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Intraday Patterns in the Returns, Bidask Spreads, and Trading Volume of Stocks Traded on the New York Stock Exchange
Chris Brooks1,ISMA Centre,University of Reading Melvin J. Hinich, University of Texasat Austin DouglasM. Patterson, Virginia Tech
Abstract Much researchhas demonstrated existenceof patterns in highfrequencyequity returns, the return volatility, bidask spreadsand trading volume. In this paper,we employ a new test for detectingperiodicities basedon a signal coherence function. The techniqueis applied to the returns,bidask spreads, and trading volume of thirty stockstradedon the NYSE. We are able to confirm previous findings of an inverseJshaped patternin spreads and volume through the day. We also demonstrate that suchintradayeffectsdominateday of the week seasonalities in spreads and volumes, while there are virtually no significant periodicities in the returns data. Our approachcan also leads to a natural method for forecastingthe time series,and we find that, particularly in the case of the volume series, the predictions are considerably more accurate than thosefrom naive methods.
October2003
J.E.L. Classifications: C32, C53, F31 Keywords: spectral analysis, periodicities, seasonality,intraday patterns, bidask spread, trading volume.
1 Chris Brooks (Corresponding author), ISMA Centre, P.O. Box 242, The University of Reading, Whiteknights, ReadingRG6 6BA, England,Tel. (+44) 1189316768; Fax: (+44) 1189314741; email: [email protected]
Chris andMel, I wonderif the forecasts would be more accurate we generated if intraday forecasts ratherthanonedayaheadforecasts. ideais that we would then expJoitthe The intradaycyclesin the data. It could be doneas follows: hold out four daysof data.Generate forecasts 39 for the first hold out day. Assumea randomwalk for the null process, a MA(1). The null i.e. forecastis thenthe currentlevel of the process. Calculatethe RMSE errorsandabsolute errorsfor eachintradayforecast. Estimatethe model againusingthe original dataplus the first hold out day. Generate another39 forecasts. Continueuntil the four daysof dataare exhausted. This will producecloseto 160forecasts eachseries. for Any thoughtson this?
1. Introduction One of the virtually indisputable stylised featuresof financial time seriesis that they exhibit periodicities, or systematicallyrecurring seasonal patterns.Such patternshave been observed in returns,return volatility, bidask spreads trading volume, and significant effects appear and to be presentat various frequencies.Early researchemployeddaily or weekly data and was focused on examining the returns themselves,including French (1980), Gibbons and Hess (1981), and Keirn and Stambaugh(1984). All three studies found closetoclosereturn on the New York Stock Exchange(NYSE) is significantly negative on Monday and significantly positive on Friday. Moreover, Rogalski (1984), and Smu Starks(1986) observedthat this negativereturn betweenthe Friday for the Dow JonesIndustrial Average (DJIA) occurson Monday itself during the 1960'sbut
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movesbackward to the period betweenthe Friday close and Monday open in th5Jf\teJ27Q'~,By contrast,Jaffe and Westerfield (1985) found that the lowest meanreturns for the Japanese and Australian stock markets ?c(;~~_()n_~~~~da~s. Harris (1986) also examined weekly and intradaypatternsin stock returns and found that most of the observeddayoftheweekeffects occur immediately after the open of the market, with a price drop on Mondays on averageat this time and rises on all other weekdays;seealso Wood, McInish and Ord (1985).
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Researchhas additionally employed intradaily data in order to determinewhether there are periodically recurring patterns at higher frequencies.Wood et at. (1985), for example, examineminutebyminute returns data for a large sample of NYSE stocks. They find that significantly positive returns are on averageearnedduring the first 30 minutes of trading and at the market close, a result echoedby Ding and Lau (2001) using a sample of 200 stocks from the Stock Exchangeof Singapore.An extensivesurveyof the literature on intraday and intraweek seasonalitiesin stock market indices and futures market contracts up to 1989 is given in Yadav and Pope(1992).
More recent studieshave also observedperiodicities in bidask spreadsand trading volume. Chan, Chung and Johnson (1995), for example,investigatebidask spreadsfor CBOE stock options and for their underlying assets tradedon the NYSE. They obtain the familiar U shape spreadpattern for the stock spreads,as McInish and Wood (1992) and Brock and Kleidon (1992) had argued previously, but the option spreadsare wide at the open and then fall rapidly, remaining flat through the day. A large spreadat the open that falls and then remains
constantfor the remainderof the day was also found by Chan, Christie and Schultz (1995) in their examination of stocks traded on the NASDAQ. The differencesin results betweenthe NYSE and the NASDAQ / CBOE has been attributedto their differing market structure,the NYSE having specialistswhile the NASDAQ is a dealermarket. Finally, Jain and Joh (1988) employhourly aggregated volume for all NYSE stocksand observethat a Vshapedpattern is also presentin trading volume. This result is corroborated Foster and Viswanathan(1993) by using volume data on individual NYSE stocks.
Many theoretical models of investor and market behaviour have also been proposed to explain these styli sed features of financial time series,including those that account for the strategicbehaviour of liquidity traders and informed traders (see, for example, Admati and Pfleiderer, 1988). An alternative method for reconciling a finding of recurring seasonal patternsin financial markets with the notion of efficient markets is the possible existenceof timevarying riskpremia, implying that expectedreturnsneednot be constantover time, and could vary in part systematicallywithout implying market inefficiency.
