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Common features between stock returns andtrading volumeMarta Regúlez & Ainhoa ZarragaPublished online: 07 Oct 2010.
To cite this article: Marta Regúlez & Ainhoa Zarraga (2002) Common features between stock returns and tradingvolume, Applied Financial Economics, 12:12, 885-893
To link to this article: http://dx.doi.org/10.1080/09603100110053317
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Common features between stock returns
and trading volume
MARTA REGUÂ LEZ and AINHOA ZARRAGA*
Dpto. de EconomõÂa Aplicada III (EconometrõÂa y EstadõÂstica), Facultad de CienciasEconoÂmicas y Empresariales, Universidad del PaõÂs Vasco, Avda. Lehendakari Aguirre83, 40815 Bilbao, Spain
This article tests for the existence of features shared in common by daily stockreturns and trading volume contributing to the empirical analysis of the relationbetween those series. Using Spanish data this study analyses this hypothesis lookingat features such as seasonality, skewness, kurtosis, non normality and serial correla-tion. This study ®nds that monthly seasonalities and distributional features such asskewness are driven by a common factor in stock returns and volume. This studyalso ®nds a non-synchronized comovement between the cycles of both variables.
I . INTRODUCTION
The analysis of the relation between stock prices and trad-ing volume has been an important topic in the ®nancial
literature, both empirically and theoretically. Authors
such as Epps (1975) or Karpo� (1987) provide a summaryof previous research on this relation. A more recent reviewcan be found in Stickel and Verrecchia (1994). Most of the
empirical studies carried out support the existence of acontemporaneous positive correlation between volume
and the absolute value of price changes as well as betweenvolume and price changes per se. The theoretical models
used to explain those empirical ®ndings take the rate ofinformation ¯ow to the market as the relevant factor.
Tauchen and Pitts (1983), Admati and P¯eiderer (1988)and Foster and Viswanathan (1990) directly relatemoments of price changes and volume data to the amount
of information arriving to the market.
One of these models is the so-called Mixture ofDistributions Model (MODM) introduced by Clark(1973), Epps and Epps (1976) and Tauchen and Pitts
(1983) and more recently modi®ed by Lamoureux andLastrapes (1994), Richardson and Smith (1994),
Andersen (1996) and Liesenfeld (1998). It provides a theor-etical framework in which returns and trading volume are
driven by the same underlying latent factor, the ¯ow of
information to the market. This suggests that features pres-ent in the ¯ow of information could be transmitted to
returns and volume. Moreover, if there exists a commonfactor there might be at least a linear combination ofreturns and volume not presenting those features.1 In this
sense, this study would like to focus on those features thatare characteristics of the ¯ow of information to test
whether they are common to returns and volume. Theproblem is that, in general the model places no distribu-
tional restrictions on the information ¯ow, that is unobser-vable to the econometrician. Therefore, it is di� cult to
select which features are to be analysed.Richardson and Smith (1994) propose a direct test of the
MODM using the generalized method of moments (GMM)
procedure of Hansen (1982) and they are able to estimatesome characteristics of the distribution of the random rate
of ¯ow of information. They focus on skewness- andkurtosis-based moment conditions because the apparent
skewness and kurtosis of price changes and volume havebeen the most studied empirical characteristics of theMODM. They ®nd several interesting stylized facts about
the unobservable information ¯ow. In particular, they ®ndpositive skewness, that is, most of the time, a small amount
of information ¯ow arrives to the market. However, onoccasion, large information ¯ow will occur on any given
day. Another characteristic of the information ¯ow found
Applied Financial Economics ISSN 0960±3107 print/ISSN 1466±4305 online # 2002 Taylor & Francis Ltd
http://www.tandf.co.uk/journalsDOI: 10.1080/0960310011005331 7
Applied Financial Economics, 2002, 12, 885±893
885
* Corresponding author: e-mail: [email protected]
There could be a nonlinear combination too, even if a linear one were not found.
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by these authors is a large kurtosis compared to a normaldistribution. This suggests that the distribution of the ¯owof information has long thick tails. With respect to themore commonly assumed distributions for the information¯ow, they ®nd evidence in favour of the lognormal distri-bution.
