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Ji ka Poměnko á, 1Roman Ma šálek*
Abs ac :
We compa e h ee i l e s commonly used o business cycle analysis: he Bax e -King, he
Ch is iano-Fi zge ald and he Hamming window i l e . Empi ical con ibu ion o he pape is
nume ical e alua ion o he app oxima ion o he ideal band-pass i l e s in he discussion o he
i l e s’ heo e ical p ope ies (gain and a enua ion wi hin he business cycle equencies, as well
as he leakage in he emaining equencies). We conside he unca ion ac o o he Bax e -
King i l e and he sample size o he la e wo. We show ha he leakage and a enua ion o he
Ch is iano-Fi zge ald and he Hamming window i l e pe o m simila ly ac oss he ange o chosen
sample sizes and be e han he Bax e -King i l e . Mo eo e , we apply he i l e s o da a o selec ed
EU coun ies and poin ou di e ences in hei es ima ion o g ow h business cycles. Ou i ndings
indica e ha Ch is iano-Fi zge ald i l e and he Hamming window bo h a e app op ia e o he
iden i i ca ion o abusiness cycle. The Hamming window i l e in oduces smalle a enua ion nea
he edges bu in case o small samples i s app oxima ion o ideal i l e is e y ough.
Keywo ds: band-pass i l e s, business cycle, equency ans e unc ion, gain, a enua ion,
leakage.
JEL Classi i ca ion: C15, C02, E32, E37
1. In oduc ion
Economic li e a u e p esen s se e al de i ni ions and me hods on how o measu e business
cycles. Bu ns and Mi chell (1946) de i ne a classical business cycle concep e e ing o
he cycle in he le els o he ou pu se ies. Ha ding and Pagan (2005) de elop a busi-
ness cycle concep dis inguishing classical and g ow h (de ia ion) cycles. G ow h cycles
a e cycles ob ained om an inpu ime se ies by emo ing he pe manen componen
(Cano a, 1998). The sensi i i y o esul s o he me hod used o isola ing he business
cycle om he inpu da a is also discussed (Ha ding and Pagan, 2005; Cano a, 1998).
The iden i i ca ion o g ow h business cycle is in he on o empi ical wo k especially
in he con ex o analysis small samples, such as ansi ion o pe iphe al economies. The
ocus is on he p ocess o economic in eg a ion, on analysis o business cycle como e-
men , synch oniza ion o business cycles du ing he c isis, analysis o in e na ional ade
linkages and many o he s.
* 1 Ji ka Poměnko á, Depa men o Radio Elec onics (DREL), B no, Uni e si y o Technology,
Czech Republic ([email p o ec ed].cz);
Roman Ma šálek, Depa men o Radio Elec onics (DREL), B no, Uni e si y o Technology,
Czech Republic ([email p o ec ed].cz).
The esea ch desc ibed in he pape was i nanced by Czech Minis e y o Educa ion in ame o
Na ional Sus ainabili y P og am unde g an LO1401. Fo esea ch, in as uc u e o he SIX Cen e
was used. We app ecia e help ul commen s om Ja ko Fid muc and S a opluk Kapounek.
P ague Economic Pape s
DOI: 10.18267/j.pep.512
Accep ed: 25. 6. 2014
Published: 24. 6. 2015
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PRAGUE ECONOMIC PAPERS 2
Ou mo i a ion o e alua ing he quali y o ideal i l e app oxima ion by a ious
band-pass i l e s de i es om he e y equen use o hese i l e s o iden i ying economic
cycles. The aim o his pape ollows consequen s eps. The i s aim/s ep is o nume ically
e alua e he quali y o he app oxima ion o selec ed band-pass i l e s o he ideal i l e
equency ans e unc ion. Fo his we selec he Bax e -King (Bax e and King, 1999),
he Ch is iano-Fi zge ald (Ch is iano and Fi zge ald, 2003) and he Hamming Window
i l e (Iacobucci and Noulez, 2005). As app o ed by many empi ical s udies, hese h ee
i l e s ep esen common widely use band-pass i l e s sui able o business cycle analy-
sis. The e also exis o he me hods such as a high-pass Hod ick-P esco i l e , ARMA
p ocesses, and i s o de di e ence (FOD) o eg ession unc ions (Bonenkamp e al.,
2001; Poměnko á, 2012) o business cycle iden i i ca ion. Bu hose a e c i icised in he
li e a u e as ha ing limi ed abili y o le only he business cycle equencies pass (Ha ey
and Jage , 1993; Guy and S -Aman , 2005). The e o e, we do no conside hem. The
quali y o he app oxima ion is assessed by using h ee me ics: undesi ed gain ( u he
deno ed as gain) in he business cycle egion, undesi ed leakage ou side he business cycle
egion ( u he deno ed as leakage) and a enua ion in he business cycle egion ( u he
deno ed as a enua ion).
In he second s ep we p o ide an empi ical e idence o ideal i l e app oxima ion.
He e we use applica ion on selec ed Eu opean coun ies, namely G eece, I eland, Po ugal,