Traditionally, studies concernedwith the detection of periodicities in financial time series would either use a regressionmodel with seasonal dummy variables (e.g., Chan, Chung and Johnson,1995) or would apply spectralanalysisto the sampleof data (e.g. Bertoneche,1979; Upson, 1972). Spectralanalysismay be defined as a processwhereby a seriesis decomposed into a set of mutually orthogonalcyclical components different frequencies.The spectrum, of a plot of the signal amplitude againstthe frequency,will be flat for a white noise process,and statistically significant amplitudes at any given frequencyare taken to indicate evidenceof periodic behaviour. In this paper, we propose and employ a new test for detecting periodicities in financial marketsbasedon a signal coherence function. Our approachcan be applied to any fairly large, evenly spacedsampleof time seriesdata that is thought to contain periodicities. A periodic signal can be predicted infinitely far into the future since it repeats exactly in every period. In fact, in economicsand finance as in nature, there are no truly deterministic signals and hence there is always some variation in the waveform over time. The notion of partial signal coherence, developedin this paper into a statistical model, is a measure how much the waveform varies over time. The coherence of measures calculatedare then employedto hone in on the frequencycomponents the Fourier transformsof the signal of
2
that are the most stable over time. By retaining only those frequencycomponentsdisplaying the leastvariation over time, we are ableto detectthe most important seasonalities the data. in
The remainder of this paper is organisedas follows. Section 2 describesthe data, while Section 3 introduces some notation, defines the test statistics employed to detect the periodicities and describesthe forecastingprocedure. Section 4 presents and analysesthe resultswhile Section5 concludesand offers suggestions extensionsand further research. for
2. Data The dataemployed this papercomprise returns,the bidaskspread, the natural in the and logarithmof tradingvolumefor a sample thirty stockstradedon theN~SE2.The TAQ of database all stockswas split into quintilesby marketcapitalisation~~i.krJanuary of 4 1999,
and ten stocksfor analysiswere selected randomly from the top, middle and bottom quintiles. Selectingstocksin this mannerallows us to examinewhether our findings are influencedby firm size. The data are sampledevery tominutes from 9:40amuntil4pm EST, making a total of 39 observationsper day. The sample covers the period 4 January 1999  24 December 2000, a total of 504 trading days, and thus there are 19,656 observationsin total on each
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series. We employ continuously compoundedmidpoint quote returns based on the la~t recordedquotation in each 10minuteperiod. Table 1 presentsthe namesof the companies selected, their ticker symbol mnemonics,and their market capitalisations.
The 2year sampleperiod is split il1to 504 nonoverlappingframes, each of length one day, with each day comprising 39 tenminutely observations.This implies that a total of 19 periodicities are examined: 39, 39/2, 39/3, ...,39/19. The autocoherence measuresare thus calculatedfor eachperiodicity acrossthe 504 frames.
3. Methodology 3.1Development a Test for SignalAutocoherence of
This paper developsbelow a model for a signal with randomly modulatedperiodicity, and a measureknown as a signal coherencefunction, which embodies the amount of random variation in each Fourier componentof the signal. Any periodic function of period T can be
2
Issuesinvolved with the analysisof suchsampledtradebytradedata are discussedin Hinich and Patterson
(1985, 1989).
3
written as a sum of weighted sine and cosine functions whose frequencies are integer multiples of the fundamentalfi'equencyliT. Thesefi'equencies called Fourier fi'equencies. are The weights, called amplitudes,are fixed constantsfor a deterministicperiodic function. The sum is called a Fourier transform of the periodic function. But a perfectly periodic function is an idealisation of a real periodic process.Each amplitude of the Fourier transform of a real periodic processis a constantplus a zero mean random time seriesthat mayor may not be stationary.The random time variationsmakesthe amplitudes"wobble" over time causingthe signal to have periodtoperiod random variation. Hinich (2000) introducesa measureof the wobble of the Fourier amplitudesas a function of frequency.This new form of spectrumis called a signal coherencespectrumand is very different from the ordinary power spectrum. Most fundamentally,it is a normalisedstatistic that is independent the height of the power of Most normalisedstatistic that spectrumat eachfrequency. Introducing somenotation to outline the approach,let {x(t), t = 0, l, 2, ...} be the time series of interest, sampled at regular intervals. The series would be said to exhibit randomly modulatedperiodicity with period T if it is of the form x(t)
= ao +
1
K
I(alk
+ Ulk(t»cos(27ifkt)+I(aZk
T k=1
1K
+uzk(t»sin(27ifkt)
(1)
T k=1
wherefi = kiT and Uik(i=1,2) are jointly dependentzero mean random processesthat are periodic block stationaryand satisfy finite dependence. Note that we do not require Uikto be Gaussian.It is apparentITom (1) that the random variation occurs in the modulation rather than being additive noise; in statistical parlance,the specification in (1) would be termed a random effects model. The signal x(t) can be expressedas the sum of a deterministic (periodic) component,a(t), and a stochasticerror term, u(t), so that (1) canbe written x(t)
where ak :;::
=ao +alk
1
K 1K I:ak exp(i27ifkt) ~>k(t)exp(i27ifkt) +
(2)
K k=l
K k=O
+ ia2k and Uk:;::Ulk + iU2k.The task at hand then becomes one of quantifying the
relative magnitudeof the modulation,ak.