Some authors like Tauchen and Pitts (1983) andLamoureux and Lastrapes (1994) generalize the previousversions of the model allowing the information arrival pro-cess to be serially correlated. Andersen (1996) modi®es thestandard MODM modelling the information ¯ow as astochastic volatility process. Andersen (1996) remarksthat, `. . . casual empiricism suggests that news arrivals arepositively correlated. When unanticipated news breaks on agiven day, more detailed disclosures tend to follow over thenext few days or weeks, and di� erent interpretations of thecircumstances leading to the event surface. This tends tokeep ``the story’’ in the headlines for an extended period oftime. Moreover, important changes in the tactical orienta-tion of a company do typically play themselves out overlonger periods and these developments are revealedthrough periodic news updates . . .’.
So far this study has evidence on non-normality, skew-ness, kurtosis, and serial correlation as possible features tobe analysed. Another feature that is considered is season-ality. Interday patterns in volume and volatility of returnshave been predicted by some theoretical models like Fosterand Viswanathan (1990). Their model predicts a weekende� ect in trading volume and return volatility because pri-vate information accumulates over a weekend. Harris(1986) shows strong intraday patterns in return volatility.Jain and Joh (1988) provide evidence of an inverted U-shape in volume and returns across days. With regard tomonthly patterns in returns, there is some importantempirical evidence for the Spanish stock market. Zarraga(1999) ®nds monthly patterns in both stock returns andvolume. Basarrate and Rubio (1994) ®nd a strongJanuary e� ect in returns which they explain in terms of®scal motives. Monthly e� ects have also been documentedby Ariel (1987) for the American stock market.
Recently, Bollerslev and Jubinski (1999) examine the be-haviour of equity trading volume and volatility for theindividual ®rms composing the Standard and Poors’ 100composite index. Using multivariate spectral methods, they®nd that fractionally integrated processes best describe thelong-run temporal dependencies in both series. They showthat, consistent with a stylized mixture-of-distribution shypothesis model in which the aggregate `news’ arrival pro-cess possesses long-memory characteristics, the long-runhyperbolic decay rates appear to be common across eachvolume-volatility pair. On the other hand Lobato andVelasco (2000), using robust semiparametric procedures,analysed the long-memory properties of trading volumefor the 30 stocks in the Dow Jones Industrial Averageindex. They ®nd that volatility and volume exhibit the
same degree of long memory for most of the stocks butthere is no evidence that both processes share the samelong-memory component.
On the light of this evidence, this article examines therelationship between daily stock returns and trading vol-ume using the common features tests introduced by Engleand Kozicki (1993). These tests make it possible to inves-tigate whether or not a particular feature (serial correla-tion, trends, seasonality, heteroscedasticity, skewness,kurtosis, etc.) that is individually present in the series isdue to a common factor, meaning that a linear combina-tion of both series fails to have the feature. The motivationof these common features tests is a latent factor model,with a factor directing the observable variables and havingparticular features which are transmitted to them. In thiscase, this latent factor would be the ¯ow of information tothe market. Using daily data for the Spanish stock marketthis study analyses whether non normality, skewness, kur-tosis, serial correlation and seasonality are features sharedin common by the trading volume and stock return series.Those features are the ones found as possible characteris-tics of the ¯ow of information. Also followed is the workby Vahid and Engle (1997) to test for codependent cyclesbetween those series. They extend the work of Engle andKozicki (1993) to test for co-movement in multiple timeseries when their cycles are not exactly synchronized. Ifcodependence exists between stock returns and volume,the common latent factor will in¯uence to the series atdi� erent times, t and t ¡ j with j > 0.
The remainder of the paper is organized as follows.Section II describes the testing procedure of common fea-tures as in Engle and Kozicki (1993). Section III presentsthe testing procedure for codependence following the workin Vahid and Engle (1997) and Tiao and Tsay (1989).Section IV describes the data used. Section V reports theresults of testing for common features and Section VI sum-marizes the results and contains the conclusions.
II . TESTING FOR COMMON FEATURES
Engle and Kozicki (1993) introduce a class of statisticaltests for the hypothesis that some feature that is presentin each of several variables is common to them.
They suppose there is a test with standard properties inorder to detect the feature of interest. Let s…y† be the teststatistic, where y represents a data series yt; t ˆ 1; . . . ; T ,and de®ne the set of probability measures under considera-tion for each of the hypotheses H0: no feature, H1: feature.For s and a particular choice of size such as 0.05, the sta-tistic must have the property that there is a critical regionfs…y† > cg where c is de®ned by PH0
‰s…y† > cŠ 4 0:05. We®nd a feature if s…y† > c.
