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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Roman Huptas 
Cracow University of Economics 

Intraday Seasonality in Analysis of UHF Financial Data: 
Models and Their Empirical Verification  

A b s t r a c t. The aim of this paper is to outline the typical characteristics of the ultra-high-
frequency financial data and to present estimation methods of intraday seasonality of trading 
activity. Ultra-high-frequency financial data (transactions data or tick-by-tick data) is defined to 
be a full record of transactions and their associated characteristics. We consider two nonparame-
tric estimation methods: cubic splines and a Nadaraya-Watson kernel estimator of regression. 
Both approaches are compared empirically and applied to financial data of stocks traded at the 
Warsaw Stock Exchange.  

K e y w o r d s: financial UHF data, intraday seasonality, diurnal pattern, cubic splines, kernel 
estimation.  

1. Introduction 
 The last dozen or so years has witnessed mounting global interest in analy-
ses of the microstructure of financial markets. Research into the microstructure 
of financial markets centres around explaining the process behind the shaping of 
the price of financial instruments and analyses of individual trade events. The 
impact of various transaction factors and mechanisms on the way in which in-
struments prices are shaped was captured by the so-called theoretical micro-
structure models. A review of those models and numerous issues collectively 
referred to as market microstructure effects was incorporated into O’Hara 
(1995) and Dacorogna et al. (2001) (cf. Doman, Doman, 2004; Bień, 2006).  
 Analysis of financial market processes and empirical verification of hy-
potheses arising from theoretical microstructure models were made possible by 
the newly-gained access to transactional databases. These databases became the 
source of specific financial time series referred to as ultra-high-frequency or 
tick-by-tick data. 
 New modelling tools for financial time series anticipate specific qualities of 
transaction data. They include, above all, asynchronous distributions of obser-



Roman Huptas 130 

vations relative to time units and discrete price changes. Additionally, the indi-
vidual events of the transaction process manifest themselves with varying fre-
quency from one time period to another. Consequently, there may be – from one 
day to the next – certain repeat pattern of intensity with which transactions are 
concluded. In pertinent literature, this pattern is referred to as intraday seasonal-
ity of durations, with durations or waiting times standing for the time spans 
between trade events. However, before one can use econometric models in 
analyses of ultra-high-frequency time series, it is essential to eliminate intraday 
seasonality, which strongly manifests itself in the time series. In estimating 
intraday seasonality use is mostly made of selected nonparametric statistical 
methods. 
 The aim of this paper is to outline the typical characteristics of UHF finan-
cial data and further to present modelling and estimation methods of intraday 
seasonality of transaction activities. Within the framework of these methods, 
two nonparametric approaches will be presented: cubic splines interpolation and 
kernel estimations of regression functions. Both approaches will be verified and 
compared empirically on the basis of data extracted from the Polish share mar-
ket. 

2. Characteristics of UHF Financial Data 
 Several years ago a new term became operative – „ultra-high frequency 
data”, also known as „tick-by-tick data” or „transaction data”. These are time 
series composed of trade event features to which the exact time of their appear-
ance was assigned. Thus observations are recorded asynchronously on time 
units. UHF financial data have a few characteristic qualities, which do not 
manifest themselves at lower frequencies. 
 The characteristic features of time series of transaction data include: non-
synchronous distribution of observations over time units, discrete transaction 
price changes, appearance of a number of transactions in the same single sec-
ond, bid-ask bounce of transaction prices and intraday seasonality i.e. transac-
tions reveal a daily periodic pattern. 
 The most important quality of UHF financial data is the nonsynchronous i.e. 
erratic distribution of observations over time units. Those data can, for instance, 
be aggregated so that they correspond to equal sequential time units (multiples 
of minutes, hours, days) and then – in analyses – enable use of a whole range of 
the GARCH models. On the other hand, such aggregation of transaction data 
and their analysis as observations made and selected at equal intervals leads to a 
loss of information furnished by the transaction process itself. Transactions or 
changes in the share price do not happen at equal intervals. Consequently, the 
durations between transactions involving shares of a given company may pro-
vide relevant information as to the intensity of their trading. Thus the assump-
tion that changes in prices or transactions are equidistant in terms of time may 



