IS THE IDEAL FILTER REALLY IDEAL: THE USAGE OF FREQUENCY FILTERING AND SPURIOUS CYCLES

Transcription

1 South Eastrn Europ Journal of Economics (005) IS THE IDEAL FILTER REALLY IDEAL: THE USAGE OF FREQUENCY FILTERING AND SPURIOUS CYCLES LYUBOMIR IVANOV Tsnov Acadmy of Economics Abstract In this papr th filtr of Hodrick and Prscott, th band-pass filtr of Baxtr and King, "th idal filtr" and th first-diffrncing ar analyzd It is shown that idal filtrs ar abl to produc spurious cycls not only whn a unit root is prsnt in th data but also whn th root is lss than unity Th main critiqus of first-diffrncing ar outlind and furthr analyzd Th conclusion is that diffrncing and mor gnrally mthods in conformity with th undrlying proprtis of non-stationary sris ar th prfrabl choic At th sam tim th application of idal filtrs and thir approximations (including HP and BP) is vry dangrous and thir usag is to b avoidd JEL Classification: C Kywords: Frquncy filtring, Hodrick-Prscott filtr; Band-Pass Filtr, Spurious Cycls Introduction On major point in th analysis of cyclical fluctuations is trnd limination, spcially whn attntion is focusd on th businss cycl (Wstlund and Ohln, 99, p 8; Zarnowitz and Ozyildirim, 00, p) and growth cycls (Zarnowitz, 99, Zarnowitz and Moor, 986, p9) Many authors mntion that conomic tim sris ar both fluctuating and trnding, i thy ar non-stationary Stock and Watson applid diffrnt mthods for trnd rmoval in analysis of th US businss cycl That includs diffrncing, th Buttrworth filtr and Baxtr and Corrsponding addrss: Dpartmnt of Mathmatics and Statistics, Em Chakarov Str, 550 Svishtov, Bulgaria, -mail: lubomir@uni-svishtovbg

2 80 L IVANOV, South-Eastrn Europ Journal of Economics (005) King filtr (Stock and Watson, 989, 990, 996) Th Hodrick-Prscott filtr is vry popular: Hamori and Kitisaka (997), Srltis and Kraus (996), Pakko (997), Olkalns (00), Grgory, Had and Raynauld (997) Th filtr suggstd by Baxtr and King was usd by Mills (000, 00), Basu and Taylor (999), Hornstin (000), Wynn and Koo (000) Wstlund and Ohln (99) applid linar dtrnding and Bvridg-Nlson dcomposition Gnrally spaking thr ar two main approachs to trnd rmoval Th first on is to modl th tndncy with an conomtric modl and thn to us it for trnd rmoval Th typical rprsntativ hr is Bjornland (Bjornland, 000, p 369) Th scond approach is frquncy filtring A numbr of filtrating mthods hav bn dvlopd to xtract and liminat non-stationarity and som of thm hav bcom vry popular, g th Hodrick-Prscott filtr (Hodrick and Prscott, 980), th bandpass filtr of Baxtr and King (Baxtr and King, 999), th filtrs of th Buttrworth family (Gomz, 00) and Winr-Kolmogorov family (Planas, 997) Whn bing utilizd ths filtrs rquir subjctiv judgmnt at a particular momnt According to Stock and Watson (996, p 7), Kaisr and Maraval (999) and Schnk-Hopp (00) this crats a srious problm - it bcoms possibl to gnrat spurious cycls in th filtrd data vn whn thr ar no cycls Th lattr is typical whn th tim sris contains on or mor unit roots (this is th Nlson and Kang (98) critiqu) Du to thir as of application, frquncy filtrs hav gaind in popularity in rcnt yars Many paprs adoptd such tchniqus in th analysis of businss cycl proprtis It may happn that th rsults ar snsitiv to dtrnding mthods and may lad to incorrct statmnts Th main goal of this papr is a partial xamination of th problm with mphasis on possibl gnration of spurious cycls Th analysis covrs th bhavior of th filtrd data and th influnc of th idal filtrs whn prsnc of a unit root or a root nar unity is possibl I shall analyz mainly th filtr of Hodrick and Prscott and th band-pass filtr in comparison with first-diffrncing Th main conclusion from this papr is that application of idal filtrs is dangrous whn unit roots or nar unit roots xist in tim sris This can lad to dtction of spurious cycls A bttr solution would b a modl-basd approach for trnd rmoval Th papr is structurd as follows In part on th Hodrick-Prscott filtr is analyzd Th band-pass filtr is discussd in th scond part In part thr "th idal filtr" is analyzd and compard with diffrncing Conclusions ar summarizd in th last part

