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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Esrella, Aruro Working Paper Exracing business cycle flucuaions: Wha do ime series filers really do? Saff Repor, Federal Reserve Bank of New York, No. 89 Provided in Cooperaion wih: Federal Reserve Bank of New York Suggesed Ciaion: Esrella, Aruro (007) : Exracing business cycle flucuaions: Wha do ime series filers really do?, Saff Repor, Federal Reserve Bank of New York, No. 89 This Version is available a: hp://hdl.handle.ne/049/6059 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 Federal Reserve Bank of New York Saff Repors Exracing Business Cycle Flucuaions: Wha Do Time Series Filers Really Do? Aruro Esrella Saff Repor no. 89 June 007 This paper presens preliminary findings and is being disribued o economiss and oher ineresed readers solely o simulae discussion and elici commens. The views expressed in he paper are hose of he auhor and are no necessarily reflecive of views a he Federal Reserve Bank of New York or he Federal Reserve Sysem. Any errors or omissions are he responsibiliy of he auhor.

3 Exracing Business Cycle Flucuaions: Wha Do Time Series Filers Really Do? Aruro Esrella Federal Reserve Bank of New York Saff Repors, no. 89 June 007 JEL classificaion: C, E3 Absrac Various mehods are available o exrac he business cycle componen of a given ime series variable. These mehods may be derived as soluions o frequency exracion or signal exracion problems and differ in boh heir handling of rends and noise and heir assumpions abou he ideal ime-series properies of a business cycle componen. The filers are frequenly illusraed by applicaion o whie noise, bu applicaions o oher processes may have very differen and possibly uninended effecs. This paper examines several frequenly used filers as hey apply o a range of dynamic process specificaions and derives some guidelines for he use of such echniques. Key words: frequency domain, specral analysis, signal exracion Esrella: Federal Reserve Bank of New York ( aruro.esrella@ny.frb.org). The views expressed in his paper are hose of he auhor and do no necessarily reflec he posiion of he Federal Reserve Bank of New York or he Federal Reserve Sysem.

4 . Inroducion The analysis of how a ypical macroeconomic ime series behaves over he business cycle is complicaed by he fac ha is movemens may conain low-frequency rends and highfrequency noise. Various mehods are available o exrac he business cycle componen of a given ime series variable. These mehods differ in heir handling of rends and noise, and in heir assumpions abou he ime-series properies of a business cycle componen. Mechanical use of hese filers wihou careful consideraion of he characerisics of he paricular problem or seing may lead o inferior resuls for a leas wo reasons. Firs, he objecives of he exercise may vary. For example, in some cases he ideal resul will only conain variaion in cerain segmens of he frequency specrum, whereas in oher cases he ideal resul will conain variaion across he full specrum. Second, he consequences of applying a given filer may vary subsanially, depending on he ime series properies of he process o which i is applied. The effecs of various filers are ofen illusraed by ploing heir gains, which essenially amouns o showing he resuls of applying he filers o whie noise. Applicaion o oher processes may produce resuls ha are very differen, boh qualiaively and quaniaively. To address hese issues, his paper examines wo general approaches o he consrucion of business cycle componens. The approaches are based on ools derived as soluions o wo differen saisical problems: frequency exracion and signal exracion. A filer designed for one of hese purposes may or may no be accepable when applied o he oher. Moreover, he analysis shows ha he consequences of applying a given filer o processes oher han whie noise may be quie differen, wih possible uninended effecs. No one filer emerges as he bes soluion across he board, bu i is possible o derive cerain guidelines for he selecion of filers in various seings. As suggesed above, he crieria depend on he exac goals of he exercise and on he ime series properies of he variables involved. Secions and 3 describe he frequency and signal exracion problems, respecively, and show how various filers are soluions o hese problems. Secion 4 applies he filers o a range of processes o see how hey inerac. Secion 5 discusses he issue of rends in he daa. Secion 6 provides some empirical illusraions and Secion 7 offers some conclusions and guidelines.

5 . The frequency exracion problem. Objecive of frequency exracion Le y be a ime series, =,,T, such ha ( ) d L y = A( L) ε, () where d is a non-negaive ineger, AL ( ) al al = is a lag polynomial wih a j <, j= and ε is whie noise. Thus, ALε ( ) is he moving average Wold represenaion of a variancesaionary purely non-deerminisic random process. The objecive in frequency exracion problems is o esimae he componen of y ha flucuaes cyclically a frequencies in a range ha corresponds o some noion of he business cycle. For insance, we may hink of he business cycle flucuaions of a given variable as conaining componens wih a range of specific frequencies, say from o 8 years per cycle. Sargen (987) proposes ha his range consiss of he frequencies a which mos aggregaes have mos of heir specral power if hey have ypical specral shapes. Focusing on variaion a individual frequencies or ranges of frequencies is accomplished mos efficienly by operaing in he frequency domain, raher han in he ime domain. Time series variaion may be decomposed ino orhogonal componens corresponding o individual frequencies. Since he relevan funcions are periodic, we may resric aenion o frequencies ranging from π o π. Moreover, symmery permis focusing on frequency values in [0, π ]. For simpliciy, assume iniially ha we associae he business cycle wih frequencies { ω :0 ω ω π} < 0 and ha we ake lower frequencies o be associaed wih rends in frequency exracion problem is hen o reain only cycles of lengh π ω 0 or less wih y. The minimum possible disorion of he variabiliy of he included individual frequency componens. We consider hree alernaive approaches: frequency domain, Baxer-King, and Chrisiano- Fizgerald. Useful surveys of frequency domain or specral analysis echniques are Brillinger (98), Koopmans (995) and Sargen (987). In he lieraure, he business cycle is mos ofen associaed wih a range of frequencies ha is also bounded above by a frequency sricly less han π, in order o censor high frequency noise. See he discussion of band-pass filers in Secion.5. An excepion is he medium-erm business cycle as defined by Comin and Gerer (006).