 Intraday Seasonality in Analysis of UHF Financial Data: Models … 131

cause us to draw false conclusions. The problem of nonsynchronous trading and 
relevant examples are dealt with more extensively in Tsay (2002, p. 207)  
(cf. Doman, Doman, 2004; Osińska, 2006). 
 An alternative way of analysis of financial data distributed asynchronously 
over time units involves using the so-called transaction-time models  (ACD, 
UHF-GARCH models etc). With those models, raw data are used for analysis. 
Owing to that, information inherent in the duration of the time between selected 
trade events can also come into focus. Duration analyses may furnish informa-
tion on the microstructure of the financial market, affording a more accurate 
insight into various market interdependencies. In pertinent literature, the most 
frequently modelled durations between trade events are trade durations, price 
durations, volume durations (cf. Engle, Russell, 1997, p. 1149). 

3. Intraday Seasonality of Durations 
 Under normal economic conditions, transactions reveal a daily seasonality 
factor. It appears that the number of transactions is higher immediately after the 
opening of business than prior to the close of the session (when the time gaps 
between transactions are the shortest), and markedly smaller during midday 
hours, i.e. in the middle of the session (so- called “lunchtime effect” when dura-
tions between transactions are also the longest). Thus, there exists a certain 
repetitious pattern of transaction intensity for each day. This is termed „intraday 
seasonality of durations”. Consequently, Engle and Russell (1997) recommend 
decomposition of duration into a deterministic component )( itφ  depending on 
moment it  of the commencement of a given duration, and a stochastic compo-
nent ix̂ , which is free from the seasonality effect and which models process 
dynamics. Pertinent literature (cf. Engle, Russell, 1997) recommends that data 
be transformed as follows: 

,
)( i

i
i t

xx
φ

=)  (1) 

where: 1−−= iii ttx - duration between transactions at time it  and 1−it , ix̂ - dura-
tion purged of the seasonality effect, )( itφ - multiplied factor of intraday 
seasonality at time it . 

 The seasonal factor )( itφ  is construed as the average duration of each time 
unit during which we made data observations (most often denoting the average 
duration for each second). The diagram which illustrates intraday seasonality 
pattern, also known as diurnal pattern or time-of-day function mostly has the 
shape of the letter U turned upside down.  
 In numerous situations researchers lack adequate information to fully spec-
ify a parametric function of intraday seasonality. Despite the fact that intraday 



Roman Huptas 132 

cyclicality is not the key issue of investigation, it still cannot be ignored, but 
much rather needs to be included in analyses. Thus, in order to estimate the 
time-of-day function use can be made of selected nonparametric statistical 
methods such as splines, Fourier series, neural networks, wavelet analyses or 
kernel methods. In most works on duration modelling use is made of cubic 
splines or kernel estimations. 
 In pertinent literature the time-of the day function is determined most com-
monly by means of splines. This is a method which allows smoothing of aver-
age durations between events in subsequent time periods. Firstly, all durations 
during all the subsequent hours of the sessions on each day are averaged. Then a 
cubic spline with knots on every full hour of the session is determined. Knots 
correspond to previously determined average durations. This version of ap-
proximation of the daily period factor was presented in the paper (Engle, Rus-
sell, 1997). With a view to ensuring enhanced elasticity, the authors added 
a knot on the half-hour of the last hour of the session to capture the fast growing 
trade activity prior to the close of the stock market. A slightly different ap-
proach to cubic spline approximation can be observed in paper (Bauwens, Giot, 
2000, p. 135). The authors note that the intraday seasonality factor may vary 
from one day of the week to the other, i.e. the shape of the periodic factor for 
Monday can be distinct from that for Tuesday etc. Consequently, the estimation 
of the intraday seasonality function was conducted separately for each day of 
the week to allow for possible seasonality within the week. In the first step, 
durations for the subsequent half-hours of the session were averaged separately 
for each day of the week and then the parameters of cubic splines were esti-
mated for knots at full hours and half hours. 
 An alternative and second most commonly practised method of estimation 
of intraday seasonality function is the kernel estimation method. The intraday 
seasonality pattern is estimated as the Nadaraya-Watson kernel estimator of 
regression of raw durations on the time of the day (cf. Bauwens, Veredas, 2004, 
p. 398): 