Spain, I aly and Aus ia. The aim is an empi ical con ibu ion o ob ained knowledge om
he s ep one. Because he ans e unc ion o he Bax e -King i l e is in l uenced by he
cu -o pa ame e K while he Ch is iano-Fi zge ald’s and he Hamming Window i l e ’s
app oxima ions bo h depend on he sample size, we also conside he pe cen age o da a
loss o se e al selec ed sample sizes.
The las hi d s ep is ocused on he compa ison o di e en i l e ing echniques app o-
p ia e o empi ical analysis ocussing on ime pe iods a ec ed by global c isis shocks.
The e o e, we also conduc a co ela ion analysis be ween he business cycles iden i i ed
o he selec ed coun ies and he business cycles iden i i ed o Ge many. We include
Ge many as he hea o he eu o a ea and i s economically mos signi i can coun y.
The empi ical analysis e eals addi ional p oblems such as edge e ec s o he Hamming
window. No e ha edge (bounda y) e ec s will be e e ed o a si ua ion a he end o he
sample size whe e es ima ed alues o i l e ed ime se ies a e biased. Ou i ndings sugges
ha he Ch is iano-Fi zge ald and Hamming window i l e s bo h a e app op ia e o iden-
i ying business cycles. The Ch is iano-Fi zge ald i l e is sui able e en o small sample
sizes, while he Bax e -King and he Hamming window i l e equi e a compa ably la ge
sample size. The Hamming window i l e also in oduces smalle a enua ion nea he edges
bu in case o small samples i s app oxima ion o ideal i l e is e y ough.
The pape is o ganized as ollows: in he nex sec ion we ou line he li e a u e e iew
and consequen ly me hodological backg ound o he h ee selec ed i l e s. The hi d sec ion
con ains a desc ip ion o he chosen da a se and e alua ion o he i
l e s. In Sec ion 4 we
p esen ou esul s and hei p ac ical and heo e ical implica ions. Sec ion 5 is ocused on
compa ison o he esul s ia co ela ion analysis. Sec ion 6 concludes and summa izes he
pape .
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PRAGUE ECONOMIC PAPERS 3
2. Li e a u e Re iew
The li e a u e paying a en ion o he i l e ing me hod use ul o he business cycle iden-
i i ca ion is ex ensi e. Gene ally, we can see wo s eams which a e nicely p esen ed by
Cano a (1998). He men ioned s a is ical p ocedu es (polynomial unc ions o ime, i s
o de di e ence, Be e idge and Nelson’s p ocedu e, equency domain me hods, unob-
se ed componen s model) and economic p ocedu es (a model o common de e minis ic
o s ochas ic ends, Hod ick-P esco i l e ). E en his ca ego isa ion applica ion o any
me hod canno be done wi hou sa is ac ion o assump ion supplemen ed abou economic
backg ound.
As we men ioned in he in oduc ion band-pass i l e s a e commonly used o i l e ing.
The sui abili y o band-pass i l e s is gi en by hei p ope y o wholly selec ing only he
da a componen belonging o a speci i c equency band (called pass-band) while elimina ing
all o he componen s o ou side his speci i c band (Bax e and King 1999). O iginally, he
use o band-pass i l e s o business cycle equency analysis was p oposed by Bu ns and
Mi chel (1946). Mode n empi ical mac oeconomics uses a a ie y o echniques o pe o m
he decomposi ion o ime se ies in o end componen s and cyclical componen s such as
de e minis ic model, s ochas ic model o i l e s. Many o hose sui able o business cycle
analysis a e based on he applica ion o a wo-sided mo ing a e age. A p ecise (pe ec )
band-pass i l e is an in i ni e o de mo ing a e age i l e ha le s only he componen s in
a gi en equency ange pass. This is only a heo e ical concep and in easible in p ac ice;
such a i l e is hus called “an ideal i l e ”. Fo p ac ical applica ions i is he e o e necessa y
o use an app oxima ion o his ideal i l e . The main p oblem hen becomes cons uc ing
he closes possible app oxima ion. As s a ed by Bax e and King (1999) i is sui able o
such an app oxima ion ha should le as much as possible o he da a o he p ede i ned band
o equency pass, while a ec ing he o he equencies as li le as possible. Then i is he
op imal app oxima ion o he ideal i l e .
Th ee band-pass i l e s commonly used o he business cycle analysis in ecen
li e a u e a e he Bax e -King i l e (Bax e and King, 1999), he Ch is iano-Fi zge ald
i l e (Ch is iano and Fi zge ald, 2003) and he Hamming Window i l e (Iacoboucci and
Noulez, 2005). Guaya and S -Aman (2005) assess he abili y o he Bax e -King i l e and
he Hod ick-P esco i l e o ex ac he business-cycle componen o mac oeconomic ime
