturbances are displayed in Table 5.
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Table 5bEstimates in a model with a single break and AR(1) disturbances.
Series Break date First sub-sample Second sub-sampled1 α1 d2 A2
DJIA Oct 2, 1987 1.121 6.6837 1.077* 6.4827(14.22) (16.96)
S&P Oct 5, 1987 1.054* 4.3347 0.972* 9.4298(13.74) (13.76)
Nasdaq Oct 5, 1987 1.100 4.5763 1.072* 3.7766(8.80) (7.32)
* Means that the unit root null (i.e. d = 1) cannot be rejected at the 5% level.The values in parenthesis are t-values.
Table 6Estimates of d for the volatility processes based on white noise disturbances.
Series No regressors (NR) An intercept (I) A linear trend (LT)
The values in brackets refer to the 95% confidence intervals.
The first thing we observe in this table is that the break date takes place at October 5th, 1987. If ut is assumed to bewhite noise, the unit root cannot be rejected in any subsample for the cases of DJIA and S&P. However, for the Nasdaqindex, the I(1) hypothesis is rejected in favor of higher orders of integration during the first subsample, but is not rejectedafter the break. Assuming autocorrelated disturbances, in Table 6, the unit root cannot be rejected for the S&P during the firstsubsample, and for the three indices after the break. These results are consistentwith those presented above (in Figs. 2 and 3)showing a slight reduction in the orders of integration of the series across time.
4.2. Long memory in the volatility series
We use two alternativemeasures of volatility: absolute returns and squared returns which have been already used in thefinancial literature. Absolute returns were employed by Ding et al. [10], Granger and Ding [48], Bollerslev and Wright [8]and Gil-Alana [49], whereas squared returns were used in Lobato and Savin [21] and Gil-Alana [50].
Table 6 displays the confidence bands and the Whittle estimates of d for the absolute and the squared returns of thethree indices, assuming that the disturbances are white noise. The first thing we observe here is that once more the valuesare robust to the three cases of no regressors, an intercept and an intercept with a linear trend. In all cases the confidencebands are constrained in the interval (0, 0.5) implying stationary longmemory volatility processes. For the DJIA and the S&P,the values are higher with the absolute returns, while the opposite happens for the Nasdaq index. Thus, for the squaredreturns, d is around 0.111 for the DJIA; it is 0.126 for the S&P, and is around 0.218 for the Nasdaq. For the absolute values,the corresponding values are around 0.164, 0.166 and 0.210 respectively for the three indices.
In Fig. 4 we display the estimates (and confidence intervals) using samples of 1-year observations and we observe thatthe behavior of the estimates are relatively stable across the sample for the three series and the two proxies. In Table 7 wereport the average values of d in the volatility processes across years.We note that d is constrained between 0.035 and 0.150for the DJIA, it is between 0.060 and 0.195 for the S&P, and is between 0.045 and 0.175 for the Nasdaq. Thus, in all cases d isabove 0 implying longmemory volatility. The results in this subsection imply that the volatility in the stockmarket returns islong memory, and thus, the use of other approaches based on autoregressive conditional heteroskedasticity models (ARCH,Engel [51]; GARCH, Bollerslev [52]) should be extended to the fractional case (e.g., FIGARCH-type models, Baillie, Bollerslev
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Fig. 4. Estimates of d in volatility based on 1-year samples with white noise disturbances.
andMikkelsen [20]). Performing themethod of Gil-Alana [43] on the volatility processes to check for the presence of breaks,we do not find evidence of structural breaks in any of the three indices.
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Table 7Average values of d in the volatility processes across years.
The values in brackets refer to the 95% confidence intervals.
4.3. ‘‘Day of the week’’ effect
The day of theweek effect is a relevant stockmarket anomaly, extensively documented in the financial literature [11–14].Thus, Osborne [11] and Cross [12] using data of the S&P 500 obtained that Monday returns were lower than those basedon Fridays. Similarly, Gibbons and Hess [14] using data from the DJIA also obtained that Friday returns were greater thanMonday returns. This anomaly in the stock market has also been observed in many countries, including Canada, Australia,Japan and the UK [16]; France [17]; South Korea, Malaysia, the Philippines, Taiwan and Thailand [53].