The test for the null hypothesis of a common feature isdeveloped by applying the feature test to the variable
886 M. ReguÂlez and A. Zarraga
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u ˆ y1¡ ¯y2 for various values of ¯. Engle and Kozicki
(1993) show that under some conditions the distributionof the minimand of s…u† over ¯ satis®es:
PH0‰s…uu† > cŠ 4 5% …1†
The test is applied to the minimized value of the statistic,s…uu†, knowing that the null will be rejected less than orequal to 5% of the time when it is true. Although thetest can be ine� cient, they show that it is consistent.
Some of the features we are going to test can be de®nedin terms of regression hypotheses. Engle and Kozicki(1993) call them regression-based tests for common fea-tures. Suppose that two series y1, y2 are each tested for afeature in the model:
y1tˆ xt 1
‡ zt®1‡ "1t
y2tˆ xt 2
‡ zt®2‡ "2t;
…2†
where in this case y1 could be daily stock return series andy2 daily volume. fx; zg are assumed to be the same for bothseries. In the serial correlation case, fxg would be simply aconstant and possibly relevant trends and fzg wouldinclude lags of the dependent variable and possibly othervariables. To test whether correlation is common to thesedata series, one has to test whether there exists a ¯ suchthat the linear combination u ˆ y1
¡ ¯y2 does not have thefeature.
Engle and Kozicki (1993) show that in order to minimizes…u† with respect to ¯ to get:
s…uu† ˆ min¯
s…y1¡ ¯y2
† ˆ uu0Mxz…z 0
Mxz†¡1z
0Mxuu=¼¼
2u; …3†
where Mxˆ I ¡ x…x 0
x†¡1x
0, uut
ˆ y1t¡ ¯y2t and ¼¼
2u is a con-
sistent estimate of the residual variance, iterative or non-linear procedures are required since ¯ appears in both thenumerator and denominator of s…uu†. However, an asymp-totically equivalent estimator can be derived by minimizingthe numerator of Equation 3. The solution is given by:
^n
ˆ ‰y 02Mxz…z 0
Mxz†¡1z
0Mxy2
Š¡1y
02Mxz…z 0
Mxz†¡1z
0Mxy1;
…4†
which is the two-stage least squares (2SLS) estimate of ¯ inthe model:
y1tˆ ¯y2t
‡ xt ‡ "t; …5†
where the instrument list is fx; zg. The test statistic canthen be computed as TR
2of the regression of e, the 2SLS
residuals, on the instruments fx; zg. The asymptotic distri-bution of the statistic under the null of a common feature isa chi-squared with degrees of freedom the number of over-identifying restrictions.
III . CODEPENDENT CYCLES
Vahid and Engle (1997) extends the work of Engle andKozicki (1993) to test for co-movement in multiple timeseries when their cycles are not exactly synchronized.
2
The concept of serial correlation common feature (SCCF)was introduced by Engle and Kozicki (1993) and it impliesthat all dependence on the past at time t can be embodiedin a factor ft, which is common between the variables so alinear combination of the series is unpredictable relative tothe past history of the variables. Vahid and Engle (1997)point out that SCCF demands a very strong relationbetween the cycles in two variables. They say it requiresthat the system shocks, which generate lasting e� ects in thevariables, should have no predictable e� ect on a particularlinear combination of the variables, not even for a veryshort period. They take SCCF to be a measure of synchro-nized comovement among variables because it requires thatthe impulse responses of the variables should be exactlycolinear.
Vahid and Engle (1997) give the following example of afactor structure that produces a comovement that is notsynchronized:
y1t
y2t
Á !ˆ
ft
ft¡1
Á !‡
"1t
"2t
Á !…6†
ft‡1ˆ ¬ft
‡ ³0 "1t
"2t
Á !…7†
where ft is the latent common factor and "1t, "2t are theinnovations in y1t and y2t. In this case the linear combina-tion y1t
¡ ¬y2t will be a process that has no correlation withall lags of both variables except their ®rst lag.
They show that codependence does indeed fall into theframework of common features developed by Engle andKozicki (1993). Following the work by Tiao and Tsay(1989) they demonstrate that testing for codependence istesting for hidden scalar components of order SCM…0; j† ( jnon-negative and small) in the data generating process. Thelinear combination of the series that is a SCM…o; j† willhave shorter memory relative to the vector process.Finding such SCMs is to ®nd linear dependence in theimpulse responses after a certain period. The test for smal-lest order SCM…0; 0† is testing for a serial correlation com-mon feature among the elements of the vector series.