3 L IVANOV, South-Eastrn Europ Journal of Economics (005) Part On Lt's first considr th ffcts of th Hodrick-Prscott filtr ovr a procss containing a unit root Th filtr is basd on th assumptions that: () th tndncy is smooth whil th shocks hav only tmporary ffct; () th procss is linar Th first can b argud on th basis of th rsults rachd by Nlson and Plossr (98) and latly by Murray and Nlson (000) and Fatas (000) Th scond has bn qustiond sinc Hamilton (989, 990) dvlopd his non-linar modl with rgim switching and subsqunt analyss Th Hodrick-Prscott filtr assums th prsnc of smooth trnd indpndnt of th cyclical componnt Considr a tim sris y t whr t =,,, n and mark th tndncy with g t and th cyclical componnt with c t W assum that th sris is sasonally adjustd and irrgular componnts play only a small rol Thus dcomposition to unobsrvd componnts is: y t = g t + c t () Hr th stimats of th tndncy g t and th cyclical componnt c t ar acquird aftr th minimization of n t n t c t g () k whr is th diffrnc oprator ( yt yt yt k ) and is a constant Th first sum accounts for th accuracy of th stimation (th rsiduals around th trnd) whil th scond rprsnts th smoothnss of th trnd Paramtr controls th wight of th two componnts, g as w incras th HP trnd bcoms smoothr and whn approachs infinity th trnd is transformd into a straight lin Th filtr has th following advantags: it is asy to us; it is implmntd in many computr packags; it allows clar intrprtation of rsults and wid spcification of th trnd, that is up to th parabolic tim trnd and up to two unit roots But th filtr also has srious drawbacks Gomz (00, p 368) and Kaisr and Maraval (999, p 75) point out that th filtrd sris is subjct to big changs whn th raw data ar rvisd or nw data ar addd Th stimats ar unstabl, which is spcially visibl at th nd of th sris Whn th filtr is usd upon a sris alrady subjctd to sasonal adjustmnt, statistical infrnc could lad to incorrct rsults As Kaisr and Maraval mntion (999, p 8) th filtr changs th autocorrlation structur of th procss and spurious corrlations can b spottd in th filtrd data t3

4 8 L IVANOV, South-Eastrn Europ Journal of Economics (005) Goms (00, p 368), Kaisr and Maraval (999, p 75) togthr with Stock and Watson (996, p 7) agr in thir conclusions that th filtr rtains significant nois in th cyclical signal that hindrs th accurat idntification of th cyclical componnts King and Rblo (993) stat that crtain charactristic of th sris and th filtrd data ar oftn diffrnt whil Cogly and Nason (995) show that whn usd upon a sris with a unit root th filtr producs sris with autocorrlation at lag as larg as 0,7 no mattr how larg was th autocorrlation at lag in th original data Finally thr is th possibility of gnrating spurious cycls, as mntiond by Kaisr and Maraval (999, p 8) Gomz (00, p 368) concluds that th mchanical usag of th filtr oftn lads to incorrct rsults and Schnk-Hopp (00, p 83) dirctly stats that th filtr crats spurious cycls vn whn th original data contains no cycls at all Whn analyzing th ffct of th combination of X and HP filtr, Kaisr and Maraval prformd a Mont Carlo simulation xprimnt and obtaind two major rsults: () th priod of th spurious cycls dpnds mainly on th ordr of th intgration of th tim sris; () whn th intgration is fixd, th priod dpnds on th valu of (Kaisr and Maraval, 999, pp 89-90) Focusing mainly on th ffct of th combination of th two filtrs (X and HP) thy conclud that "th criticism that HP filtring inducs a spurious cycl in th sris is unwarrantd" (p 75) bcaus th analyst "can choos th appropriat valu of " (p 9) But it is th subjctiv choic of which contains th possibility of gtting cycls with any priod dsird just by slcting th appropriat valu of Schnk-Hopp (00) has xprimntd on a diffrnt ground Basd on th conomic thory, h crats a modl of an artificial conomy and tracks down th trac of a priori dtrmind tndncy and cycls Nxt h applis th HP and th BP filtrs to gnratd tim sris and compars both th ral trnd and cycls with thos xtractd by th filtrs His rsults show that both filtrs wr unabl to trac th ral cycls Instad thy xtract spurious cycls with priods btwn 3 and 5 yars Schnk-Hopp finishs his analysis with th conclusion that "any businss cycl dating basd on ths filtrs lads to incorrct statmnts" (p 83) and maks suggstions for th dvlopmnt of a nw mthodology or th modification of th xisting on (p 85) Th main problm is idntifid in th application of th filtr ovr intgratd sris so I bgin my analysis with this issu To achiv my rsults I shall us th spctral analysis (s Hamilton, 995, part 5, for xampl) Lt's hav a linar stationary procss x t with th following ARMA rprsntation:

5 L IVANOV, South-Eastrn Europ Journal of Economics (005) whr (L) and (L) ar th polynomials of th lag oprator whil t is a whit nois Th spctral dnsity of this procss is givn by: whr is th varianc of th whit nois t In th prsnc of a unit root th procss is no longr stationary and its spctrum is undfind But it is still possibl to driv th psudo-spctrum, which is: whr th dominating root is markd with and is not unity but approachs unity from blow Whn quals unity th statmnt bcoms: which shows an infinit maximum in th spctral dnsity at th zro frquncy (th spctrum is not dfind thr) To trac th ffct of th application of th HP filtr ovr a sris containing a unit root I us th fact that th application of th filtr ovr a sris lads to nw sris with spctral dnsity qual to th dnsity of th original data multiplid by th transfr function of th filtr calld squard gain function Th cyclical componnt xtractd by th HP filtr is givn by: whr L is th lag oprator (L k x t = x t-k ) (L)x t = (L) t, (3) Sx( ) i i i i Substituting L with -i I gt th following gain function: i i G (8) i i whos squar is th transfr function: () Sx (5) Sx (6) L L L L (7) Ct x t G (9)

6 8 L IVANOV, South-Eastrn Europ Journal of Economics (005) (Th rsults up to hr follow King and Rblo, using th -ik = k - isin k quality and th appropriat transformations) Now th application of th HP filtr ovr a sris x t, which contains unit root: z t = C HP (x t ) lads to th following spctral dnsity of th filtrd sris z t : whr: Th last rsult () shows that: () th spctrum is dfind in th ntir rgion [0;] contrary to th spctrum of th non-stationary sris it's a spctrum not psudospctrum; () th spctral dnsity at th zro frquncy is zro th HP filtr cuts off th infinit maximum thr This is on of th faturs of th HP filtr as mntiond abov th limination of up to two unit roots At th sam tim it is clar that th spctral dnsity of th filtrd data dpnds upon th valu of paramtr Th lattr can b asily obsrvd in figur (0) ; 6 3 f S G S x z () 3 ; f ()

7 L IVANOV, South-Eastrn Europ Journal of Economics (005) Figur Spctral dnsity of th cyclical componnt whn th HP filtr is usd ovr sris with a unit root lambda=00 lambda=000 lambda=0000 lambda=00000 Th rsult shows that in th spctral dnsity thr is a maximum at th lowr frquncis and it dpnds on th valu of Thus it is possibl for fals cycls to aris at low frquncis (businss cycl frquncis) To dtrmin th xact location of th maximum I furthr analyz function () for xtrms bcaus th othr paramtrs in () ar constants Trating as a constant th first drivativ of () conditionally on is: f 3 sin 3 3 Aftr quating to zro and solving th quation I gt th following roots in th rgion [0;]: = 0; = (3) 3 whr - is th invrs in function 3

8 86 L IVANOV, South-Eastrn Europ Journal of Economics (005) Th valus of th spctral dnsity in th first two points (which ar at th nds of th rgion) ar 0 and 3 6 rspctivly, whil th third on is insid th intrval and valus As th third valu is th biggst for any positiv valu of th maximum is achivd at that point Thus if th HP filtr is applid ovr a tim sris with a unit root th rsult is a sris with cyclical fluctuations at th frquncy 3 This finding supports th simulation rsults of Kaisr and Maraval (999, p 90), as thr is almost prfct fit of thortical valus from th formula with th ons producd by th simulation Th two authors stat that "th rlationship btwn this priod and is highly nonlinar" Th rlationship is indd nonlinar although not vry complicatd If th valus for, suggstd by Hodrick and Prscott (000 for monthly data and 600 for quartrly data) ar put into th formula th rsults show that thr will b maximums at 9,98 months and 30, quartrs rspctivly Both cass ar btwn 7 and 8 yars It is not surprising that th application of th filtr in many mpirical studis lads to cycls with such or narly th sam priod This is bcaus many conomic tim sris ithr contain unit root or at last unit root offrs good approximation to th tru form of non-stationarity At th nxt stag w ar intrstd in th first autorgrssiv root that is lss than unity in modulus Lt's considr th simplst cas with an AR() procss: y t = y t- + t () whr is lss than but positiv Th spctral dnsity of th procss () is givn by: S y (5) Th application of th HP filtr lads to cyclical componnt z t with th following spctral dnsity:

9 L IVANOV, South-Eastrn Europ Journal of Economics (005) S z G S (6) y Th imag of th spctral dnsity for = 05 and diffrnt valus of can b sn in figur Again a maximum xists at th low frquncis Figur Spctral dnsity of th cyclical componnt whn th HP filtr is usd ovr sris with first autorgrssiv root of lambda=00 lambda=000 lambda=0000 lambda=00000 Unfortunatly th xact functional form of th rlationship btwn th frquncy at th maximum and th paramtrs and cannot b drivd analytically Aftr tdious computations th first drivativ turns out to b an quation of fifth powr A bttr solution is to us numrical mthods to comput both th maximum and th corrspondnt frquncy for any givn valus of and This dos not chang th situation bcaus th shap of th spctral dnsity and th conclusions rmain th sam Spurious cyclical fluctuations ar gnratd by th HP filtr not only whn th first root is unity but also whn th fist autorgrssiv root is larg Th maximums in spctral dnsity disappar only whn th valu of bcoms ngligibly small Th rason for such a bhavior is th form of th spctral dnsity of th original procss, which is dcrasing Th HP filtr cuts off part of it around th zro frquncy but th rst rmains and gnrats an artificial maximum

10 88 L IVANOV, South-Eastrn Europ Journal of Economics (005) Part II Th nxt filtr is band-pass (BP) filtr of Baxtr and King It's a symmtric moving avrag filtr with wights that sum to zro It nsurs th limination of th infinit maximum at th zro frquncy as shown by Baxtr and King (999, p 59, Appndix A) In thir papr Baxtr and King dvlopd th wights for thr cass: two for monthly data and on for yarly data Bcaus of th tim consuming computations I driv th spctral function analytically only for th third cas, with wights of 0050, 035, 000, 077, 000, -035 and (p 59, Tabl, Column ) Th filtrd sris (pur cyclical componnt) is givn by: (7) z t = (L)x t whr th filtr is: (L) = 0050L L- 000L L 0 000L - 035L 0050L 3 (8) Substituting L with -i and simplifying I gt th transfr function: G (9) Th application of th BP filtr ovr sris with a unit root lads to th following spctral dnsity: S z S x G S z ( ) (0) Aftr computing th first drivativ and quating to zro I gt th nxt thr ral solutions in th rgion [0;]: = 0; = ; 3 = 377

11 L IVANOV, South-Eastrn Europ Journal of Economics (005) Again th first two ar at th nds of th intrval with valus of 0 and Th third on is insid th rgion and thr th spctral dnsity rachs its maximum of 3305 This frquncy corrsponds to cycls with priod of 59 yars Part III Th Hodrick-Prscott filtr can b drivd from both th Winr-Kolmogorov and th Buttrworth familis (Planas, 997; Gomz, 00) I shall not analyz th filtrs from ths familis as thy dpnd on many mor paramtrs Instad I shall analyz th idal filtr and show that th conclusions drivd in prvious parts still hold Baxtr and King dfin th idal filtr as a filtr that kps unchangd th spctral dnsity in som intrval [ ; ] whil making dnsity at all th othr frquncis narly zro Usually th rsarchr dfins this intrval Th graphical form of th idal filtr is rprsntd in figur 3, whr its transfr function is shown it valus for th dsird intrval and zro lswhr Figur 3 Transfr function of th idal filtr,,0 0,8 0,6 0, 0, 0,0 0,03 0,3 0,60 0,88,6,5,73,0,9,58,86 3,