se ies by using wo di e en de i ni ions o he business-cycle componen . They show ha
bo h i l e s do ela i ely well when applied o se ies ha ha e a peak in hei spec a a
business-cycle equencies. Bu hey do poo ly wi h se ies whose spec a dec ease sha ply
and mono onically a highe equencies. The e o e, as w o e Ha ey and Jage (1993) o
Haug and De ald (2004) he Ch is iano-Fi zge ald i l e can be aken as imp o emen o
Bax e -King i l e , because i chose he weigh s o he i l e in equency domain, i.e. i
uses spec a es ima ions as weigh ed unc ion.
Haug and De ald (2012) also use band-pass i l e s o ex ac cycles (no e en business
cycles) in p e-de i ne ange (2-8 and 8-40 yea s) om ime se ies. They use l uc ua ions
o co ela ion analysis and o assessmen o como emen be ween se ies. Because he
band-pass i l e s p o ide possibili y o i l e p ede i ned equency ange, hey dis inguish
he long- e m componen and sho - e m componen by speci i ca ion o i l e s bands. This
app oach allows analysis o pa s o ime se ies sepa a ely. F om he g oup o band-pass
i l e s Haug and De ald (2012) chose he Ch is iano-Fi zge ald i l e . Fo obus ness hey
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PRAGUE ECONOMIC PAPERS 4
checked hei esul s wi h he Bax e -King i l e . They ound ha he i l e ed componen s
we e almos iden ical and ha he phase shi (which can occu in he Ch is iano-Fi zge ald
i l e case due o i s non-symme y) in he i l e ed se ies is likely negligible. Acco ding o
hei i ndings he Ch is iano-Fi zge ald i l e p o ides he closes app oxima ion o he ideal
i l e . Ou i ndings in his a icle suppo his s a emen .
Iden i i ca ion o cyclical l uc ua ion in he con ex o con e gence analysis is used in
D ake and Mills (2011). They ocused on examina ion o p ope ies o GDP in he eu o
a ea wi h he s ess o he adop ion eu o in 1999. D ake and Mills (2011) ha e pa icula
in e es in he ime se ies decomposi ion in o end and cyclical componen s using Ch is-
iano-Fi zge ald i l e , and he Bax e -King i l e . They ake he Ch is iano-Fi zge ald i l e
as supe io o he adi ional Hod ick-P esco i l e . They suppo hei decision by he ac
ha he asymme ic e sion is be e a es ima ing cycle in eal ime and nea a he end o
he sample. As hey also men ioned, he e a e wo app oaches o con e gence analysis,
compa ison o cycles be ween hemsel es o compa ison o cycles wi h he benchma k
coun ies such as Ge many. We a e going o use his idea in di e en way. On he basis o
known empi ical esul s we can e alua e he sui abili y o selec ed band-pass i l e s i s
acco ding o he le el o he measu emen o ideal i l e app oxima ion and in he con ex
o he esul s o como emen analysis wi h Ge many.
C oux e al. (2001) ocused on heo y and empi ics o como emen o economic
a iables asking whe he i can be explained by la ge agg ega e shocks o i he answe
should be ound in non-linea p opaga ion mechanism. They p opose dynamic co ela ion
and cohesion as he ele an measu emen o como emen analysis. Mac oeconomic
li e a u e o en p esen s s anda d app oach o co ela ion p e- i l e ed (high-pass o
band-pass i l e applica ion) da a. C oux e al. (2001) discuss he di e ence be ween
co ela ion o p e- i l e ed da a and applica ion o dynamic measu e. They p e e wo-sided
i l e which elimina es all he inapp op ia e wa es. As inapp op ia e wa es hey deno e
all he wa es whose equency is ou o he ele an in e al and lea es unchanged he
ampli ude o he wa es wi hin he in e al.
F om a me hodical poin o iew in he las decade he ime domain and he equency
domain (Iacobucci, 2003; Iacobucci and Noullez, 2005) analysis has been ex ended o an
in eg a ed iew o he ime- equency domain (C oux e al., 2001; Halle and Rich e ,
2007; Rua, 2010; Ma šálek e al., 2013). In all hese i elds he impo ance o app op ia ely
iden i ying business cycle phases o l uc ua ions in economic ac i i y a ises. The e o e he
abili y o p ecisely iden i y business cycles can inc ease he e i ciency o economic policy
ins umen s and he assessmen o he como emen o economies.
3. Selec ed Band Pass Fil e s
We can analyse a ime se ies, yn, n = 1, ..., N ei he in he ime o he equency domain.
The app oach p oposed by Bax e and King (1999) o Ch is iano and Fi zge ald (2003)
is o pe o m he i l e ing in he ime domain, while he equi emen s a e speci i ed in
he equency domain. In he ime domain ep esen a ion o he ideal, hough in easible,
wo-sided linea i l e is gi en by he in i ni e mo ing a e age p oducing i l e ed ime
se ies un:
,
njnj
j
uby



