Q¨
UESTII´
O, ol. 26, 1-2, p. 197-211, 2002
SMOOTHING THE CATALAN TOURISM
MICRO-DATA TIME SERIES
M. ART´
IS ORTU ˜
NO
J. L. CARRION I SILVESTRE
`
A. COSTA S ´
AENZ DE SAN PEDRO
J. SURI˜
NACH CARALT
In his pape we p opose a me hod o smoo hing he Ca alan ou ism i-
me se ies be ween 1997 and 2000. These ime se ies, buil upon a mic o
da abase d awn om a su ey conduc ed by he S a is ical Ins i u e o Ca-
alonia, a e somewha ola ile due, i would seem, o he incomple e na u e
o he in o ma ion. The applica ion o a smoo hing p ocedu e based on he
combina ion o classical echniques and weigh ed mo ing a e ages allows
us o o e come he p oblems caused by his lack o in o ma ion and o ob-
ain ime se ies ha e ol e smoo hly o e ime.
Keywo ds: Smo hing mic o-da a, ou ism ime se ies
AMS Classi ica ion (MSC 2000): 62P20
This pape is a join unde aking be ween he S a is ical Ins i u e o Ca alonia (Idesca ) and he An`alisi
Quan i a i a Regional Resea ch G oup o he Uni e si y o Ba celona.
An`alisi Quan i a i a Regional (AQR) Resea ch G oup Depa amen d’Econome ia, Es ad´ıs ica i Economia
Espanyola Uni e si a de Ba celona. A . Diagonal, 690, 08034 Ba celona.
Ins i u d’Es ad´ıs ica de Ca alunya, Via Laie ana, 58, 08003 Ba celona.
– Recei ed Oc obe 2001.
– Accep ed Janua y 2002.
197
1. INTRODUCTION
S udies o he ou ism sec o in Ca alonia ha e adi ionally d awn on mac o agg ega es
co esponding o each ou is season, such as p i a e consump ion and g oss domes ic
p oduc . In so doing hey ha e ended o ely on one o he main da a sou ces o his
sec o i.e. he su ey o he supply o ho el accommoda ion in each o he Spanish
egions. This su ey, unde aken by he Spanish S a is ical Ins i u e (INE), p o ides
in o ma ion abou ho el occupancy, in o he wo ds, in o ma ion p o ided by he supply
side o he ou ism ma ke .
In 1997, he S a is ical Ins i u e o Ca alonia (Idesca ) in oduced a new su ey o Ca-
alan ou ism, bu in con as o he su ey desc ibed abo e his sough o ob ain in o -
ma ion om he demand side. This su ey p o ides analys s wi h aluable in o ma ion
abou isi o s o Ca alonia, whose poin o o igin is one o he o he Spanish egions.
Gi en i s ecen in oduc ion, hese s a is ics o e in o ma ion o he mos ecen ou-
ism seasons only and compa a i e da a is only p o ided wi h he p e ious ou ism
seasons and, consequen ly, no ime se ies is de ined. I should, howe e , be no ed ha
he ime in e al o each ou is season has changed since he in oduc ion o he su ey
which hinde s he de ini ion o an app op ia e ime se ies. Thus, h ee ou ism seasons
we e iden i ied o 1997 and 1998: Janua y o May, June o Augus and Sep embe o
Decembe ; while o 1999and 2000 ou seasons we eiden i ied: Janua y o Ap il, May
o June, July o Augus and Sep embe o Decembe .
The ecen in oduc ion o he su ey and he a ying ime in e als used in de ining he
ou is season hinde compa isons. Fu he mo e, al hough he su ey was designed o
emb ace all he Spanish egions, only a ew obse a ions a e e en uallyincluded wi hin
he da abase, and so he in o ma ion desc ibing indi idual cha ac e is ics ends o be
highly he e ogeneous. This he e ogenei y becomes e en mo e ma ked when he da a
a e aised o he en i e popula ion.
Consequen ly using his da abase o calcula e g ow h a es gi es highly ola ile ime
se ies. The e o e, he aim o his pape is o p esen a me hodology o compu ing ime
se ies om he mic o da a ( he su ey) bu , in con as wi h he o iginal (popula ion-
aised) ime se ies, wi h a smoo hed empo al pa e n.