12 90 L IVANOV, South-Eastrn Europ Journal of Economics (005) 79-96

13 L IVANOV, South-Eastrn Europ Journal of Economics (005) To strngthn that conclusion lt's considr th cas whn in th original data thr ar cycls at frquncy 0 nar but outsid th intrval [ ; ] In such a situation th idal filtr will transfr cycls from th ral frquncy 0 to th lowst possibl frquncy if < or to th highst possibl if > Th lattr 0 0 is clarly sn in figur 5 Figur 5 Rsult from th application of th idal filtr ovr th sris containing cycls outsid th rgion of intrst (a) (b)

14 9 L IVANOV, South-Eastrn Europ Journal of Economics (005) Thus th application of th "idal filtr" lads to spurious cycls not only in th prsnc of non-stationarity but also in th cas of stationary data with cycls Any filtr that approximats th idal filtr lads to th sam rsult, i artificial cycls with a priod dtrmind by th rsarchr To ovrcom this problm it is rcommndd that w analyz sris for th trnds and th cycls and only thn apply th filtr Naturally aftr such an analysis th application of th idal filtr is uslss To rmov th non-stationarity Box, Jnkins and Rinsl (99) suggstd th us of th diffrnc oprator It can b considrd as a filtr lik thos alrady xamind, for th purpos of comparison Baxtr and King compar it with thir filtr and dtrmin its main drawbacks in two dirctions First, thy conclud that th filtr is not symmtric which can caus problms if applid to mor than on variabl linkd btwn thm Th filtr gnrats phas changs in th spctrums of th variabls and th xisting rlationships can b dilutd Scond, a critiqu also supportd by Hodrick and Prscott, Maraval, Gomz, th filtr drastically changs th powr of th spctral dnsity by frquncis It givs biggr wight to th high frquncis whil dcrasing th powr of th low and liminating compltly th zro frquncy Th conclusion of Baxtr and King is that "If th goal of a businsscycl filtr is to isolat fluctuations in th data that occur btwn spcific priodicitis, without spcial mphasis on any particular frquncy, th first-diffrnc filtr is a poor choic" (pp ) In my opinion such a conclusion is not right Lt's first considr th transfr function of th diffrnc filtr, as did Baxtr and King Th function is: i i G () and indicats th xtnt to which th filtr raiss (lowrs) th varianc of th filtrd sris rlativ to that of th original sris As can b sn th influnc of th filtr is lik that dscribd abov, i th varianc at high frquncis is xpandd four tims whil at th low ons it is dcrasd But this is th influnc whn th filtr is applid ovr stationary sris whil th main purpos of th filtr is to b applid ovr non-stationary ons As corrctly mntiond by dn Haan (pp 9-0) whn comparing th rsults from th diffrnt filtrs th rsarchrs usually compar diffrnt things Hr th application of th diffrnc filtr ovr stationary sris (whr it givs th worst rsults) is compard with th rsults of th filtr dscribd (HP and BP) whn applid ovr stationary sris (thy ar cratd for such a purpos) Lt's first compar th prformanc of th filtrs ovr non-stationary sris using th rsults from th prvious parts

15 L IVANOV, South-Eastrn Europ Journal of Economics (005) To s th ffct of th diffrnc filtr lt's considr a linar stochastic procss from th ARIMA family: ARIMA (p, d, q), whr d = for simplicity Th procss can b prsntd as: with a spctral dnsity: S ( ) It's obvious that th spctrum is not dfind at th zro frquncy This is psudospctrum Th application of th first-diffrnc filtr: lads to a sris with th following spctral dnsity: S x ( ) But this is xactly th spctral dnsity of th ARMA (p, q) procss, i th stationary part of th original procss No incras of th high frquncis or dcras of th low ons is obsrvd Th filtr just rmovs th influnc of th unit root It rmovs th tndncy not only at th zro frquncy as th othr filtrs do but also rmovs th influnc of th tndncy in both th low frquncis (whr th tndncy has high influnc) and th high ons (whr th ffct of th tndncy is ngligibl) If th sris looks rough aftr such a transformation it's bcaus th tndncy smooths th sris It can b concludd that th main argumnt that th filtr incrass th high frquncis is not consistnt bcaus it should b usd only whn a unit root is prsnt or such an assumption cannot b rjctd Th situation is diffrnt whn th dominating root is lss thn unity In this cas th spctral dnsity of th original data is: and th dnsity of th filtrd sris is: y L Ly t L t () t y t i i i i (3) x L () i i i i S y ( ) S x ( ) i i i i i i i i i i i i (5) (6) (7)