 (1)
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PRAGUE ECONOMIC PAPERS 5
whe e yn is he inpu ime se ies and bj a e i l e weigh s (Ch is iano and Fi zge ald,
2003). This linea ans o m selec s only he da a componen s in he speci i ed band o
angula equencies [ω1, ω2], called pass-band. The componen s ou side o his band a e
elimina ed. The adjec i e “ideal” co esponds o he equi emen o an in i ni e amoun o
da a. The equencies speci ying he pass-band ( he so-called cu -o equencies) a e
112 2
2/ , 2/ ,qq
 

whe e q1 and q2 deno e he longes and sho es pe iod o
cycles passed h ough he i l e .
In he equency domain, he ideal band-pass i l e is de i ned by he equency ans-
e unc ion G(ω) equal o 1 o equencies in he ange [ω1, ω2], and ze o o all o he
equencies. The powe spec um SU , (ω), o he i l e ed ime se ies can be compu ed as
  
2,
Uy
SGS

 (2)
whe e Sy , (ω) is he spec um o he inpu ime se ies. The i l e equency ans e unc-
ion can be decomposed as G(ω) = |G (ω)|ei

(ω) whe e he absolu e alue o G(ω) deno ed
as |G(ω)| is a module cha ac e is ic ep esen ing how he ampli ude o equency compo-
nen s a e al e ed by he i l e ,

(ω) is he phase shi caused by he i l e and desc ibing
how di e en equency componen s a e delayed and i is he complex uni . The squa ed
module |G(ω)|2 hus de e mines he weigh s co esponding o he componen s o he
powe spec um Sy (ω) a he angula equencies ω. Fo mo e de ails ega ding he module
and phase cha ac e is ics, we e e eade s e.g. o Pollock (2009). No e ha he symme ic
i l e s ha e linea phase cha ac e is ics as a unc ion o equency. Consequen ly, a g oup
delay o all equency componen s is l a (cons an ). This is ad an ageous, as in such
a case all componen s a he i l e inpu , ega dless hei equency a e delayed by he
same amoun . Thus he phase dis o ion is a oided.
The main ep esen a i es o i l e s based on a easible app oxima ion o he in i ni e
mo ing a e age a e he Bax e -King, he Ch is iano-Fi zge ald and he Hamming window
i l e s.
3.1 The Bax e -King (BK) i l e
The BK i l e is a wo-sided linea mo ing a e age band-pass i l e . In he case o business
cycle equencies i s pass-band co esponds o cycle pe iods be ween six qua e s and eigh
yea s. The componen s ou side his ange o equencies a e emo ed. Bax e and King
(1999) p opose he app oxima ion o an ideal i l e by he i ni e symme ic linea mo ing
a e age i l e o he odd o de M = 2K+1 such ha
ˆ
ˆ.
K
njnj
jK
uby