A common approach to processing signals with a periodic structure is to portion the observationsinto M frames, each of length T, so that there is exactly one waveform in each sampling frame. There could alternatively be an integer multiple of T observationsin each frame. The periodic componentof aCt)is the mean componentof x(t). In order to determine
4
how stablethe signal is at eachfrequencyacrossthe frames,the notion of signal coherence is employed. Signal coherence is loosely analogous to the standard R2 measure used in regressionanalysis, and quantifies the degreeof associationbetween two componentsfor eachgiven frequency.It is worth noting that the methodologythat we proposehere is based on the coherenceof the signal acrossthe framesfor a single time series(which may also be termed autocoherence). This is quite different from the tests for signal coherenceacross marketsused,for example,by Hilliard (1979) and Smith (1999l
The discreteFourier transform of the mthframe,beginningat observationfJm=«ml)1)+1 and endingat observationmT, for frequency = kiT is given by fi + t)exp( i2ifkt) =ak + U m (k) (3)
(4) where Cu (tptz) = E[um'(t\)um(tz)], and the varianceis of order OCT).Provided that um(t)is weakly stationary,(4) can be written (J~(k) = T[Su(Ik) + O(lIT)] whereSu(f)is the spectrumof u(t). (5)
The signal coherencefunction, rx(k), measures variability of the signal acrossthe ftames, the and is definedas follows for eachfrequency fi yx(k) = 21ai 2 lakl +CTu(k) (6)
It is fairly obvious ftom the constructionof yx(k)in (6) that it is boundedto lie on the (0,1)
interval. The endpoint case rx(k) = 1 will occur if ak'l'Oand CTu 2(k)=0, which is the case where
the signal componentat ftequencyfi has a constantamplitude and phaseover time, so that there is no random variation acrossthe ftames at that ftequency (perfect coherence).The
3
Both of these papers employ the frequency domain approachin order to examine the extent to which stock
marketscomove acrosscountries. Our techniqueis also distinct from that proposedby Durlauf (1991) and used by Fong and Ouliaris (1995) to detect departuresfrom a random walk in five weekly US dollar exchangerate series.
5
other endpoint,rx(k) = 0, will occur if ak=Oand o"u 2(k);t:0,when J!!temean value of the meanvalue of the
component at frequencyfi is zero, so that all of the variation across the frames at that frequencyis pure noise (no coherence).
The signal coherence function is estimated ITom the actual data by taking the Fourier transformof the mean frame and for eachof the M frames.The meanframe will be given by

1
M
x(t)=Lx(Pm+t) M m=l
,
t=O,l,...,Tl
(7)
Letting aCt) denote the mean frame estimate,with its Fourier transform being A(k), and lettingXm(k) denotethe Fourier transform for the mthframe, then Dm(k) = Xm(k)A(k) is a
measureof the difference between the Fourier transforms of the mth frame and the mean frame for eachfrequency.The signal coherence function canthen be estimatedby
yx(k)=
(8)
and 0 ~ Y:t(k)2 1. It can be shown (see Hinich, 2000) that the null hypothesis of zero coherence at frequency fi can be tested using the statistic M Yx5k)22' which is lYx(k)
asymptotically distributed under the null as a noncentral chisquaredwith two degreesof freedomand noncentrality parametergiven by ILk= Ma; , whereSl./k)is the spectrum of TSu(!k) {u(t)} at the frequency We also employ a joint test of the null hypothesisthat there is zero fic. coherenceacross the M frames for all K./2 frequenciesexamined. This test statistic will asymptoticallyfollow a noncentralChisquared distribution with K degrees freedom. of
2.2 Forecast Production One of the primary advantages the method that we propose is that a method for outofof sampleforecastingof seasonal time seriesarisesnaturally from it. This method is explained in detail in Li and Hinich (2002), who demonstrate that seasonal ARMA models can produce inaccuratelongterm forecastsof timeseriesthat are subject to random fluctuations in their periodicities. Thus we focus on those periodic componentsthat are the most stable over the
6
sample,whereasseasonal ARMA modelsfocus upon the most recent seasonal patterns,which arenot necessarilystableover time. Explaining the approachintuitively, supposethat the mean frame is computedfrom the nonoverlapping frames and is subtractedfrom each frame. The Fourier transform of the mean frame is computed along with the Fourier transforms of each residual frame. The signal coherence spectrumis computedfrom theseFourier transform amplitudes.The coherentpart of the mean frame (COPAM) is the inverseFourier transform of the Fourier transform of the mean frame where those amplitudeswhose coherence values are less than a threshold are set to zero. Thus the COPAM is a "clean" version of the mean frame purged of the noisy amplitudes. Only frequenciesthat are statistically significant at the 1% level or lower are retainedfor use in forecastproduction. Oncethe COPAM is computed,the amplitudesof the nonzeroedcomponentsof the Fourier transformsof the residual frames are forecastedusing a V AR with a lag selectedby the user. The dimension of the V AR is twice the number of nonzeroamplitudes
)
;<usedto compute!the COPAM. The one step aheadforecastfrom the V
V AR of the residual framesis addedto the COP AM to producea forecastof the next frame to be observedif the data segmentcan be extended. Further details of the approachcanbe found in Li and Hinich (2002).