I is no , howe e , ou aim o supply he analys wi h a speci ic se o smoo hed ime
se ies bu a he o design a me hodology ha allows p ac i ione s o ob ain smoo hed
ime se ies au oma ically wha e e he concep s c ossed in he da abase. The success ul
achie emen o his goal depends on he applica ion o simple smoo hing me hods ha
can be adequa ely employed in all cases.
This pape is o ganised as ollows. In Sec ion 2 we desc ibe he da abase p o ided by
he su ey ca ied ou by Idesca . We desc ibe some o he cha ac e is ics o his da a-
base and de ine he ime se ies ha cons i u e he ocus o his pape . Sec ion 3 ou lines
198
he me hodology ha is applied in o de o smoo h hese ime se ies. This sec ion con-
ains h ee sub-sec ions ha o e a de ailed desc ip ion o he ans o ma ions in ol ed
a each s age o ou me hodological p oposal. Sec ion 4 p esen s he esul s ob ained.
Finally, Sec ion 5 concludes.
2. DESCRIPTION OF THE DATABASE
The a ailabili y o a da abase buil upon he conduc ingo a su eya di e en poin s in
ime allows us o unde ake he analysis a di e en ime in e als. As a las eso , he
in o ma ion con ained in he da abase can always be used o de ine a daily ime se ies.
Howe e , p oblems a ise owing o he absence o obse a ions and dis o ions in he
signi icance o he indings as he ime equency o he analysis inc eases.
Fo hese wo easons, in his pape , he empo al e e ence is ixed a mon hly in e -
als and he mon hly ime se ies is he basic in o ma ion o which ou me hodology is
applied. This speci ica ion allows us o use classical smoo hing echniques including
exponen ial smoo hing and Hol -Win e s smoo hing p ocedu es. In addi ion, we can
ob ain ime se ies o a ying empo al equency (qua e ly and annual) by agg ega ing
mon hly ime se ies.
The mic o da abase used he e p o ides in o ma ion abou indi idual pe sonal cha ac e-
is ics, including age, p o ession,ma i al s a us and egiono esidence. I also con ains
de ails abou hei holidays: numbe and cha ac e is ics o he o he g oup membe s,
des ina ion, ype o accommoda ion, amoun o expendi u e, numbe o days spen in
Ca alonia and he numbe o o e nigh s ays, among o he s. Howe e ,he e we ocus on
jus wo o hese a iables: (1) he numbe o o e nigh s ays and (2) he numbe o ou-
is s. Bo h a iables a e classi ied by ype o accommoda ion(ho el, amily and iends’
households, o he ypes o accommoda ion and o al) and by des ina ion (Ba celona,
Cos a Dau ada, o he des ina ions in Ca alonia and all des ina ions in Ca alonia). The
di e en combina ions gi e ise o o y ime se ies.
The de ini ion o hese ime se ies is s ongly condi ioned by he quali y o he in o ma-
ion comp ising he ou ism mic o-da abase. Thus, i s ly, al hough in agg ega ing he
in o ma ion we ha e ied o a oid missing o ze o alues, his has been una oidable in
ce ain pe iods o some ime se ies. This migh ha e a de imen al e ec on he quali y
o he ou pu ollowing he applica ion o he smoo hing p ocedu e. Secondly, g aphic
inspec ion o he ime se ies indica es ha he e migh be some ou lie s, he p esence
o which implies g ow h a es o doub ul alidi y. Finally, he e would seem o be an
Eas e Week e ec due o he ac i is a mo eable eas and as such does no always
occu in he same ime pe iod.
199
In his analysis hese i s wo p oblems wi h he in o ma ion a e le o u u e consi-
de a ion, pa icula ly gi en ha Idesca is planning o modi y some o hese anomalies.
The hi d p oblem is discussed below.
3. METHODOLOGICAL PROPOSAL
In his sec ion we p esen he me hodology adop ed in smoo hing he ime se ies des-
c ibed in he p e ious sec ion. One o he easons why hese ime se ies a e appa en ly
so e a ic is ha he su ey loses p ecision as he geog aphical and concep ual ange is
inc eased. Ou p oposal ies o compensa e o his absence o obse a ions by inc ea-
sing he amoun o in o ma ion used when es ima ing he mic o da a o one pa icula
mon h.