Table 8 displays the estimates of d using the same procedures as before, splitting the data by the day of week. Startingwith the white noise case (in Table 8a) we see that for the DJIA and S&P, the estimated values of d are smaller than 1 forthe five days of the week, and, moreover, for Mondays and Tuesdays, the unit root null is rejected in favor of smaller ordersof integration. Thus, we obtain evidence of mean reversion for these two days of the week. On the other hand, the Nasdaqindex shows strong evidence of no mean-reversion, with values of d strictly above 1 for all days of the week.
Allowing AR(1) disturbances the values are higher in all cases and evidence of d < 1 is only obtained for the DJIA indexon Mondays. The estimates are also smaller than 1 for Tuesdays and Wednesdays on the Dow Jones, and for Mondays andTuesdays in the S&P. Similarly to the white noise case, evidence of d > 1 is obtained for the Nasdaq index across all days. Inany case, even for the Nasdaq index, smaller orders of integration are obtained on Mondays and Tuesdays compared withthe other days of the week.
The results presented above indicate that the degree of persistence is different depending on the day of the week,observing lower orders of integration on Mondays and Tuesdays in all series. In what follows, we consider a model wherethe long memory dynamic effects are supposed to be dependent on the day of the week, while other short run dynamicsare described through the I(0) structure of the disturbance term. In particular, we suppose the data follow a model of thefollowing form:
yt = β ′ zt + xt , t = 1, 2, . . . , (5)
(1 − L5)dxt = ut , t = 1, 2, . . . , (6)ut = ρut−1 + εt , t = 1, 2, . . . , (7)
where zt in (5) again adopts the three functional forms of no deterministic terms, an intercept, and an intercept with a lineartime trend. This model given by (5)–(7) implies that the present value of the series (yt ) depends in the long run on its value
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Table 9aValues of the test statistic for the ‘‘day of the week’’ with white noise ut .
In bold, significant statistics at the 5% level. NR = No deterministic terms; I = Intercept. LT = Linear trend.The values in brackets refer to the 95% confidence intervals.
Table 9bEstimates of d for the ‘‘day of the week’’ based on white noise ut .
Series No regressors (NR) An intercept (I) A linear trend (LT)
NR = No deterministic terms; I = Intercept. LT = Linear trend.The values in brackets refer to the 95% confidence intervals.
five periods before (yt−5), but also on all past observations which are backwards multiples of 5, i.e., yt−10, yt−15, . . . .2 Here,we employ another version of Robinson’s [42] parametric approach, which is also asymptotically N(0, 1)-distributed. Thistype of model is relevant in the context of daily financial data, where the value of an asset on a given day of the weekmay bestrongly influenced by its value on the same day of the previous week. There is in fact an extensive literature documentingthe presence of calendar anomalies (such as theweekend effect, the day of theweek effect, and the January effect) in financialseries, both in the US and in other developed markets, dating back to Osborne [11]. Negative Monday returns were found,inter alia, by Cross [12], French [13] and Gibbons and Hess [14], the former two analyzing the S&P index, the latter the DJIA.Similar findings have been reported for other US financial markets, such as the futures, bond and Treasury Bill markets [54,55], foreign exchangemarkets [56], and for Australian, Canadian, Japanese and UK financial markets [16,57]. Effects on stockmarket volatility have also been documented [58].