16 9 L IVANOV, South-Eastrn Europ Journal of Economics (005) Th diffrnc filtr rmovs th maximum at zro and quadrupls th high frquncis varianc Th scond critiqu was about th phas shift du to th asymmtry of th diffrnc filtr I do not think this will gnrat problms in practic If mor than on sris ar analyzd and th filtr is applid to all of thm, th phas shift will b prsnt in all sris and th xisting rlationships will rmain Th potntial dangr lis in th possibility of applying th filtr only in part of th sris whil th othrs (which ar stationary) ar analyzd without transformation Rgarding th idas about cointgration and th rlationships btwn intgratd procsss, th possibility of rlationships btwn stationary and intgratd variabls is vry unlikly Thus th scond critiqu is not problmatic Part IV I can summariz th following propositions: First, whn a unit root is prsnt in th data first-diffrncing rsults in a stationary procss whil "idal" filtrs lad to spurious cycls Scond, whn th procss is stationary diffrncing boosts th high frquncis On th othr hand, idal filtrs will ithr gnrat spurious cycls whn a significant first autorgrssiv root xists or may transfr cycls if thy ar prsnt Th dangr of fals cycls xists in both cass whn applying th idal filtr From this point of viw th application of diffrncing is rcommndd spcially whn combind with th appropriat tsting for unit roots If w hav a stationary sris, but du to th low powr of th unit root tsts w accpt that th sris is a non-stationary on, th application of th filtr will intnsify th high frquncis But such a dcision is not as dangrous as th cas whr missing accommodation for th unit root will gnrat spurious cycls In this papr w usd first-diffrncing as an xampl of dtrnding whn nonstationarity taks a particular form - g unit root In gnral, sris may b nonstationary but not with a unit root (fractional intgration, polynomial trnds) As long as thr is larg autorgrssiv root in th ARIMA rprsntation of th procss th application of th idal filtr will gnrat fals cycls Instad of frquncy filtring or diffrncing othr trnd rmoval procdurs may b ndd in accordanc with th data gnrating procss Th main rquirmnt for th transfr function of th procdur is to b monotonic Thus th possibility of gnrating spurious cycls is liminatd unconditionally bcaus transfr function contains no xtrms Anothr possibl solution is to us both mthods (filtring and diffrncing) at th sam tim If aftr th trnd rmoval th spctral stimats diffr only in thir amplituds in th high frquncy rgion, no problms xist with th application of

17 L IVANOV, South-Eastrn Europ Journal of Economics (005) th idal filtrs But if th mthods rcord diffrnt cycls, th sris probably contains a unit root, a larg positiv autorgrssiv root or cycls at frquncy nar th rgion of intrst Situations lik ths ar oftn mt in practic and som rsarch paprs dfinitly nd rxamination bcaus th nar idal filtrs hav bn usd Rfrncs Basu, S, A M Taylor, 999, "Businss Cycls in Intrnational Historical Prspctiv", NBER Working papr 7090 Baxtr, M and King, R G, 999, "Masuring businss cycls: Approximat band-pass filtrs for conomic tim sris", Rviw of Economics and Statistics, 8, Bjornland, H C, 000, "Dtrnding mthods and stylizd facts of businss cycl in Norway - an intrnational comparison", Empirical Economics, 5, Box, G E P, Jnkins, G M and Rinsl, G C, 99, Tim Sris Analysis: Forcasting and Control, Nw Jrsy : Prntic Hall Canova, F, 998, "Dtrnding and Businss Cycl Facts", Journal of Montary Economics,, 75-5 Cogly, T and Nason, J M, 995, "Effcts of th Hodrick-Prscott filtr on trnd and diffrnc stationary tim sris: Implications for businss cycl rsarch", Journal of Economic Dynamics and Control, 9, Fatas, A, 000, "Endognous growth and stochastic trnds", Journal of Montary Economics, 5, 07-8 Gomz, V, 00, "Th Us of Buttrworth Filtrs for Trnd and Cycl Estimation in Economic Tim Sris", Journal of Businss and Economic Statistics, 9, Grgory, A W, Had, A C and Raynauld, J, 997, "Masuring World Businss Cycls", Intrnational Economic Rviw, 38, dn Haan, W J, 996, "Th comovmnts btwn ral activity and prics at diffrnt businss cycl frquncis", NBER Working papr 5553 Hamilton, J D, 989, "A nw approach to th conomic analysis of non-stationary tim sris and th businss cycl", Economtrica, 57, Hamilton, J D, 990, "Analysis of Tim Sris Subjct to Changs in Rgim", Journal of Economtrics, 5, Hamilton, J D, 995, Tim Sris Analysis, Princton: Princton Univrsity Prss Hamori, S and Kitisaka, S-I, 997, "Th Charactristics of th Businss Cycl in Japan", Applid Economics, 7, 05-3 Hodrick, R J and Prscott, E C, 980, "Post-war US Businss Cycls: An Empirical Invstigation", Working papr, Carngi-Mllon Univrsity Hodrick, R J and Prscott, E C, 997, "Post-war US Businss Cycls: An Empirical Invstigation", Journal of Mony, Crdit and Banking, 9, - 6 Hornstin, A, 000, "Th Businss Cycl and Industry Comovmnt", Economic Quartrly, Fdral Rsrv Bank of Richmond, 86, 7-8 Kaisr, R and Maraval, A, 999, "Estimation of th businss cycl: A modifid Hodrick-Prscott filtr", Spanish Economic Rviw,, 75-06