The prediction framework that is employedin this paperis organisedas follows. The coherent part of the mean frame is constructedfrom the first 403 frames (days), amountingto 15,717 observations and then forecastsare producedfor one whole frame (one day) ahead.The outofsample forecasting period begins on 7 August 2000. That day's observationsare then addedto the insample estimationperiod and an updatedestimateof the coherentpart of the meanframe is calculated.A further day of forecastsis producedand so on until the sampleis exhausted. total of 101 frames (trading days) are forecast,and the root mean squarederror A (RMSE) and mean absolute error (MAE) are computed in the usual way. The forecast accuraciesare comparedwith naIve forecastsconstructedon the basis of the unconditional mean of the series over the insample estimation window. A more complete forecasting exerciseencompassinga wider range of potential models is left for future research.Since forecasts are produced for whole frames in advance (in our case, a day of 10minutely observations),the procedure would be of particular use to those requiring multistep ahead forecasts,and over such a long horizon, the majority of stationaryforecastingmodels would
7
have producepredictions that convergedon the longtenn mean of the series.Therefore,we conjecturethat the longtenn mean is likely to representa reasonablecomparatormodel in this case4.
4. Results 4.1 Testing for the Presenceof Periodicities in Returns, Spreads and Volumes Table 2 gives the pvalues for testsof the joint null hypothesisthat there is zero coherence at all 19 frequencies examined, together with the number of frequencies with significant coherence, each of the returns, spreadand volume series.The returns show somelimited for evidence of coherence at one or more frequencieswith most firms' returns having no significantly coherent periodicities at all. A nonrejection from the joint test does not in practice imply that there is actually no coherence any frequency,however, since the effect at of significanceat one or two frequenciescould be diluted by many insignificant frequencies. A case in point is the Firstenergy (FE) returns series, where there is one frequency with statisticallysignificant coherence, wherethejoint test is very far from a rejection. but
The results for the returns are in stark contrastto those for the bidask spreadsand volume series,all of which havepvalues for the joint test that are zero to four significant figures. It is wholly consistentwith both existing empirical evidenceand theoretical intuition that these quantities would show a greater degree of seasonality than the returns. There is little consistentevidence of either increasingor decreasing numbers of coherentperiodicities as firm size increases any of the returns,spreads volume. for or
However,the number of significant periodicities gives no real guide as to how strong eachof the individual seasonal componentsare, and which of them dominatein the joint test. Hence Table 3 presents the periodicities and the coefficients of autocoherencefor which the individual autocoherenceestimates are statistically significant. Since there are so many significant periodicities, we employ the considerablystricter statistical significance criterion of 0.01% (i.e. a pvalue of 0.0001 or less) for inclusion in this table. This has the effect of highlighting only the very strongest periodic signals, and requires an autocoherence
4 Brooks (1997) also observedthat the longterm meanof financial serieswas usually the best predictor among severalmodels testedacrossa range of forecasthorizons.
8
coefficient (which, like a correlation coefficient, scaled lie within is to 0.134 beforeit wouldbe included thetable. in
+ 1) of at least
:<~
idenc~ft . Several features of Table 3 are worthy of comment. First, there is again little e)'l
, "
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periodicity in the returns  only the Birmingham Steel Corp (BIR)
~~." (EOGy'~
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Resources
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have significant autocoherence at~
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.
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l".~ b '('etelY ~its Q3~)" f. f! .s, pt2 I til, (: I or r " .~,~riminu 1,
, ", ..., "
.. ,
anal .5{eft
minatory'
')BIR and EOGJxlhese perio icltlescorrespond t6 6 and a half
hours (one trading day) and 3 and a quarter hours (half a trading day) respectively,which cF\)
correspond 1 cycleand2~~~:~ per da~.~;th~;~ngle to
series
periodicitywhereall 301/t"(
"~mes
show significant coherence simultaneously (except the
periodicity of 39, correspondingto a daily frequency),there are several common features across the firms. First, the daily and halfdaily periodicities dominate in terms of their coherenceacrossthe 2 years of daily windows for both the spreadsand the volume series. Second,examiningrelationshipbetweenthe extent of coherence firm size,there appearto and be slightly stronger coherent seasonalpatterns for the small cap stocks than the large cap stocks,although there is an overwhelming degreeof idiosyncratic firm behaviour. As for the returns, it seems to be the 39 and 19.5 period seasonalitiesthat are the most common, although the majority firms also have 13 unit periodicities in their bidask spreadsand volume, correspondingto 3 cycles per day. The coefficients of autocoherence (which are standardised fall on the 0, I interval) are in many casesvery high for both the spreadsand to the volume series typically of the order of 0.2 to 0.45 for the daily and halfdaily cycles. This demonstrates remarkable degreeof stability of these relatively low frequency signal a componentsso that there is surprisingly little variation in the waveform over the frames for the most coherentparts of the signal.