The inc ease in he amoun o in o ma ion is achie ed by he join conside a ion o
he mic o-obse a ions e e ing o he same mon h in wo consecu i e yea s. Thus,
we compu e he a e age numbe o ou is s and o e nigh s ays in he same mon h o
wo consecu i e yea s and assign his mean alue o hese mon hs. Hence, we ake in o
accoun in o ma ion ha e e s o wo simila pe iods (mon h) and, as a consequence,
we a e able o educe he ola ili y o he ime se ies.
Figu e 1. B ie desc ip ion o he me hodological p oposal.
This simple me hodallows us o ob ain ime se ies ha ha e a smoo he pa e n h oug-
hou he ime pe iod unde conside a ion. The main p oblem a ises, howe e , when
deciding he weigh ings ha should be applied when compu ing his mean alue. One
possibili y is he speci ica ion o equal weigh s o each ime pe iod. Ye , i migh be a -
gued ha a weigh ingsys em ha gi es g ea e weigh ing o mo e ecen in o ma ion is
p e e able o a sys em ha a aches he same impo ance o he wo se s o in o ma ion.
200
I his is he case, he analys needs o selec hese weigh ings. This is no , howe e , a
s aigh o wa d decision, gi en ha di e en weigh ings will esul in di e en num-
be s o ou is s and o e nigh s ays. We y o o e come his d awback by sugges ing a
me hod by which he weigh ings can be es ima ed. The me hod comp ises h ee s ages.
3.1. Fi s s age: The compu a ion o he o iginal ime se ies
The app oach elies on he de ini ion o wo se s o ime se ies. The i s se is he one
de ined by he o iginal ime se ies, ha is, he ime se ies ha a e de i ed om aising
he da a o he su ey o he popula ion. As men ioned in he p e ious sec ion, he o y
ime se ies hus ob ained a e highly ola ile o e ime, which is he p oblem ha his
pape seeks o ec i y. The la ge numbe o obse a ions a ailable o he sho pe iod
unde analysis ( o y-eigh obse a ions in jus ou yea s)means ha he applica ion o
he s ochas ic app oach o he modelling o hese p ocesses is no he mos app op ia e
and ha he classical app oach should be he one o be adop ed.
3.2. Second s age: De ini ion o he ime se ies o e e ence
In he second s age o he analysis we ob ain he se o o y ime se ies ollowing he
applica ion o a classical smoo hing p ocedu e o he o iginal ime se ies. This second
se o smoo hed ime se ies se es as a e e en o de ining he sys em o weigh ings o
be used when compu ing he a e age.
Be o e applying he classical me hodological app oach o he modelling o he ime
se ies we need o know he ype o ime se ies ha is being deal wi h. He e, he cha ac-
e isa ion o he ime se ies was pe o med using wo es s a is ics. In o de o decide
he conside a ion o a ime end we applied he Daniel es , while o he seasonal
componen we applied he K uskal-Wallis es . These es s indica ed ha in mos cases
he pa e ns o he ime se ies a e gi en by bo h componen s. Ne e heless, i should
be no ed ha hese esul s a e no en i ely eliable since hese s a is ical ools a e mo e
sui ed o mode a e o la ge sample sizes. Table A.1 in he Appendix shows he esul s
o he applica ion o bo h es s. The mos app op ia e smoo hing p ocedu e o hese
da a is ha o Hol -Win e s since he e a e end and seasonal componen s in mos o
he o y ime se ies. G aphical inspec ion indica es ha he addi i e model can p o i-
de a good i , al hough his conclusion migh need o be e ised as u he in o ma ion
comes a ailable.
Be o e he Hol -Win e s smoo hing p ocedu e can be applied o he ime se ies unde
conside a ion, we need o analyse he e ec o Eas e Week on hese ime se ies. The
only pe iod ha migh ha e hadan in luence on he ime se ies was in 1997. In his yea
Eas e ell in he mon h o Ma ch while o he emainingyea s i ell in Ap il. In o de
201
o a oid dis o ions ha migh a ec he ou pu o he smoo hing p ocedu e we decided
o es o he p esence o a 1997 Eas e Week e ec and, i he e was ound o be such
an e ec o co ec he ime se ies o ake i in o accoun .