For instance, suppose we ®nd an SCM…0; 1†. Considerthe Wold representation of yt in terms of its innovation:
ytˆ "t
‡X1
jˆ1
Cj"t¡j…8†
Common features between stock returns and trading volume 887
2We thank an anonymous referee for pointing us to take into account this in our analysis.
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If ¬00yt is an SCM…0; 1†, we have
¬00Cj
6ˆ 0 if j ˆ 1
ˆ 0 if j > 1
(…9†
Vahid and Engle (1997) show that the Tiao and Tsay(1989) proposed test for scalar components models oforder …0; j† can be seen as a test for codependent cyclesbased on a consistent, but suboptimal, estimate of thecofeature vector. They propose an e� cient GMM estima-tor of the codependence vector and develop a moment testbased on that estimator. They perform a series of simula-tions to study the ®nite sample performance of the Tiaoand Tsay test and the moment test based on the optimalGMM estimator. Their results show that the Tiao and Tsaytest performs better in small samples. So their recomenda-tion is to use Tiao and Tsay test in the testing stage andthen use GMM in the estimation stage.
Given that this study’s goal is to test for codependence,rather than to estimate the codependence vector e� ciently,it will apply the Tiao and Tsay testing procedure. Their testfor the null hypothesis that there are s scalar componentsof type SCM…0; j† is a test for s zero canonical correlationsbetween Y0;t
ˆ yt and Yh;t¡j¡1ˆ …y 0
t¡y¡1; . . . ; y0t¡j¡1¡h
† 0.
The testing procedure is as follows:
(1) Let yt be the bivariate vector with E…yt† ˆ 0 that
follows an ARMA…p; q† process.(2) For t ˆ h ‡ j ‡ 1; . . . ; T compute the following
backward and forward autoregressions
Y0
0;tˆ Y
0h;t¡j¡1 …0; h; j† ‡ et
Y0
h;t¡j¡1ˆ Y
00;t
¤…0; h; j† ‡ e¤t
…10†
(3) For h 5 0 and j 5 0, the squared sample canonicalcorrelations between Y0;t and Yh;t¡j¡1 are the eigen-values of
AA…0; h; j† ˆ ¤…0; h; j† …0; h; j†
where ¤…0; h; j† and …0; h; j† are the ordinary least
squares estimates of the forward and backward auto-regressions.
(4) Let ¶¶i… j† be the ith smallest eigenvalue of AA…0; h; j†,
…i ˆ 1; 2†, Then we may check for the number ofzero canonical correlations of A…0; h; j† by testingAA…0; h; j† sequentially. To do so, Tiao and Tsayde®ne the statistic
C… j; s† ˆ ¡…T ¡ h ¡ j†Xs
iˆ1
loge 1 ¡ ¶¶i… j†
di… j†
( )…11†
where di… j† is de®ned as
di… j† ˆ 1 ‡ 2
Xj
vˆ1
»»v…ww1t
†»»v…ww2t
† …12†
where »»v…wwut
† is the lag v sample autocorrelation ofwwut
…u ˆ 1; 2†, ww1tˆ rr
0Y0;t, ww2t
ˆ gg0Yh;t, and rr and gg
are respectively the right and left eigenvectors ofAA…0; h; j† corresponding to the eigenvalue ¶¶i
… j†.The left eigenvectors of AA…0; h; j† are the right eigen-vectors of the same eigenvalue from the matrixAA]…0; h; j† ˆ …0; h; j† ¤…0; h; j†.
Tiao and Tsay show that if yt is a k-dimensionalARMA…p; q† process with Gaussian innovations at,then for h 5 0 and j 5 0 the statistic C… j; s† isasymptotically a chi-squared random variable with
degrees of freedom fsx…hxk ‡ s†g provided that
¶1… j† ˆ ¶2
… j† ˆ ¢ ¢ ¢ ˆ ¶s… j† but ¶s‡1
… j† 6ˆ 0 and fori ˆ 1; . . . ; s wit is uncorrelated with at¡v for v > j.
(5) Choice of h. Tiao and Tsay suggests h ˆ …q ¡ j† atthe …0; j† position, where q is the order of the MApart in the ARMA process speci®ed for the vectorprocess.