,)(

1

1

∑

∑

=

=

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛ −

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛ −

=
n

i n

i

n

i n

i
i

h
ttK

h
ttKx

tφ

 
where: t - number of seconds since the midnight of each day (or since the start 

of a session), ix - durations corresponding to moments it  ( ix is a de-
pendent variable), it - number of seconds since the midnight of each day 
(or since the start of a session) until the moment of a given transaction, 
K - kernel function, nh - bandwidth, s - standard deviation of  sample 

it , n - number of observations. 



 Intraday Seasonality in Analysis of UHF Financial Data: Models … 133

 As far as the kernel function is concerned, the paper (Bauwens, Veredas, 
2004, p. 398) makes use of the quartic kernel (with optimal bandwidth of 

5/178,2 −sn ) which has the following shape: 

,
1||,0

1||,)1(
16
15

)(
22

⎪⎩

⎪
⎨
⎧

>

≤−=
xdla

xdlaxxK

 
and in paper (Bauwens, Giot, 2002, p. 13) use is made of the gamma kernel 
function.  
 In the case of paper (Bauwens, Veredas, 2004, p. 398) the estimation of the 
intraday seasonality function is made separately for each day of the week to 
incorporate possible seasonality arising from the transaction repetition patterns 
also over a week-long period. 

4. Empirical Example 
 The empirical verification of the methods presented above was carried out 
on the basis of time series involving trades in the shares of three companies 
listed in the WIG20 index: Telekomunikacja Polska S.A. (TPSA)/Polish Tele-
com/, Agora S.A. (Agora) and CEZ S.A. (CEZ) over a period between 22 
March 2009 and 25 June 2009. The analysis covers transactions closed during 
the continuous quotation phase. On the basis of such time series, durations be-
tween each transaction were determined. Additionally, the time lags between 
the close of the session and the opening of next day’s trading were removed. 

Tabel 1. Descriptive statistics of transaction durations 

 CEZ Agora TPSA 
Number of observations 13919 19183 65166 

 Mean 98.930 84.840 
 

25.110 
 Standard deviation (SD) 224.720 176.560 

 
43.640 

 Dispersion index ( =Mean/SD) 2.270 2.080 1.740 
Minimum 1 1 1 
Maximum 4196 4003 833 

ACF(1) 0.220 0.225 
 

0.212 
 ACF(2) 0.157 

 
0.176 

 
0.168 

 Q(5) 1805.430 
 

2799.010 
 

8948.820 
 Q(10) 2569.120 

 
4111.970 

 
13294.120 

 Q(15) 3056.570 
 

5179.310 
 

16546.610 
 Q(20) 3298.350 

 
5866.790 

 
19483.450 

 
Note: ACF(k) – the value of the k-th order autocorrelation coefficient, Q(k) – the value of the 
Ljung-Box Q-statistic of k-th order, descriptive statistics in seconds. 



Roman Huptas 134 

 The basic descriptive statistics of transaction durations for the shares in 
question are illustrated in Table 1. In our example, we witness three companies 
experiencing different trading activity patterns. The majority of the transactions 
involved TPSA, for which the average duration between transactions is 25 sec-
onds. CEZ reported the fewest transactions and the average transaction time is 
99 seconds. Agora, in turn, is a company of average liquidity and the average 
duration is around 85 seconds. In an analysis of the features of the distribution 
of durations our attention is momentarily attracted to marked overdispersion, 
i.e. the standard deviation exceeds the mean. The dispersion indexes (the ratio 
standard deviation to mean) are generally very high, which may imply great 
dynamics of the series in question. It is worth noting that the greater the fre-
quency of trades, the lower the value of the dispersion index. 
 The values of the Q Ljung-Box statistics in Table 1 formally test the null 
hypothesis whereby there is no autocorrelation of durations respectively for the 
fifth, the tenth, the fifteenth and the twentieth order. Clearly, on the basis of the 
determined test statistics, the null hypothesis whereby there is no autocorrela-
tion is definitely rejected for all three companies. Duration autocorrelation is 
thus extremely strong. 