Various explanations have been offered for the observed patterns. Some focus on delays between trading and settlementin stocks [14]: buying on Fridays creates a two day interest free loan until settlement; hence, there are higher transactionvolumes on Fridays, resulting in higher prices, which decline over the weekend as this incentive disappears. Othersemphasize a shift in the broker investor balance in buying–selling decisions which occurs during weekends, when investorshave more time to study the market themselves (rather than rely on brokers); this typically results in net sales onMondays,when liquidity is low in the absence of institutional trading [59]. It has also been suggested that the Monday effect largelyreflects the fact that, when daily returns are calculated, the clustering of dividend payments around Mondays is normallyignored; alternatively, it could be a consequence of positive news typically being releasedduring theweek, andnegative onesover the weekend [60]. Additional factors which could be relevant are serial correlation, with Monday prices being affectedby Friday ones, and a negative stock performance on Fridays being given more weight [61]; measurement errors [62];size [63] and volume [64].
Initially, we suppose that there are no short run components (i.e., ρ = 0 in (7)) and therefore all the dynamic behaviorof the series is described through the fractional differencing polynomial in (6). Table 9 displays the values of the teststatistic, testing again Ho (3) now in the model given by (5) and (6) for do-values from 0 to 1.50 with 0.10 increments(do = 0, 0.10, 0.20, . . . , 1.40 and 1.50). The first thing we observe is that the value of the test statistic monotonically
2 Note that for the three indices, if therewas no value at a given day, the arithmeticmean using the previous and the following observationwas computed.
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Table 10Values of the test statistic for the ‘‘day of the week’’ with AR(1) disturbances.
In bold, significant statistics at the 5% level. NR = No deterministic terms; I = Intercept. LT = Linear trend.
decreases with d in all cases across the threemodels presented. It is also observed that the only non-rejection value reportedin the table takes place at d = 1 for the Nasdaq index in the case of no deterministic terms. However, other non-rejectionsmay occur at values of d in the intervals across the points. Thus, it seems that some non-rejections could take place at valuesof d between 0.9 and 1 for the DJIA and S&P, and at values constrained between 1 and 1.1 for the Nasdaq index.3 Note that,these are the values where the test statistic changes its sign and thus we can find values within the N(0, 1) confidenceinterval. Table 9b reports the 95% confidence band of non-rejection values along with the estimate of d. It is observed thatd is below 1 for the DJIA and S&P, while it is slightly above 1 for the Nasdaq index.
Table 10 is similar to Table 9 but imposing an AR(1) structure for the error term. Herewe observe that the null hypothesisis rejected in all cases except when d = 0, implying that a simple AR(1) model could be an adequate specification for theseries. However, though not reported, the AR coefficients were in these cases extremely close to 1, implying once more thatthe fractional differencing polynomial is competing with the AR parameter in describing the long run effect. Note that thepolynomial (1− L5) can be decomposed into (1− L)(1+ L2 + L3 + L4) implying then the existence of a unit root at the longrun or zero frequency, which may compete with the AR coefficient if this is close to 1. Thus, we also employ an alternativeto the AR model, which is based on Bloomfield [67] exponential spectral model. This is an approach of modeling the I(0)disturbances, where the spectral density function is given by:
f (λ; τ) =σ 2
2πexp
2
mr=1
τr cos (λ r)
, (8)
where m is the number of parameters required to describe the short run dynamics of the series. Bloomfield [67] showedthat the logarithm of an estimated spectral density function is often found to be a fairly well-behaved function and canthus be approximated by a truncated Fourier series. He showed that the spectral density of an ARMA process can be wellapproximated by (8). Moreover, this model is stationary across all values of τ , the model accommodates extremely wellin the context of Robinson’s [42] tests, and no matrix inversion is required in the specification of the test statistic [68]. Theresults using themodel of Bloomfield [67] (withm = 1) are displayed in Table 11. Higher values formwere also tried and theresults were practically the same in terms of the non-rejection values of d. First, we observe in Table 11a that monotonicityis achieved in all cases and the non-rejections take place at values of d constrained between 0.70 and 0.80 for the DJIA andS&P, and between 0.80 and 0.90 for the Nasdaq. The estimated values of d are higher in the case of a linear time trend andin all cases d is between 0.74 and 0.81, implying long memory and mean reversion for the ‘‘day of the week’’ effect.