IV. DATA
The data used in this work come from 100 ®nancial stocksof the Spanish continuous stock market. They are dailydata corresponding to the logarithm3 of trading volume…Vit
† and stock return …Rit† for each stock i ˆ 1; . . . ; 100,
being the former the number of shares traded daily and thelatter is calculated from the price series adjusted for divi-dends and equity issues in the following way:
Ritˆ ln…Pit
‡ DIit‡ SUit
† ¡ ln…Pit¡1† …13†
where Rit is the daily return on stock i, Pit is the closingprice, DIit are the dividends and SUit the equity issues.
The sample period considered goes from 20/04/1990 to
29/01/1996, which means a total of 1452 observations ofreturn and volume for each stock. In principle, it seemedbetter to consider individual stocks. The problem is how toselect them from the 100 stocks available. There are very
few stocks that are traded every day in the sample. On theother hand, not all the ®nancial stocks are present in themarket for the whole sample period since some of themwere incorporated in the market after 20/04/1990, they
stopped trading in the market before 29/01/1996 or evendisappeared temporarily from the market to be incorpo-rated again after a period of time.
888 M. ReguÂlez and A. Zarraga
3Even though the standard MODM considers the conditional distribution of volume as being normal, following Richardson and Smith
(1994) `. . . a better ®t may be to assume that, conditional on the information ¯ow, volume is skewed to the right. One possible candidatealong these lines is to replace volume with the log of volume . . .’.
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In order to avoid those problems and to use as muchinformation as possible, this study takes as an alternativethe following procedure. For each day in the sample, thesample average of Rit and Vit de®ned above for the non-zero volume stocks is computed.4 That is, this studyconsiders:
Rtˆ 1
Nt
X100
iˆ1
IitRit Vtˆ 1
Nt
X100
iˆ1
IitVit
where
Iitˆ
1 if Vit > 0
0 otherwiseNt
ˆX100
iˆ1
Iit:
(
V. EMPIRICAL RESULTS
The main goal is to analyse whether stock returns andtrading volume present some features that are shared incommon. In order to select which features to be analysed
Common features between stock returns and trading volume 889
Fig. 1. Stock return Fig. 2. Trading volume
Fig. 3. Adjusted stock return Fig. 4. Adjusted trading volume
4Although Rt and Vt are stationary, they present daily and monthly calendar e� ects, which are removed from using the two-step
procedure of Gallant et al. (1992). The results are not represented but can be obtained from the authors upon request. Figures 1 to 4show the graphs for both the raw and seasonally adjusted stock returns and volume series. As Figs 2 and 4 show, the series of the tradingvolume exhibit jumps toward zero, which are typically not present in volume series of other countries used in the literature. These jumpsresult from the fact that many people take holidays around a festive day, implying that trading volume would be lower during those days.
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as common, the features present in both series are consid-ered ®rst.
5Secondly, this study focuses on those features
for which there is some empirical or theoretical guide ascharacteristics of the information ¯ow to be transmitted tovolume and returns. As already pointed out in the intro-duction, the features that some authors have found to char-acterize this latent variable are skewness, kurtosis, serialcorrelation and seasonality.
First, this study proceeds to test for non-normality,skewness, kurtosis, and serial correlation as common fea-tures in the adjusted data and ®nally, we consider the rawdata to test for daily and monthly seasonalities.
Non normality test is carried out using the statistic pro-posed by Jarque and Bera (1980). Under the null hypoth-esis of normality for a series y the test statistic, JB…y†, hasthe following asymptotic distribution:
JB…y† ˆ Tb1
6‡ …b2
¡ 3†2
24
" #¹a X2
…2†; …14†
where���b
p1
ˆ …m3=¼3†, b2
ˆ …m4=¼4†, mr
ˆ …1=T†P
Tiˆ1
…yi¡ ·yy†r
and ¼2 ˆ m2 is the variance of the series, being
T the sample size.Table 1 presents the testing results. They show evidence
of non normality in both series. In order to test whethernon normality is driven by a common factor, an optimiza-tion procedure was used, which gives the value of ¯ suchthat the JB…u† statistic for the liner combination
u ˆ y1¡ ¯y2 is minimized, where y1 and y2 are return
and volume series, respectively. Using a grid searchmethod, the selected value of ¯ is ¡0:01, for which the
test statistic JB…u† takes the value 780.7 with a p-value of
0.0. This means that the linear combination of u ˆ y1¡ ¯y2
does not present normality and therefore, the null hypoth-
esis of common feature is rejected knowing that at least the
type I error is not greater than 5%.