 
Figure 1. Autocorrelation functions of transaction durations for CEZ, Agora and TPSA 

 Graphs representing the autocorrelation function of the durations for the 
companies in question are to be found in Figure 1. Regardless of the company, 
the first values of the autocorrelation function are surprisingly low and stand at 
around 0.22. For CEZ shares the ACF function fairly soon shrinks to zero for 
the first several dozen delays, only to level off. As far as TPSA shares are con-
cerned, the autocorrelation function takes much more time to decrease, ap-
proximately at hyperbolic speed (rate), which is typical of long memory proc-
esses. This evidences high „stability” (persistence) of the process. Moreover, 
the high values of lower order autocorrelations indicate stronger clustering of 
transaction activities. The dynamics of duration processes and the effect of clus-
tering of transaction activities may be noted in figure 2 containing graphs of the 
series of the first 5000 observations. 

ACF - CEZ

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0 100 200 300 400 500

Lags

ACF - Agora

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0 100 200 300 400 500

Lags

ACF - TPSA

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0 100 200 300 400 500

Lags



 Intraday Seasonality in Analysis of UHF Financial Data: Models … 135

     
Figure 2. Plots of trade durations for Agora and TPSA – first 5000 observations 

 The existence of very powerful autocorrelation of durations may result from 
the midday seasonality of transactional activity. With this in mind, patterns of 
intraday seasonality were estimated by means of four methods, and further veri-
fied empirically to determine whether the method selected translates into effec-
tive elimination of high autocorrelation of the series under research. The follow-
ing approaches, described extensively in point 3, were applied: 

1. Nadaraya-Watson estimator of regression of the duration on the time of the 
day, determined separately for each day of the week (NW_days); 

2. Nadaraya-Watson estimator of regression of the duration on the time of the 
day, ignoring possible seasonality over a week-long period (NW); 

3. cubic splines with knots on every full hour and every half-hour of the ses-
sion, determined separately for each day of the week (CS_days); 

4. cubic splines with knots on every full hour and every half-hour of the ses-
sion, ignoring possible seasonality over a week-long period (CS).  

In the event of kernel estimators, use was made of the quartic kernel with 
the optimal bandwidth of 5/178,2 −sn . Next, after the intraday seasonality factor 
was estimated, durations purged of the seasonality effect were determined pur-
suant to formula (1). All algorithms and procedures were implemented using the 
GAUSS software.  

 
Figure 3. Intraday seasonality patterns for CEZ, Agora and TPSA 

 Figure 3 depicts the shape of intraday seasonality patterns (NW and CS 
methods) for the three companies under analysis, without regard for the effect 
that the different days of the week have. Figure 4 shows plots of the estimated 
time-of-day functions for subsequent days of the week for CEZ, Agora and 

02000

0 2000 4000
S
e
c
o

Observations

Trade durations - Agora

01000

-1000 1000 3000 5000
S
e
c
o

Observations

Trade durations - TPSA

-20180
9S

e
c
o

Hours

Intraday seasonality -
CEZ

N
W

C
S

-20180
9S

e
c
o

Hours

Intraday seasonality -
Agora

N
W

C
S

0
9S

e
c
o

Hours

Intraday seasonality -
TPSA

N
W

C
S



Roman Huptas 136 

TPSA respectively, obtained by means of the kernel estimator (NW) and cubic 
splines (CS). 

  

  

  
Figure 4. Intraday seasonality patterns for the days of the week for CEZ, Agora, TPSA 