This mean he es ima ion o a eg ession model ha speci ies he ime se ies as a unc-
ion o an independen e m, a ime end, a seasonal dummies se , and an impulse
dummy ha cap u es he e ec ha can be assigned o Ma ch 1997. Only in h ee cases
was his impulse dummy ound o ha e a s a is ical signi icance o 10%, and he e ec
was co ec ed in each case. The h ee ime se ies we e he o al numbe s o o e nigh
s ays in Ca alonia (CATPE T), o e nigh s ays wi h amily o in a iend’s household in
Ca alonia (CATPE F) and o e nigh s ays wi h amily o in a iends’ household using
his minimisa ion c i e ion in Ba celona (BCNPE F).
Table 1. Es ima ed coe icien s o he Hol -Win e s smoo hing p ocedu e.
O e nigh s ays Tou is s
ALFA BETA GAMMA ALFA BETA GAMMA
CAT T 00 000 0
CAT H0.02 0.03 0 0.03 0 0
CAT F0 0 0 0.01 0 0
CAT R00 000 0
CAT NH 0 0 0 0.01 0 0
BCN T0 0 0 0.02 0 0
BCN H0.01 0.09 0 0.26 0 0
BCN F00 000 0
BCN R0 0 0 0.01 0.18 0
BCN NH 0 0 0 0.01 0.07 0
CD T00 000 0
CD H0.01 0.08 0 0 0 0
CD F00 000 0
CD R00 000 0
CD NH 00 000 0
RD T00 000 0
RD H00 000 0
RD F00 000 0
RD R00 000 0
RD NH 0.1 0 0 0 0 0
No e: CAT T e e s o all ypes o accommoda ion used in Ca alonia (CAT). CAT H indica es hose
people s aying in a ho el. CAT F hose s aying wi h amily o in a iend’s household. CAT R deno es
he o he ypes o accommoda ion used. Finally, CAT NH deno es hoses aying in accommoda ion o -
he han a ho el. This no a ion is epea ed o he e i o ial di ision conside ed he e: BCN-Ba celona,
CD-Cos a Dau ada and RD- emaining des ina ions.
202
No e: CATPE T deno es he aw ime se ies o he o al numbe o o e nigh s aysin Ca alonia, CATPE TSMAE deno es he smoo hed ime se ies using he Hol -Win e s p o-
cedu e wi h he es ima ed coe icien s and CATPE TSMAM is he smoo hed ime se ies using he Hol -Win e s p ocedu e wi h a ixed alue o he coe icien s. GCATPE T,
GCATPE TSMAE and GCATPE SMAM a e he co esponding g ow h a es. CATTU T, CATTU TSMAE and CATTU TSMAM e e o he ou is se ies.
Figu e 2. O e nigh s ays and ou is s in Ca alonia. Le els and g ow h a es o he o iginal and smoo hed ime se ies.
Once he ime se ies a ec ed by he Eas e Week had been modi ied, we applied he
Hol -Win e s smoo hing p ocedu e o es ima e he alue o he pa ame e s o he inde-
penden e m (α
, he slope (β
and he seasonal pa ame e (γ
using he c i e ia o he
minimisa ion o he sum o squa ed esiduals. The es ima ed coe icien s o each ime
se ies a e p esen ed in Table 1. No e ha al hough i is possible o ix he alue o hese
pa ame e s, he es ima ion p o ides a be e i .
As can be seen om Table 1, in mos cases he es ima ed pa ame e s equal ze o, indi-
ca ing ha he co esponding componen —independen e m, end and seasonali y—
is s able, ha is, i does no a y du ing he ime pe iod unde analysis.
The mon hly ime se ies o he le el and a e o g ow h o he numbe o ou is s
isi ing Ca alonia and he numbe o o e nigh s ays ollowing he applica ion o he
Hol -Win e s smoo hing p ocedu e a e gi en in Figu e 2. We deno e hese ime se ies
as he smoo hed-HW ime se ies. Each igu e con ains in o ma ion abou he o iginal
ime se ies, he smoo hed-HW ime se ies wi h manual selec ion o he pa ame e s and
he smoo hed-HW ime se ies wi h he es ima ed pa ame e s ob ained using he mini-
misa ion o he squa ed sum o e o s’ c i e ia.