Tables2 and 3 show the frequenciesof the most stableperiodic signalsfor eachof the series, but they do not show the amplitudesof thesestablesignals.An, idea of the spectralamplitude can be gleanedby plotting the coherentpart of the mean frame for eachof the series,giving the averagesizes of the periodic movementsin terms of the heights of the peaksand troughs of the coherentperiodicities. Whilst autocoherence quantifies how stable theseperiodicities are, the amplitude measuresthe size of the cyclical fluctuations. Figures 1 to 6 plot the coherentpart of the mean frame for framesof length one week for a sampleof 2 firms from eachsize quintile, with returns and the bidask spread being plotted on the lefthand scaleand
9
the natural logarithm of volume on the righthand scales. Note that the mean frame has been purged of all frequencieswith higher al1lountsof random variation, and the numbershave been standardised have zero mean acrossthe week. One might expect the graphsto look to very different from one another since different frequencieshave been retained for different stocks, and even when the same frequencies are included, differences in their relative amplitudeswould alter the shapeof the plot. ill all cases, however,the cyclical patternsquite similar, acrossfirms and both for the spreadand for the log of volume. In Figure 1, which showsthe coherentpart of the meanframe for Shandong HuanengPower Development(SH), the bidask spreadis slightly higher in the first 10 minutes of the trading day and then is largely flat through the rest of the day. Volume is also highest from 9:309:40am,and above its daily average until 11:00, before falling rapidly and then rising again to reacha peak at the end of the trading day. No interesting and stablepatternsare present in the returns over the day for SH, although this contrastswith the returnsline in Figure 2 for OsmonicsInc (OSM). In this latter case, a simple cycle with small amplitude has been identified, with returns peaking at around lOam and 1:40pm. A very similar daily returns pattern is observedin Figure 3 (Toll Brothers) and Figure 6 (Firstenergy).In this latter case,the inverted hockey stick patternin the spreadand the ushapein volume becomemore apparent.
Only one coherent frequency was significant for Western Gas Resources(WGR) returns, plotted in Figure 4, and this leads to the single trough in returns midway through the day with symmetrical highest levels at the open and the close. No less than seven coherent frequencieswere retained in the caseof International Paper (IP), however, which leads the plot of the mean frame.over the day to be very jagged as a number of cycles overlay one another. Finally, we can observethat for all six series,the volume cycles are much more volatile through the day than those of the spread or returns, in part reflecting the larger numberof coherentfrequenciesof the former. i 4.2 Forecast Production using Periodicities Tables4 to 6 give the root meansquared error and meanabsoluteerror for the forecastsof the returns, spreadsand log volume respectively for the signal coherenceapproachdescribed .
5
Only a small sampleof firms is examinedand the three quantitiesfor eachfirm are plotted in the samefigure in
the interestsof maintaining a manageable number of plots; the intraday patterns for other firms are qualitatively identical to those shown.
10
above and for forecasts produced using the longterm mean of the series. The results describedabove for the insample coherence statisticssuggested that there is relatively little periodicity in the returns themselvesto be used for forecasting, and therefore one would expect only minor improvementson the naive model in such cases.This is exactly what we find
 indeed, for many of the seriessuch as Coles Myer (CM) and Timberland (TBL), no
significant frequenciesat all were observedand therefore,none would remain after the noisy amplitudes are purged. In these instances,the forecasts (and therefore the forecast error measures) will be exactly identical to thoseof the unconditionalmean.The signal coherencebased approach is still able to lead to modest improvements in forecast accuracy over a simple average rule for 4 of the series.
The picture is rather different for the bidask spreads and in particular for the volume series. In the caseof the spreads,small reductionsin both the RMSE and MAE occur for 8 of the series,including Coles Myer and StatenIsland Bancorp(SIB). The method is able to improve upon the naIve approach in 28 of the 30 instances for the volume series, and.Jhe~
~ /'\



" 

improvementsare typically quite large  for example,the RMSE and MA!UrU.he j;:1!~!i~L_""~.
~'~_."=:: ~  ~ ,=== ~ 
EOG Resources ar~'200 and 1.43 7o;fue "si~l co~ce
approach,while they are 2.23 and
1.73 for the simple mean forecasts.Theserepresentreductionsof the order of 11% and 17% respectively. 5. Conclusions This paper has proposed and employed a new method for evaluating and quantifying the autocoherence financial time series,which was then testedon a set of tenminutely returns, of bidask spreads,and volume for a sampleof 30 NYSE stocks. Significant coherencefor at least one frequency acrossframes was revealedfor firms for the spreadand volume series, althoughthere is far less seasonality the returns.Overall we find the signal coherence be in to maximal at the daily frequency,with spreads mostly following an inverse Jshapethrough the day and volume being high at the open and at the close and lowest in the middle of the day. Theseresults for the spreadsare consistentwith the arguments forward in the theoretical put literature (Brock and Kleidon, 1992, for example)that the market power of specialistsnear the open and close combined with inelastic demand for sharesat these times. The similar patternsobservedover the day for trading volume are also consistentwith theoriesof strategic behaviour of liquidity traders and informed traders, such as that of Admati and Pfleiderer
11
(1988), as well as featuresof the market suchas settlementtiming that is affectedby the date of tradesbut not their timing within the day. Suchmodels suggestno role for seasonalities in returns, which is to a large extent what we find, since the theories imply that prices should follow a martingale. We find no differencesin the presenceor strengthof seasonal patterns accordingto market capitalisation.An investigationusing longer frame lengths of one week6 suggested intradaily effects completelyswampany lower frequencyseasonalities that suchas day of the week effects. Such a statement could not havebeenmadecategoricallyon the basis of existing tools for time seriesanalysis.