The JB statistic for normality compares a distribution to
the normal distribution by comparing its skewness to zeroand its kurtosis to three. The rejection of normality as a
common factor, however, does not imply that skewness
and or kurtosis are not common factors. More concreteresults can be obtained by testing for skewness and kurtosis
separately.
Skewness and kurtosis tests are based on the statistics
SK and KU respectively, whose asymptotic distributions
under the null hypothesis of no skewness and no kurtosisare the following
SK…y† ˆ T2
…T ¡ 1†…T ¡ 2†
Á !m3
¼3
³ ´" #
£��������������������������������…T ¡ 1†…T ¡ 2†
6T
r¹a N …0; 1† …15†
and
890 M. ReguÂlez and A. Zarraga
5If the feature is only present in one of the original series, then the test proposed for a common feature will always suggest that there is a
common feature because a linear combination that puts all weight on the series that does not contain the feature will trivially not showevidence of the feature.
Table 1. Testing results
Return Volume Common features
H0: normality H0: normality H0: normalityJB…y1
† ˆ 1944:4 JB…y2† ˆ 27979:9 JB…u† ˆ 780:7
(0.000) (0.000) (0.000)
H0: no skewness H0: no skewness H0: no skewnessSK…y1
† ˆ ¡5:85 SK…y2† ˆ ¡47:33 SK…u† ˆ ¡1:28
(0.000) (0.000) (0.2)
H0: no kurtosis H0: no kurtosis H0: no kurtosisKU…y1
† ˆ 43:78 KU…y2† ˆ 160:64 KU…u† ˆ 27:88
(0.000) (0.000) (0.000)
H0: no serial correlation H0: no serial correlation H0: no serial correlationTR
2…y1† ˆ 70:57 TR2…y2† ˆ 369:1 TR
2…u† ˆ 68:8(0.002) (0.000) (0.002)
H0: no daily e� ect H0: no daily e� ect H0: no daily e� ectW …y1† ˆ 5:879 W …y2† ˆ 16:196(0.208) (0.003)
H0: no monthly e� ect H0: no monthly e� ect H0: no monthly e� ectW …y1† ˆ 38:88 W …y2† ˆ 53:52 W…u† ˆ 19:31(0.000) (0.000) (0.06)
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KU…y† ˆ T2
…T ¡ 1†…T ¡ 2†…T ¡ 3†
Á" !
…T ‡ 1†m4¡ 3…T ¡ 1†m2
2
¼4
Á !#
£
�����������������������������������������������…T ¡ 1†…T ¡ 2†…T ¡ 3†
24T…T ‡ 1†
s
¹a N …0; 1† …16†
Application of the tests to the data series gives evidenceof negative skewness and leptokurtosis for each return andvolume series. These results, shown in Table 1, are similarto those obtained in previous research.
6The negative skew-
ness means that on occasion, very low returns and very lowtrading volume occur in the market while most of the timethe magnitude of them is above the mean. On the otherhand, the leptokurtosis found on both series implies thatthe tails of both distributions are fatter than those of thenormal distribution. Nonetheless, the result of kurtosismust be interpreted with caution because the distributionis not symmetric but under the null, we are taking theNormal distribution, which is symmetric, as the reference.
As both data series present skewness and kurtosis, thisstudy analyses if some of these features are shared in com-mon. Again we use a search method to ®nd the value of ¯ inthe linear combination u ˆ y1
¡ ¯y2 such that SK…u† andKU…u† statistics are minimized. For the skewness test, theselected value of ¯ is ¡0:007 for which the test statistictakes the value SK…u† ˆ ¡1:3 with a p-value of 0.2.Therefore, for a 5% signi®cance level the null hypothesisof common feature cannot be rejected, indicating thatskewness is a feature common to stock returns and trading
volume.On the contrary, we cannot ®nd a ¯ for the linear com-
bination u ˆ y1¡ ¯y2 such that the value of the KU statis-
tic does not reject the null hypothesis of common feature.In fact, the value of ¯ that minimizes the KU statistic is
¯ ˆ ¡0:01, for which KU…u† ˆ 27:8, which shows that kur-
tosis is not common to the series.Next, this study analyses whether stock returns and vol-
ume have serial correlation as a common feature. That is,
do lagged returns and volume actually provide informationuseful for forecasting both returns and volume in a bivari-ate system? Do the series share the dynamics in common?