18 96 L IVANOV, South-Eastrn Europ Journal of Economics (005) King, R G and Rblo, S T, 993, "Low frquncy Filtring and Ral Businss Cycls", Journal of Economic Dynamics and Control, 7, Mills, T C, 000, "Businss cycl volatility and conomic growth: a rassssmnt", Journal of Post Kynsian Economics, 3, 07-6 Mills, T C, 00, "Businss cycl asymmtry and duration dpndnc: an intrnational prspctiv", Journal of Applid Statistics, 8, 73-7 Murray, C J and Nlson, C R, 000, "Th uncrtain trnd in US GDP", Journal of Montary Economics, 6, Nlson, C R and Kang, H, 98, "Spurious priodicity in inappropriatly dtrndd tim sris", Economtrica, 9, 7-75 Nlson, C R and Plossr, C I, 98, "Trnds and random walks in macroconomic tim sris", Journal of Montary Economics, 0, 9-6 Olkalns, N, 00, "Cyclical asymmtris in Australian macroconomic data", Applid Economics Lttrs, 8, 5-8 Pakko, M R, 997, "Th Businss Cycl and Chain-Wightd GDP: Has Our Prspctiv Changd?", Rviw of Fdral Rsrv Bank of St Louis, 79, 39-9 Planas, C, 997, Applid tim sris analysis, Luxmburg: Eurostat Schnk-Hopp, K R, 00, "Economic Growth and Businss Cycls: A Critical Commnt on Dtrnding Tim Sris", Studis in Nonlinar Dynamics and Economtrics, 5, Srltis, A and Kraus, D, 996, "Nominal Stylizd facts of US Businss Cycl", Fdral Rsrv Bank of St Louis Rviw, 78, 9-5 Stock, J H and Watson, M W, 989, Nw Indxs of Coincidnt and Lading Economic Indicators, in O J Blanchard and S Fishr (ds), NBER Macroconomics Annual, Cambridg: MIT Prss, Stock, J H and Watson, M W, 990, "Businss Cycl Proprtis of Slctd U S Economic Tim Sris, ", NBER Working papr 3376 Stock, J H and Watson, M W, 996, "Evidnc on Structural Instability in Macroconomic Tim Sris Rlations", Journal of Businss and Economic Statistics,, - 30 Wstlund, A H and Ohln, S, 99, "On Tsting for Symmtry in Businss Cycls", Empirical Economics, 6, Wynn, M A and Koo, J, 000, "Businss Cycls undr Montary Union: A Comparison of th EU and US", Economica, 67, 37-7 Zarnowitz, V and Moor, G, 986, Major Changs in Cyclical Bhavior, in R J Gordon (d), Th Amrican Businss Cycl: Continuity and Chang, Chicago: Univrsity of Chicago Prss Zarnowitz, V, 99, "What is a businss cycl?", in M T Blongia and M R Garfinkl (ds), Th Businss Cycl: Thory and Evidnc, Boston : Kluwr Acadmic Publishrs Zarnowitz, V and Ozyildirim, A, 00, "Tim Sris Dcomposition and Masurmnt of Businss Cycl, Trnds and Growth Cycls", NBER Working Papr 8736