Finally, the approachto measuringthe extent of periodicities in data proposedhere can also be employedas a method for forecastingthe series.A comparisonof the forecastsfrom this model was made with those from a simple longterm mean rule. In the case of the spread series, reasonable improvements in forecast accuracy were made in some cases, while considerableimprovements were possible for the volume data. This improvement did not, however,also apply to the returnsor spreadseries.We conjecturethat the approachemployed in this paper could be a useful tool for researchers detect and to quantify the various to periodic components other time seriesdata. in
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Jain, P. and Joh, G. (1988) The Dependence betweenHourly Prices and Trading Volume Journal of Financial and Quantitative Analysis 23, 269284. Keirn, D.B. and RF. Stambaugh(1984) A Further Investigation of the Weekend Effect in Stock Returns,Journal a/Finance 39, 819840. Li, TH. and Hinich, M.J. (2002) A Filter Bank Approach for Modelling and Forecasting Seasonal PatternsTechnometrics 44(1), 114. McInish, T. and Wood, R. (1992) An Analysis of Intraday Patternsin Bid/Ask Spreadsfor NYSE StocksJournal of Finance 47, 753764. Rogalski, RJ. (1984) New Findings Regarding Day of the Week Returns over Trading and Nontrading Periods:A Note Journal of Finance Vol. 39, pp. 16031614.
13
Smirlock, M, and Starks, L. (1986) DayoftheWeekEffects and Intraday Effects in Stock Returns Journal of Financial EconomicsVol. 17, pp. 197210. Smith, K.L. (1999) Major World Equity Market Interdependence Decade After the October 1987 a Crash: Evidence from CrossSpectralAnalysis Journal of BusinessFinance and Accounting 26, 365392. Upson, R.B. (1972) Random Walk and Forward ExchangeRates: A Spectral Analysis Journal of Financial and Quantitative Analysis 7(4), 18971905. Wood, RA., McUnish, T.H. and Ord, J.K. (1985) An Investigation of TransactionsData for NYSE StocksJournal of Finance 40(3), 723739.
Yadav,P.K. and Pope,P.F. (1992) Intraweekand IntradaySeasonalities Stock Market Risk on Premia: Cash Futures and Journalof Banking Finance16,233270. and
14
Table 1: List of Stocks Employed and their Market Capitalisations
CompanyName
Mnemonic Market Capitalisation.
Panel A: Small Stocks urn SRI SinopeeShanghaiPetroleum 19 GPM Getty PetroleumMarketing CM Coles Myer BIR Brimingham Steel Corp OsmonicsInc U::SM OsmoniesIne Dover:Downs Entertainment DVD Dover Downs Entertainment Dan River Inc DRF Dan River Ine ShandongHuanengPower Development SH Starrett L S SCX Doncasters DCS
Panel B: MidCap Stock

.. 
..  .
h (frOD ).
..
34269 54985 61832 76829 108560 145145 145933 146906 148299 159599
Imation Western Resources Gas Oakley Staten IslandBancorp PhilippineLongDistance Tele Toll Brothers Cooper Tire andRubber Orthodontic Centres America of HellerFinancial Timberland
.