Under the null hypothesis of `no feature’ returns andvolume are orthogonal to the information set includinglagged stock returns and volume. This orthogonality con-
dition can be tested estimating by least squares each equa-tion of 2 and using the statistic TR
2of the estimated
regressions, which asymptotically follows a chi-squareddistribution with degrees of freedom the number of over-
identifying restrictions.In this case, Equations 2 form a bivariate VAR…20†
model, where the order, p, has been selected not onlyaccording to the Final Prediction Error, FPE…p†,Akaike’s Information, AIC…p†, Hannan-Quinn, HQ…p†,and Schwarz, SC…p†, criteria but also imposing that thecorresponding residuals are white noise.7 The results of
serial correlation tests applied to each of the data seriesare presented in Table 1. They show the value of the TR
2
statistic for both series, stock returns and volume, with thecorresponding p-value. This statistic follows asymptoticallya chi-squared distribution with 40 degrees of freedomunder the null hypothesis of no serial correlation. Thereis evidence that both series present this feature.
Given that the null of no serial correlation for both serieshas been rejected, this study proceeds to test for a serialcorrelation common feature (SCCF). The regression-basedtest explained in Section II is followed. The list of instru-ments used for the 2SLS estimation is a constant term and
20 lags of y1 and y2 respectively. In this case, the statisticTR
2follows asymptotically a chi-squared distribution with
39 degrees of freedom. It takes the value of TR2 ˆ 68:8
with a signi®cance level of 0.002. Therefore, stock returnsand trading volume do not present serial correlation as acommon feature.
As Vahid and Engle (1997) point out SCCF demands avery strong relation between the cycles of both variables, asynchronized comovement. Therefore, Tiao and Tsay test-ing procedure as explained in Section III is applied to testfor unsynchronized comovement or codependence.
As suggested by Vahid and Engle (1997) this study starts
with smallest j and then increase j by 1. If an SCM…O; j†exists, then the procedure should be ®nished since any testfor SCM of order …0; j
0† with j0> j will not be rejected.
Note that the test for SCM…0; 0† is testing for a serial cor-relation common feature between stock returns and tradingvolume. The codependence tests for di� erent values of q,
the order of the MA part in the ARMA process of thevector process, are conducted. However, the results donot di� er signi®cantly.
The results of the codependence tests are presented inTable 2. They show no evidence of serial correlation as acommon feature when both series are synchronized sincethe existence of an SCM…0; 0† is rejected. These results are
Common features between stock returns and trading volume 891
6Peiro (1999), using distribution-free tests, ®nds for the Spanish stock exchange di� erences between negative and positive excess returns.
This type of asymmetry occurs in the markets with the lowest capitalizations and trading volumes from all the analysed. We have notused those tests because they assume observations to be independent and identically distributed. If for returns is not so hard to assumethat, it is a very strong assumption for volume.7
The concrete expression for each criterion can be found in LuÈ tkepohl (1991). Testing whether the residuals are white noise is based onthe Ljung±Box statistic.
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similar to those we obtain when using the procedure ofEngle and Kozicki (1993). On the contrary, an SCM struc-ture of order …0; 1† in the data generating process of thevector process cannot be rejected at a 5% signi®cancelevel.8 It means that there is a non-synchronized comove-ment between stock returns and trading volume. Theimpulse responses of the two variables are linearly depen-dent after one period.
Finally, using the stock returns and volume series with-out removing the calendar e� ects this study has tested ineach series for daily and monthly e� ects.
To test for seasonal e� ects we consider the followingOLS regression:
ytˆ 0
‡XJ
jˆ1
jDjt‡ "t; …17†
where Djt are daily (monthly) dummies from Mondaythrough Thursday (January through November). The null
hypothesis of no feature is H0 : 1ˆ 0; . . . ; J
ˆ 0 and thestatistic considered is the Wald statistic estimating the cov-ariance matrix of the OLS coe� cients following Newey
and West (1987). Under the null, the Wald statistic, W ,follows asymptotically a chi-squared distribution with Jdegrees of freedom.
Looking at the results in Table 1, there is only evidenceof monthly seasonality in both stock returns and volume.
9
Therefore this study only tests for monthly seasonality as a
possible common feature between those series. Using a gridsearch method, the selected value of ¯ is 100, for which thetest statistic W…u† takes the value 19.31 with a p-value of
0.06. This means that there is a common factor in monthlyseasonality.
In summary, this study ®nds skewness and monthly sea-
sonality as common features between the series of stockreturns and trading volume considered. Also found isthat stock returns and volume cycles are not completely
synchronized, and their response to a shock is similarfrom one day after the shock has occurred.