 The graphs presenting the estimated time-of-day functions have the shape of 
the letter U turned upside down and reveal unequivocally that durations are 
subject to daily seasonality. The durations between transactions are markedly 
shorter after the opening and before the close of the session than at midday. The 
extent of trading activity between 12:00 a.m. and 2:00 p.m. is noticeably less, 
due, amongst others, to the lunchtime effect. It should be noted that in the case 
of companies listed on the American and West European Stock markets, the 
effect is less manifest than in the case of their Australian counterpart, where the 
effect is more pronounced, i.e. the hump in the graph is manifestly spikier (cf. 
(Bauwens, Giot, 2001; Hautsch 2004)). Similarly, trading activity at the open-
ing of the session is more intense as traders begin to accommodate the informa-
tion of the night before (macroeconomic data, etc). Trading activity at the close 
of trade can be explained in terms of some investors’ attempt to close their open 
positions. It is worth noting that intraday seasonality will vary from one day of 
the week to the next (Figure 4). It seems that regardless of the type of company, 
trading activity is most intense on Tuesdays and Wednesdays than on any of the 
other days. 
 Duration statistics purged of intraday seasonality by means of the four 
above-named methods were included in Table 2. Deseasonalisation of the data 

20220
9S

e
c
o

Hours

Intraday seasonality (NW) 
The days of the week - Agora

Mo
n

Tue

20220

9
S
e
c
o

Hours

Intraday seasonality (CS)
The days of the week - Agora

Mo
n

Tue

-200
300

9
S
e
c
o

Hours

Intraday seasonality (NW)
The days of the week - CEZ

Mo
n

Tue

0500
9S

e
c
o

Hours

Intraday seasonality (CS)
The days of the week - CEZ

Mo
n

Tue

-40
60

9
S
e
c
o

Hours

Intraday seasonality (NW)
The days of the week - TPSA

Mo
n

Tue

0100

9S
e
c

Hours

Intraday seasonality (CS)
The days of the week - TPSA

Mo
n

Tue



 Intraday Seasonality in Analysis of UHF Financial Data: Models … 137

partly reduced the autocorrelation of transaction durations. From the point of 
view of effective elimination of seasonality impact on autocorrelation “meas-
ured” in terms of the values of the Ljung-Box test statistics, the NW_days (Ta-
ble 2) appears to be the most effective method. In the case of TPSA and Agora 
it was definitely the most successful, i.e. the values of test statistics are the low-
est of all four methods used. On the other hand, in the case of CEZ company, it 
ranked number two, with the CS_days approach ranked the highest. 

Tabel 2. Descriptive statistics of adjusted transaction durations after deseasonalisation 
by means of the four methods       

Stock Method Mean SD Disp. 
Index ACF(1) Q(5) Q(10) Q(15) Q(20) 

CEZ 

NW_days 0.990 2.320 2.340 0.205 1360.150 1897.080 2101.860 2240.300 
NW 0.990 2.200 2.220 0.215 1537.830 2111.380 2475.990 2676.680 

CS_days 0.980 2.280 2.320 0.190 1290.340 1782.540 1972.950 2121.000 
CS 0.970 2.130 2.190 0.209 1535.590 2137.890 2530.380 2759.580 

Agora 

NW_days 0.990 1.980 2.000 0.215 2518.120 3521.420 4333.470 4886.400 
NW 0.990 1.980 2.000 0.211 2675.970 3719.670 4600.870 5169.360 

CS_days 0.990 1.950 1.970 0.209 2599.490 3632.710 4520.890 5097.310 
CS 0.990 1.950 1.970 0.215 2553.890 3604.400 4480.420 5078.150 

TPSA 

NW_days 0.990 1.660 1.670 0.195 7357.560 10568.660 12843.060 14864.480 
NW 0.990 1.680 1.700 0.197 7614.770 10964.070 13401.270 15598.530 

CS_days 0.990 1.670 1.690 0.197 7527.590 10874.810 13294.540 15400.190 
CS 0.990 1.680 1.700 0.200 7793.400 11248.550 13802.000 16085.570 

Note: SD – standard deviation; ACF(k) – the value of the k-th order autocorrelation coefficient, 
Q(k) – the value of the Ljung-Box Q-statistic of k-th order, descriptive statistics in seconds. 

 An analysis of the values of Ljung-Box statistics implies that the NW_days 
and CS_days approaches should be used. Consequently, in eliminating intraday 
seasonality,  the “day of the week” effect i.e. possible seasonality arising from 
variations in trading activity over the entire week should be taken into account. 
Regardless of the deseasonalisation method used, the values of Ljung-Box test 
statistics for the companies in question dropped by approximately 15%-25%, 
but still continued to be very high. Thus, the null hypothesis implying lack of 
autocorrelation continues to be rejected on each reasonable level of signifi-
cance. This bears witness to the fact that the dynamics of transaction durations 
are influenced by factors other than the purely deterministic seasonality effect, 
which in turn is due to the structure of the share market. 