IMN WGR 00 SIB PHI TOL CTB OCA HF TBL
628772 637058 804035 837379 971035 1158727 1160123 1225163 1259811 1279885
PanelC: LargeStocks
EOG UPC FE EPG FPL IP NCC WAG MO XOM 4531390 5501656 7455382 10471071 11919726 16707546 20735387 35715995 114045117 239997400
EOG Resources Union Planters Firstenergy El PasoEnergy FPL Group International Paper National City Walgreen Philp Morris Exxon Mobil
Note: Market capitalisation is measured US dollars as at 24 December2000. in
15
Table2: Pvaluesfor Joint Testof Null Hypothesis that there is no signalcoherence all 19 for Frequencies Number of Frequencies and with SignificantCoherence the 1% Level at CompanyMnemonic Returns BidAsk Spread Volume vvalue No. Sig.Freas. vvalue No. Sig.Freas. vvalue No. gig. Freas. PanelA: SmallStocks sm 0.1569 0 0.0000 15 0.0000 17 GPM 0.0033 0 0.0000 6 0.0000 19 CM 0.4599 0 0.0000 19 0.0000 4 BIR 0.0000 5 0.0000 16 0.0000 14 OSM 0.0892 1 0.0000 7 0.0000 18 DVD 0.0000 4 0.0000 9 0.0000 17 DRF 0.0000 3 0.0000 19 0.0000 17 SH 0.0000 5 0.0000 18 0.0000 19 sex 0.0452 0 0.0005 3 0.0000 9 DCS 0.4877 0 0.0000 8 0.0000 9 IMN WGR 00
Panel B: MidCap Stocks 0 0.0000 1 0.0000 0 0.0000 10 0.0000 8 0.0000 2 0.0000 0 0.0000 5 0.0000 1 0.0000 0 0.0000
sm
PIll TOL CTB OCA HF TBL
0.2960 0.0000 0.5156 0.0000 0.0000 0.0000 0.0009 0.0000 0.0000 0.0327
7 9 11 18 19 19 14 15 14 7
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
8 13 12 11 11 6 8 5 7 8
16
Table 3: Periodicities with Coherence Statistics that are Significant at the 0.01% Level CompanyMnemonic Returns BidAsk Spread Volume Period Autocoherence Period Autocoherence Period Autocoherence PanelA: SmallStocks ~nl J:Jf V.l:Jfl J:Jf V.JLO 8HI 39 0.191 39 Ou'i2R

19.5 19.5 9.75 9.75 7.8 7.8 6.5 6.5 4.875 4.875 4.333 4.333 2.785 2.785 2.437 2.437
0.168 0.140 0.144 0.137 0.168 0.145 0.138 0.141
GPM


39
13
0.181
0.174
19.5 13 9.75 7.8 6.5 5.571 4.875 4.333 3.9 3.545 3.25 3 2.6 2.29
39
0.351 0.329 0.283 0.239 0.244 0.207 0.230 0.195 0.169 0.184 0.146 0.217 0.140 0.149
0.293
CM


39
19.5 13 9.75 7.8 6.5 5.57 4.875 4.333 3.9 3.545 3.25 3 2.786 2.6 2.438 2.294 2.167 2.053 39 19.5 13 6.5 5.57 3.9
0.352
0.354 0.365 0.343 0.290 0.282 0.293 0.285 0.339 0.308 0.277 0.262 0.273 0.305 0.283 0.291 0.276 0.307 0.282 0.271 0.229 0.185 0.148 0.140 0.157
19.5 13 9.75 7.8 6.5 5.57 4.875 3.9 3.545 3.25 3 2.438 2.053
39
0.173 0.208 0.167 0.188 0.190 0.173 0.141 0.140 0.180 0.153 0.177 0.194 0.145 0.141
0.211
19.5
BIR
39 19.5 4.875 4.333
0.180 0.167 0.157 0.150
39 19.5 13 5.571 4.875 4.333
0.523 0.322 0.183 0.188 0.253 0.184
17
18
13 9.75 7.8 6.5 5.571
0.199 0.158 0.150 0.159 0.142
SIB
13 9.75
0.210 0.157
PHI
6.5
0.148
39 19.5 13 9.75 7.8 4.875 4.333 3.9 3.25 2.785 2.294 2.167 39 19.5
0.221 0.181 0.168 0.215 0.163 0.191 0.159 0.165 0.138 0.137 0.146 0.151 0.250 0.260
3.545 39 19.5 13 9.75 5.571
0.151 0.476 0.224 0.209 0.162 0.153
39 19.5
0.505 0.273
19
HF
19.5
0.150
4.875 39
0.136 0.155
39
0.454
20
WAG


39 6.5
0.292 0.123
39 19.5
3.9
0.145
0.381 0.285
21
2.167 Note:Weemploya'considerably stricterstatistical significance criterionfor inclusionin this tablecompared the with previous in orderto keepit at a manageable one size.
0.162
22
Table 4: Forecasts of Returns using Signal CoherenceApproach and Simple Average SienalCoherence Approach SimpleAveraeeApproach RMSE MAE RMSE MAE
Panel A: Small Stocks
SRI GPM CM BIR OSM DVD DRF SH scx DCS
0.480 0.974 0.327 2.204 v' 0.307 0.481 0.998 0.140 0.344 0.510 .