The evidence found in favour of skewness as a commonfeature could be interpreted as follows. This study hasobtained negative skewness in both series. Hence, if the¯ow of information is positively skewed, as Richardsonand Smith (1994) suggest, the large information ¯ow arriv-ing to the market on occasions could be interpreted as badnews. The reason is that this large amount of informationis associated with low volume and low returns also onoccasions. On the contrary, most of the time a smallamount of information ¯ow arrives to the market, andvolume and returns are higher than the mean.
On the other hand, the result found of codependenceafter one period can be read as follows. The response ofstock returns and volume to the latent factor, the ¯ow ofinformation, is not simultaneous but with one lag of a day.Once both variables react to a shock in the ¯ow informa-tion then they will respond in the same fashion and theimpulse response functions will be colinear. This study con-siders that from this analysis it cannot be determined whichvariable responds ®rst. A Granger causality study of theseseries done in Zarraga (1998) proves that returns causevolume in the Granger sense but rejects the causality inthe other direction. This ®nding could suggest that returnswill react ®rst to a shock in the information ¯ow.
The interpretation of the result of monthly seasonality interms of the rate of information to the market being thecommon factor is more di� cult because this study does nothave a theoretical reason, as far as is known, for the ¯ow ofinformation to present monthly patterns. One interpret-ation of what has been obtained could be that in a givenmonth it would by systematically more pro®table to releaseinformation to the market than in the rest of the months.
VI. CONCLUSIONS
This article describes the theory and implementation ofmethods for testing for common features in a bivariatesystem for Spanish daily return and trading volume.Data properties such as skewness, kurtosis, serial correla-tion, and monthly seasonalities that are in fact present inthe data series are tested to be common to them.
The results of the tests show that although non normal-ity and kurtosis are features present in stock return andvolume series, they are not common to them since thereis no linear combination of the data series such that fails tohave the feature. Even though evidence is not found onserial correlation as a common feature, there is a non-synchronized comovement between the cycles of both ser-ies, so that there is a linear dependence in the impulse
892 M. ReguÂlez and A. Zarraga
8The codependence tests could be a� ected by the seasonal adjustment, see Hecq (1998). The test has been carried out considering the
data without seasonal adjustment and the results do not change.9
There is only signi®cant daily seasonality in the volume series but not in the stock returns.
Table 2. Codependence results
q SCM…m; j† C… j; 1† Degrees of freedom p-value
1 SCM…0; 0† 24.85 3 0.001 SCM…0; 1† 0.09 1 0.762 SCM…0; 0† 25.70 5 0.002 SCM…0; 1† 0.79 3 0.853 SCM…0; 0† 31.87 7 0.003 SCM…0; 1† 5.94 5 0.31
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responses after one period. Moreover, monthly e� ects andskewness are also shared in common by both returns andvolume series. This evidence indicates the existence of acommon factor driving both the returns and volume interms of the dynamics, seasonality and skewness.
ACKNOWLEDGEMENTS
We would like to thank Javier Gardeazabal, Amado Peiro ,Ignacio PenÄ a, Gonzalo Rubio, Rafael Santamarõ a,Yochanan Shachmurove, David Stolin, Fernando Tusell,Jesu s Va zquez and an anonymous referee for their helpfulcomments. We also received valuable comments from semi-nar participants at ESADE, 1998 EFMA-FMA FinanceConference, XXIII Simposio de Ana lisis Econo mico andVII Foro de Finanzas. This work was supported byDireccio n General de EnsenÄ anza Superior del MinisterioEspanÄ ol de Educacio n y Cultura under research grantsPB97-0620, PB98-0149, by Universidad del Paõ s Vasco(UPV/EHU) under research grant UPV 035.321-G54/98and by Gobierno Vasco under research grant PI-1999-131. Ainhoa Zarraga acknowledges the ®nancial supportprovided by Departamento de Educacio n, Universidades eInvestigacio n del Gobierno Vasco through BecaPostdoctoral para Formacio n de Investigadores. MartaRegu lez would like to thank Yin-Wong Cheung for hiscomments when she was visiting UC Santa Cruz with afellowship from the Spanish Ministerio de Educacio n,Cultuna y Deportes and the hospitality of the EconomicsDepartment at UCSC. The usual disclaimer applies.
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