Next, we wonder which may be the best model specifications in the context of ‘‘day of the week’’ effects, and uselikelihood criteria along with t-tests to determine the best models. It is obtained that the best specification for the DJIAis the following:
3 The day of the week effect may not be the same across markets. For example, Wang et al. [65] and Chang et al. [66], show that the weekend effect wasmore intense in the Nasdaq than in other US stock market indices.
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Table 11aValues of the test statistic for the ‘‘day of the week’’ with Bloomfield ut .
In bold, significant statistics at the 5% level. NR = No deterministic terms; I = Intercept. LT = Linear trend.The values in brackets refer to the 95% confidence intervals.
Table 11bEstimates of d for the ‘‘day of the week’’ based on Bloomfield ut .
Series No regressors (NR) An intercept (I) A linear trend (LT)
and the 95% confidence intervals for the fractional differencing parameters are (0.782, 0.796), (0.773, 0.783) and (0.797,0.817), respectively for the three indices. Thus, once more we obtain a higher degree of persistence for the Nasdaq indexthan for the other two indices, and evidence of mean reversion in the three cases.
5. Conclusions
In this paper, we have examined the long range dependence in the US stock market by means of investigating the ordersof integration of the DJIA, the S&P and the Nasdaq indices over the period February 5, 1971 to February 7, 2007. The volatilityprocesses are also examined by looking at the degrees of integration of the absolute and squared returns, which are usedas proxies for the volatility. In the final part of the paper we have examined the ‘‘day of the week’’ effect, first by looking atthe orders of integration separately for each day of the week, and then proposing a newmodeling framework, based on thelong run dependence for the week-day effect.
The results can be summarized as follows: first, we observe very similar patterns for the DJIA and the S&P indicescompared with the Nasdaq index. The Nasdaq index only includes technology stocks whereas the DJIA and the S&P indicesinclude stocks from the industrial sector and other sectors and thus are usually good indicators of the stock market as awhole. Thus, the similarities observed in these two indices may also be translated to their degrees of persistence. Startingwith the original log-series, we observe that if all the dependence is captured through the fractional differencing polynomial,the order of integration is slightly above 1, and the unit root cannot be rejected for the DJIA and the S&P. However, allowingweak dependence (AR and Bloomfield) the orders of integration are significantly smaller than 1 (and thus showing meanreversion) for these two indices, while it is statistically significantly above 1 for the Nasdaq. In general, we observe a slightdecrease in the degree of dependence across the years in the sample.
Further, we also examine the existence of possible structural breaks in the three stockmarket indices. The results suggestthe presence of one structural break in the series related with the stock market crash in October 1987 when the DJIA lose22%, the S&P also dropped 20% and 11% in the Nasdaq in one day. The unit root cannot be rejected for any of the three indicesfor the period after the stock market crash.
L.A. Gil-Alana et al. / Physica A 392 (2013) 3198–3212 3211
With respect to the volatility processes, there is strong evidence of some degree of stationary longmemory (0 < d < 0.5)for the two proxies in the three series, being oncemore higher in the case of the Nasdaq index. If we separate the data by theday of the week, the lowest degrees of dependence are obtained in the three series onMondays and Tuesdays. Modeling the‘‘day of the week’’ effect by means of a long memory model, mean reversion is obtained in practically all cases, with highervalues of the differencing parameter observed again for the Nasdaq index. Thus, the Nasdaq seems to be the closest marketto efficiency while the DJIA and S&P500 seem to present a small degree of mean reversion.
Acknowledgments
Luis A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnología (ECO2011-2014ECON Y FINANZAS, Spain) and from a Jeronimo de Ayanz project of the Government of Navarra. Juncal Cunado and FernandoPerez de Gracia gratefully acknowledge financial support from the Ministerio de Economía y Competitividad (ECO2011-25422). The authors thank the editor, H. E. Stanley and two anonymous referees for their helpful comments and suggestions.
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