5. Summary 
 Based on the results of empirical data, the application of the kernel estima-
tor of regression separately for each day of the week appeared to be the most 
effective method of elimination of intraday seasonality impact on the autocorre-
lation of transaction durations.  It is noteworthy, though, that the results for all 
analytical methods used are highly similar. In the case of splines, their prelimi-



Roman Huptas 138 

nary averaging of durations between events for the subsequent full or half-hours 
may become something of a drawback. The extent of data aggregation and the 
elasticity of the estimated function can be reduced by increasing the number of 
knots used in the spline.  So it appears that the inclusion of intraday seasonality 
models in the base models is a natural step, and wins over earlier data filtrations 
and testing if the use of a two-step or one-step approaches will have identical 
impact on the quality of the estimators determined. 

References 
Bauwens, L., Giot, P. (2000), The Logarithmic ACD Model: An Application to the Bid-ask Quote 

Process of Three NYSE Socks, Annales d’Économie et de Statistique, 60, 117–149. 
Bauwens, L., Giot, P. (2001), Econometric Modelling of Stock Market Intraday Activity, Kluwer 

Academic Publishers, Boston. 
Bauwens, L., Giot, P. (2002), Asymmetric ACD Models: Introducing Price Information in ACD 

Models, CORE Discussion Paper 9844. 
Bauwens, L., Veredas, D. (2004), The Stochastic Conditional Duration Model: A Latent Variable 

Model for the Analysis of Financial Durations, Journal of Econometrics, 119, 381–412.  
Bień, K. (2006), Model ACD – podstawowa specyfikacja i przykład zastosowania (ACD Model – 

Basic Specification and Example of Application), Przegląd Statystyczny (Statistical Sur-
vey), t.53, z. 3, 83-97. 

Dacorogna, M. M., Gençay, R., Müller, U., Olsen, R. B., Pictet, O. V. (2001), An Introduction to 
High-Frequency Finance, Academic Press, San Diego.    

Doman, M., Doman, R. (2004), Ekonometryczne modelowanie dynamiki polskiego rynku finan-
sowego (Econometric Modelling of Dynamics of Polish Financial Market), Wydawnictwo 
AE w Poznaniu, Poznań. 

Engle, R. F., Russell, J. R. (1997), Autoregressive Conditional Duration: A New Model for Irre-
gularly Spaced Transaction Data, Econometrica, 66, 1127–1162. 

Hautsch N. (2004), Modelling Irregularly Spaced Financial Data, Springer-Verlag, Berlin, Hei-
delberg. 

O’Hara, M. (1995), Market Microstructure Theory, Blackwell Inc., Oxford. 
Osińska, M. (2006), Ekonometria finansowa (Financial Econometrics), PWE, Warszawa. 
Tsay, R.S. (2002), Analysis of Financial Time Series, Wiley Series in Probability and Statistics, 

John Wiley& Sons, New York. 

Wewnątrzdzienna sezonowość w analizie danych finansowych UHF: 
modele i ich empiryczna weryfikacja 

Z a r y s  t r e ś c i. Celem artykułu jest krótkie przedstawienie cech charakterystycznych dla 
danych finansowych UHF oraz prezentacja metod modelowania i szacowania wewnątrzdziennej 
sezonowości aktywności transakcyjnej. W ramach tych metod są przedstawione dwa podejścia 
nieparametryczne: interpolacja za pomocą kubicznych funkcji sklejanych oraz estymacja jądrowa 
funkcji regresji. Oba prezentowane podejścia są zweryfikowane i porównane empirycznie na 
podstawie danych z polskiego rynku akcji.  

S ł o w a  k l u c z o w e: dane finansowe UHF, wewnątrzdzienna sezonowość, funkcje sklejane, 
estymator Nadaraya-Watsona. 