 (3)
The weigh s o he i l e ˆj
b a e compu ed in he equency domain by minimizing he loss
unc ion Q (Bax e and King, 1999; Ch is iano and Fi zge ald, 2003) o he di e ences
be ween he ideal i l e G(ω) and he easible i l e H(ω):
2
1() () ,
2
QGHd






 (4)
whe e () .
Kij
j
jK
Hbe




 Acco ding o Kowal (2005), his i l e has a numbe o
desi able p ope ies. Fi s , since i is eal and symme ic, i does no in oduce a phase shi

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PRAGUE ECONOMIC PAPERS 6
and lea es he ex ac ed componen s una ec ed excep o hei ampli udes. Second, being
o cons an i ni e leng h and ime-in a ian , i is s a iona y.
3.2 The Ch is iano-Fi zge ald (CF) i l e
Ano he i l e which well app oxima es he ideal i l e is he CF i l e (Ch is iano and
Fi zge ald, 2003). A i l e ed es ima e ˆn
u o he N-obse a ions long da a se yn can be
w i en as
,
,
ˆ
ˆ
p
p
njnj
j
uby


 (5)
whe e = N – n and p = n – 1 o n = 1, ..., N and ,
ˆp
j
ba e he ime- a ying i l e weigh s.
Simila ly o he BK i l e , he CF i l e weigh s a e designed wi h he aim o minimizing
he mean squa e e o be ween he ou pu o he ideal i l e and i s app oxima ion. In he
equency domain his p oblem can be w i en in he o m
2
,
1() () () ,
2
p
y
QGBSd






 (6)
whe e ,,
() p
p p i j
j
j
Bbe




. The i l e s weigh s a e se wi h espec o he impo ance
o he spec um in he gi en equency and he e o e depend on he p ope y o he
analysed ime se ies. I possible, he end componen should be emo ed om he o iginal
ime se ies p io o he CF i l e applica ion (Ch is iano and Fi zge ald, 2003; Haug and
De ald, 2004; Iacobucci and Noullez, 2005). We ha e ollowed his ecommenda ion in
ou analysis.
3.3 The Hamming-Window (HW) i l e
In he case o i ni e-leng h sample sizes i is impossible o design an ideal band-pass i l e .
An in e es ing i l e ha p o ides e y simple (easily applicable) and e i cien solu ion
o an ideal i l e app oxima ion de eloped wi h espec o applica ion on a sho ime
se ies was p oposed by Iacoboucci and Noulez (2005). In con as o CF i l e , he HW
i l e is symme ic and s a iona y, in con as o BK i l e i s applica ion esul s in no da a
loss. As e i i ed u he (see Figu e 1) i s equency ans e unc ion is much l a e han
in he case o CF and BK i l e s. I consis s in smea ing he ideal i l e esponse wi h
a selec ed window and i leads o good a enua ion o he spec al powe ou side he pass-
band, allowing almos comple e emo al o undesi ed equency componen . The use o
a window is ad an ageous o supp ession o a Gibbs phenomenon, Haykin and an Veen
(2003), ha can esul in he appea ance o spu ious oscilla ions a he end o ime se ies
(edge e ec s). In con as o he wo abo e-men ioned (BK, CF) i l e s, he HW i l e
is applied in he equency domain. Thus, al hough he o he wo i l e s can be de i ned
easily in he ime domain, he HW i l e has o be de i ned in he equency domain. In he
ollowing we adop ed he o iginal HW i l e de i ni ion p o ided in Iacoboucci and Noulez
(2005). Fi s a ime se ies syn is con e ed o he equency domain by he calcula ion o
i s disc e e Fou ie ans o m yn ,
1
2/
0
, 0,1... / 2
N
iknN
kn
n
Yye k N






. (7)
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PRAGUE ECONOMIC PAPERS 7
A e ha , he Hamming windowed i l e is applied acco ding o he o mula

*
kkkk
UWGY, (8)
whe e Gk ep esen s he equency ans e unc ion o he ideal i l e , i.e. he i l e ha
i l e s ou all he equency componen s ou side he business cycle band, Wk co esponds o
he selec ed window unc ion, e.g. Hamming window (Iacobucci, 2003) and he ope a o *
deno es a linea con olu ion. Finally, he i l e ed ime se ies is compu ed coming back
om he equency domain o he ime domain using he In e se Disc e e Fou ie T ans-
o m, Haykin and an Veen (2003) o Uk