0.141 0.245 0.050 0.940 0.088 0.217 0.338 0.069 0.089 0.159
0.477 0.972 0.327 2.212 0.307 0.481 0.998 0.125 0.343 0.510
0.127 0.221 0.050 0.890 0.071 0.195 0.305 0.029 0.078 0.159
IMN WGR
0.540 0.421
Panel B: MidCap Stocks 0.248 0.540 0.237 0.420
/
0.248 0.224
00
sm
0.603
0.272,/
0.349
0.142 0.157 0.266 0.342 0.381 0.232 0.335
~
0.603
0.273 0.410 0.457 0.546 0.659 0.404 0.566
0.349
0.127 0.143 0.262 0.331 0.371 0.219 0.335
PHI TOL
CTB OCA HF TBL
0.408J 0.456/
0.547 , 0.660 0.405 0.566
EOG UPC FE EPG FPL IP NCC WAG MO XOM
0.454I 0.309 0.329 . 0.372 0.316' 0.477 . 0.384 1 0.387 . 0.417 ' 0.256
Panel0.285 C: Larfle Stocks 0.456
0.278
0.195 0.217 0.248 0.204 0.313 0.251 0.253 0.261 0.172
0.195 0.219 0.248 0.206 0.318 0.256 0.253 0.269 0.172
0.309 0.329 0.372 0.316 0.477 0.384 0.387 0.414 0.256
23
~
Table 5: Forecastsof BidAsk Spreads using Signal CoherenceApproach and Simple Average SbmalCoherence Approach SimpleAvera!!eApproach
DM~1i' M A Ii' DM~1i' M A Ii'
IMN WGR 00
sm
PHI TOL CTB OCA HF TBL
0.052 0.076 0.060 0.062 0.054 0.104 0.049 0.091 0.078 0.091
Panel C: Large Stocks
0.048 0.074 0.062 0.063 0.054 0.105 0.044 0.092 0.072 0.104
>
0.037 0.063 0.049 0.051 0.042 0.083 0.039. 0.072 0.063 0.086...,
0.054 0.045 0.039 0.045 0.047 0.040 0.037 0.041 0.031 0.038
EOG UPC FE EPG FPL IP NCC WAG MO XOM
0.075 0.053 0.044 0.063 0.061 0.056 0.057 0.051 0.039 0.051
0.055 0.045 0.039 0.049 0.047 0.045 0.048 0.041 0.035 0.038
0.076 0.053 0.044 0.062 0.062 0.050 0.039 0.050 0.035 0.049
24
able 6: Forecastsof Volume using Signal CoherenceApproach and Simple Average Si2llal Coherence MAE Apjn'oach Simple RMSE RMSE A veral!eApproach' MAE
'
SHI GPM CM
4.179"; 4.783 V" 2.314
Panel A: Small Stocks 3.665 4.288 3.04V' 4.839 1.346 2.297t
v:
~
3.904 'f 3.115
BIR OSM
DVD DRF
5.989V 4.112 ../
~'
5.737/
2.933'
6.186'1.
'/
"
1.202 6.097
3.043' .
SH SCX DCS
IMN
5.390 4.829
5.321\/ 3.798 5.308
5.210 ~ 4.066 ./
~I
4.18rt~ .
4.782,,/ 2.883. 4.0151
3.402 t/
5.488 4.923
5.464 3.771 5.376
5.392 4.214
'
5.003 2.789,. 3.931
'
4.888 t,.
Panel B: MidCap Stocks
4.964
3.645
"
WGR 00
5.594\1" 5.133
,
5.340v/ 4.697./ '
5.748 5.281
5594 4.943
sm PHI TaL
CTB
5.729 <l 6.015 4.611
3.4421/
5.306 5.7 12 4.090 v
,
~

5.764
6.176 4.847
3.523
5.470
6.018 4.426
2.174
2.018
OCA
3.619
2.536
J
:;;
3.786
2.842 >
'
HF TBL EOG OPC FE EPG FPL IP
NCC
4.880 3.963 2.004 2.168 1.9691, 1.490 1.589 1.347
1.477
4.019 .J 3.359/
5.041 4.160
4.367 v 3.646 1.734
1.254 1.611 .
PanelC: LargeStocks. 1.431V 2.231 t 1.136v 2.221 1.102 V 2.047 . 1.003;;/ 1.969 0.947./ 1.690 0.741.1 1.415
0.885
1.231
1.097 0.839.
I
1.545
0.975
WAG MO XOM
1.338.1" 1.299 1.225
/
0.733 ~(1
0.6581
1.373,1'
1.~~?.
0.796,
0.73~..
,
0.5461
1.275
0.620),.
25
~
Figure 1 : Coherent Part of the Mean Frame for a Day  Shangdong Huaneng Power Development
0.2 12
 Returns
0.15
+ Spread
LogVolume
10
Q1
'C ell e aU)
8
CII E ::I "S > C)
0 ...J
Ii
c ... ~
0.05 0
&
0
4
0.05 0
2
41
0
Figure 2: Coherent Part of the MeanFramefor a Day Osmonies Ine

'C ell Cb ... aU) Ii
c 5
&
26
Figure 3: Coherent Part of the Mean Frame for a Day  Toll Brothers
0.25
14
0.2
12
0.15
't:I III I!! Q. (/) Iii c ... :::I Gi 0::
~
10
8 0.1 6 0.05 4 0
CII
0 ...J
E =' '0 > CI
1
0.05
3
5
7
15
21
23
25
2
0
Figure 4: Coherent Part of the Mean Frame for a Day  WesternGasResources
't:I
1"11
e Q. I/) iii c .. ::I
&
0 ...J
CD E ::J '0 > CI
27
/;;...
Figure 5: Coherent Part of the Mean Frame for a Day  International Paper
"tI '" 2! Q. C/) iii c ... :I
~
Figure 6: Coherent Part of the Mean Frame for a Day Firstenergy

0.16 0.14 Returns
14.5
 Spread
0.12"
0.1 h
'0 C\\ ~ Co (/) Vi s:: .. ::: G) II::
... LogVolume
14
0.08 0.06 0.04
""""" "'" 13
G)
E ::I
~
CI 0 ..J
12.5 0.02
0 1 0.02
0.04 11.5
3
19
21
23
12
28
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