A numbe o commen s should be made. Fi s o all, i can be seen ha he smoo hed
ime se ies buil on he use o he es ima ed coe icien s show a smoo he beha iou
han hose in which he alue o such coe icien s is imposed (in his case he pa ame e
alues we e ixed a 0.1). Second, hese esul s indica e ha he es ima ion o he ini ial
alues used in ob aining he smoo hed-HW ime se ies in luences he compu a ion o
he g ow h a es. Thus, o ins ance, we encoun e a con adic ion o he ime se ies
o ou is s coming o Ca alonia in which he ype o accommoda ion is no speci ied
(CATTU T). In his case he g ow h a es compu ed using he o iginal ime se ies a e
nega i e, while wi h he smoo hed-HW ime se ies hey a e posi i e. This is also he
case o he ime se ies o ou is s coming o Ca alonia and s aying in o he ypes o
accommoda ion (CATTU RD) and o e nigh s ays in ho el accommoda ion in Ca alo-
nia (CATPE T). Thi d, and in con as o he smoo hed-HW ime se ies, o some ime
se ies and pe iods he o iginal ime se ies p esen null alues which means he g ow h
a es a e discon inuous in hese cases.
3.3. Thi d s age: Applica ion o he Seasonal Weigh ed Mo ing A e age (SWMA)
smoo hing p ocedu e
In he hi d s age o he analysis we selec he weigh ings ha bes i he ime se ies
smoo hed in he p e ious s age. The es ima ion o hese weigh ings is ca ied ou by
speci ying he c i e ia o minimisa iono he sum o squa ede o s, whe e he e o s a e
gi en by he di e ence be ween he smoo hed-HW mac o ime se ies and he weigh ed
a e age ime se ies —he ea e smoo hed-mic o ime se ies. This sec ion desc ibes he
me hodology adop ed in his op imisa ion p ocedu e.
204
Once he mac o ime se ies in ques ion has been smoo hed using he Hol -Win e s p o-
cedu e, employing an addi i e speci ica ion and by es ima ing he pa ame e s o he
model, we p oceeded o selec he se o weigh s used in he p ocedu e applied in his
pape o he mic o se ies. This p ocedu e can be unde s ood as he compu a ion o sea-
sonal weigh ed mo ing a e ages (SWMA). Fo ins ance, in compu ing he smoo hed
ime se ies o ou is s o Janua y 1998 using he SWMA p ocedu e we need o ake
in o accoun he in o ma ion o he o iginal ime se ies o ou is s ha co esponds o
Janua y 1997 and Janua y 1998. To compu e he obse a ion o Feb ua y 1998 o he
smoo hed ime se ies we need o look a he obse a ions o he o iginal ime se ies
e e ing o Feb ua y 1997 and Feb ua y 1998, and so on.
The impo an aspec o ou p oposal is he sys em o weigh ings applied in compu ing
he a e age. As men ioned abo e, di e en smoo hed ime se ies a e ob ained depen-
ding on he se o weigh s used. The g ea e he weigh gi en o mo e ecen alues in
he ime se ies, he mo e he smoo hed-mic o ime se ies ends o esemble he o igi-
nal ime se ies. In he i s s age we smoo hed he o iginal ime se ies by applying i e
se s o weigh s: 50/50, 40/60, 30/70, 20/80 and 10/90. Ye , in o de o a oid being sub-
jec i e when selec ing he sys em o weigh ings, we es ima ed his pa ame e h ough
he minimisa ion o he sum o squa ed e o o he di e ence be ween he smoo hed
ime se ies using he SWMA p ocedu e — he smoo hed-mic o ime se ies— and he
smoo hed-HW ime se ies. This es ima ion can be ou lined as ollows.
I we deno e he o iginal ime se ies by Y, he smoo hed-HW ime se ies by Y
and he
smoo hed-mic o ime se ies using he SWMA p ocedu e by ˆ
Y, he a ge unc ion o be
minimised is he unc ion gi en by:
p
T
∑
i
s
j
y
i
ˆyi
2
j
1
2
s
, whe esdeno es he o de o seasonali y —he e s
12. Thesmoo hed-
mic o ime se ies is compu ed om:
ˆyi
pyi
1
p
yi
s
whe e pis he weigh (pa ame e ) o be es ima ed. The e o e, he op imisa ion p og am
can be ep esen ed by:
min T
∑
i
s
j
y
i
pyi
1
p
yi
s
2
subjec o 0
p
1
The necessa y condi ion es ablishes ha :
∂
p
∂p
T
∑
i
s
j2
y
i
pyi
1
p
yi
s
yi
yi
s
0
205