/2
2/ *2/
1
0
0
ˆ, 0,1... 1
N
iknN iknN
nkk
N
k
uU Ue Ue n N





   



, (9)
whe e * in he supe sc ip means complex conjuga ion.
4. E alua ion o he Fil e s
4.1 Da a
The e alua ion is done wi h espec o he in l uencing ac o , which is he unca ion ac o
o he BK i l e and he sample size o he CF and HW i l e . Fo he applica ion we use
GDP da a in qua e ly alues, deno ed in millions o na ional cu ency, ans o med in o
chain-linked olumes and based on he e e ence yea 2000 (including ‘eu o i xed’ se ies
o eu o a ea coun ies). The da a a e seasonally adjus ed and ans o med by na u al
loga i hm o business cycle iden i i ca ion. The sou ce o ou da a is he s a is ical o i ce
o he Eu opean Union (Eu os a , 2012). We choose Aus ia as a benchma k coun y, ep e-
sen ing a s able and de eloped economy in he EU co e, and pe iphe al eu o a ea coun ies
in a ailable sample sizes. Fo compa ing he app oxima ion esul s o he i l e s we also
use Ge many as ano he ep esen a i e o a co e EU coun y. A ailable sample sizes a e:
Po ugal (N =68) om 1995/Q1–2011/Q4, I eland (N=59) om 1997/Q1–2011/Q3, I aly
(N=84) om 1991/Q1–2011/Q4, G eece (N=45) om 2000/Q1–2011/Q11, Spain (N=68)
om 1995/Q1–2011/Q4, Aus ia (N=96) om 1988/Q1–2011/Q4 and Ge many (N=84)
om 1991/Q1–2011/Q4. These coun ies we e selec ed due o opicali y o he economic
si ua ion du ing he deb c isis. Ano he impo an ac o is he di e en sample size o
he selec ed coun ies.
4.2 Quali y o app oxima ion
As s a ed in Ch is iano and Fi zge ald (2003) i he aw da a be o e applica ion o band-
pass i l e s ha e a non-ze o mean o a e co a iance s a iona y abou a end, hen he end
has o be emo ed p io o analysis o op imal i l e design. This s a emen is ollowed
by Iacobucci and Noullez (2005) in wo k ocused on a equency selec i e i l e used o
sho -leng h ime se ies. They also ecommend as one solu ion o op imal i l e design
o sub ac de e minis ic end be o e i l e ing. The e o e, we use o his he high-pass
Hod ick-P esco (HP) i l e (Hod ick and P esco , 1997). The use o he BK i l e does
no equi e his s ep.
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The equency ans e unc ion plo s (Figu e 1, Figu e 3 in Appendix A) a e
p esen ed in he no malized equency ange (0, 0.5). In he ange (0.5, 1) he equency
ans e unc ion o he band-pass i l e o business cycle equencies is ze o. No e ha
he no malized equency 1 s ands o hal o he sampling equency and ha he e is
a s aigh o wa d ela ionship be ween he no malized equency and he cyclic compo-
nen pe iod T: no malized = 2/T. No malized equencies o 0.0625 (T1 = 32 qua e s) and
0.33 (T2 = 6 qua e s) co espond o he pe iods delimi a ing he business cycle equency
band (Bu nsch and Mi chell, 1946).
Figu e 1 | Example o he Ideal Fil e App oxima ion Using he Band-Pass Fil e s, K=10, N=45 (Solid
Line A ea: The Gain, Dashed and Do ed Line A ea: The A enua ion, Do ed Line A ea: The Leakage)
To de e mine he quali y o app oxima ion, we quan i y he (undesi ed) gain o he
i l e app oxima ion in he business cycle equency ange, he (undesi ed) a enua ion in
he business cycle equency ange and he (undesi ed) leakage o he i l e app oxima ion
ou side he business cycle equency ange. In all cases he me ic alue is calcula ed as
he a ea unde /abo e he i l e equency ans e unc ion wi h espec o he ideal i l e .
The mo i a ion o selec hese h ee me ics is gi en by he shape o he ec angle o
ideal i l e . Looking a he Figu e 1 we decided o e alua e how big he a ea o leakage is,
because in case o big leakage he i l e gi es he pass o such cyclical mo emen s close o
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PRAGUE ECONOMIC PAPERS 9
he es ablished bands which does no belong o de i ned ange. We also decided o e alua e
a enua ion and he gain o he cyclical componen belonging o he de i ned ange (in ou
case business cycle ange). In case o big gain (a enua ion) he i l e ed ime se ies can
excauda e (inhibi ) impo ance o cyclical componen . The e o e, in all h ee cases o
app oxima ion o ideal i l e ec angle consecu i e analysis (como emen analysis e c.)
using ime se ies i l e ed by such i l e can indica e bias esul s.
As he ans e unc ion o he CF i l e is ime a ian , hese me ics we e compu ed
in he ime ins an n = N/2 ( he bes ideal i l e app oxima ion). Fo he desc ip ion and
illus a ion o he selec ed me ics see Figu e 1 abo e Among hese h ee obse ed quan i-
ies, he leakage and he a enua ion a e mo e c i ical o business cycle isola ion han he
undesi ed gain in he business cycle equency ange. The gain magni i es he iden i i ed
componen s so in some sense i can be bene i cial o his applica ion.
4.2.1 Bax e -King i l e app oxima ion
The desc ip ion o he BK i l e in Equa ion 3 shows ha bo h he leng h o i s impulse
esponse and i s equency ans e unc ion as well depend on he unca ion ac o
K only and no on he inpu sample size. The e o e, he measu es o he app oxima ion
o he ideal i l e using he BK i l e will be he same o an a bi a y sample size. The
a ia ion o sample size esul s only in a ela i e loss o da a change. This loss is caused
by sho ening he analysed ime se ies by K obse a ions om bo h sides o he da a
se . The e o e, we de i ne he indica o o ela i e da a loss (RDL) as a a io be ween he
numbe s o educed da a om bo h sides (2K) and he sample size (N):
2100
K
N
RDL  . (10)
We a ied he unca ion ac o in a ange o K = 5, …, 19. The esul s a e p esen ed in
Table 1.
A de ailed g aphical ep esen a ion o he esul s in Table 1 can be ound in Appen-
dix A (Figu e 3). As we men ioned abo e, he equency ans e unc ion does no depend
on he analysed da a o on i s sample size, bu only on he unca ion ac o K. Thus, he
op imum alue o he pa ame e K is he alue o which he app oxima ion o he ideal
i l e esul s in minimum alues o (undesi ed) gain, a enua ion and leakage wi h espec
o minimized da a loss. We can s a e ha his condi ion is ul i lled o he alue K=10.
Fo K g ea e han 10, he alue o leakage has a descending endency, bu on he con a y
he ela i e loss o da a is ascending. Fo p ac ical easons du ing he applica ion on eal
da a we equi e a loss o da a as small as possible. We can obse e ha undesi ed gain is
also ising o highe K alues and he di e ence be ween he leakage o app oxima ion
o pa ame e K=10 (0.0125) and K=11 (0.0107) is no signi i can . I is hus p e e able o
accep a ma ginally highe alue o leakage in o de o achie e a smalle da a loss. No e
ha his esul is in good acco dance wi h he o iginal ecommenda ion o an op imum
selec ion o he K alue (Bax e and King, 1999) wi hou conside ing da a loss.
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PRAGUE ECONOMIC PAPERS 16
loss close o a minimum. This esul is in good acco dance wi h he o iginal ecommen-
da ion o op imum K alue selec ion (Bax e and King, 1999). Ou heo e ical i ndings
show ha ega ding leakage and a enua ion he Ch is iano-Fi zge ald and he Hamming
window i l e pe o m simila ly ac oss he ange o chosen sample sizes, while yielding
be e esul s han he Bax e -King i l e .
Secondly, we apply he i l e s o GDP da a o selec ed EU coun ies. The empi ical
analysis e eals addi ional p oblems such as edge e ec s and shape o equency ans-
e unc ion o i l e s. We sugges ha he Ch is iano-Fi zge ald i l e migh be he mos
app op ia e o he iden i i ca ion o business cycles e en o small sample sizes, while he
Hamming window i l e o Bax e -King especially equi e a compa ably la ge sample size.
Ou i ndings a e suppo ed by he esul s o a co ela ion analysis be ween Ge many and
I eland, G eece, Spain, Po ugal, I aly and Aus ia.
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Appendix A
Figu e 3 | Fil e App oxima ion Using he Bax e -King Fil e wi h T unca ion Fac o K= 5, …, 19
(Solid Line A ea: The Gain, Dashed and Do ed Line A ea: The A enua ion, Do ed Line A ea: The
Leakage), (X-Label: No malized F equency, Y-Label: The F equency T ans e Func ion)