6 . Frequency domain filer (FD) The FD filer is he ideal soluion o he frequency exracion problem and is defined as follows. For π ω π, le f ( ω ) be a funcion such ha 0 if ω < ω0 f ( ω) = if ω0 ω π () and le F( L) be is inverse Fourier ransform. The cyclical componen of c y is hen defined as = F( L) y. (3) I conains frequency componens only in he range ω0 ω π, wih he same weighs ha hose componens have in y. F( L) is he ime-domain represenaion of he high-pass frequency-domain filer f ( ω ), and is coefficiens are given by ω0 sin( ω0h) h h F( L) = ( L + L ). (4) π πh h= Raher han applying he ime domain filer (4), which in principle requires an infinie sample, a sraighforward way of esimaing and o calculae c is o ake he Fourier ransform of y, say y ( ω) c as he inverse Fourier ransform of he produc f ( ω) y ( ω). Noe ha his filer may be applied o inegraed processes because i annihilaes he specrum in a neighborhood of ω = 0, where he specrum is singular when d > 0..3 Baxer-King filer (BK) Baxer and King (999) propose a ime-domain approximaion o he frequency domain filer f ( ω ). They define an approximaion B( L ) ha is opimal in he sense ha i minimizes he equally-weighed average square modulus of he difference beween he frequency domain represenaion of he FD filer, f ( ω ), and he frequency domain represenaion of he approximaion, b( ω ). Specifically, he objecive is o minimize subjec o resricions ha () B( L ) be symmerical π f ( ω) b( ω) dω (5) π 3

7 and ha () is coefficiens sum o zero The soluion o he opimizaion problem is K h= ( ) h h B( L) = b0 + bh L + L (6) K B() = b + b = 0. (7) h= h b = ω π θ (8) b = sin( hω ) ( hπ) θ (9) h 0 for h =,, K, where θ is chosen o saisfy condiion (7). The cyclical componen of y is esimaed as cˆ = B( L) y. (0) The BK filer is opimal under he saed condiions in he sense ha i minimizes he mean squared error when he series y is whie noise. The wo resricions (6) and (7) imply ha he BK filer may be applied o inegraed series up o I() and sill produce saionary resuls, a feaure shared wih filers designed for signal exracion, as discussed in Secion 3 below. If he zero-sum resricion (7) is no imposed, θ = 0 and he coefficiens, h = 0,, K, are he same as in he ime domain represenaion of he ideal FD filer F( L). b h.4 Chrisiano-Fizgerald filer (CF) Chrisiano and Fizgerald (003) propose a filer ha solves an opimizaion problem similar o he one ha leads o he BK filer, wih a few noable differences. The objecive funcion for he CF filer is also a mean squared error, bu i differs from he one used by Baxer and King (999) in ha squared deviaions beween he approximae filer c( ω ) and he ideal filer f ( ω ) are weighed by s ( ω ), he specrum of y : yy π f ( ω) c( ω) syy ( ω) dω. () π The wo crieria are he same if y is whie noise, which has a consan specrum. The CF filer, however, is derived under he assumpion ha y follows a random walk. As in he BK filer, he CF ime domain coefficiens are consrained o sum o zero, so he filer can deal wih he single 4

8 uni roo implici in he random walk assumpion. In conras o he BK filer, he symmery resricion is no imposed on he coefficiens of y, =,,T, all of which may be nonzero. { A simple way o hink abou he calculaion of he CF filer is o exend he daa sample y, =,,T} infiniely in boh direcions by aking y = y for < and y = yt for > T. This exension is moivaed by he predicive properies of he random walk assumpion. The ideal weighs (4) are hen applied o he exended sample. I follows direcly from his descripion of he filer ha i is asympoically ideal in he sense ha i approaches he ideal filer F( L) as he sample size approaches infiniy in boh direcions..5 Band-pass filers The business cycle is frequenly associaed in he lieraure wih a range of frequencies ha is bounded boh below and above, in order o censor boh low frequency rends and high frequency noise. 3 The upper and lower bounds may be implemened using band-pass filers. Le Φ ( L) be a high-pass filer (FD, BK or CF), calibraed o reain frequencies ω and above. Then ω Φ ( L) Φ ( L) () ω ω 0 is a band-pass filer ha exracs (reains) frequencies ω in he range 0 < ω0 ω < ω < π. The bulk of he analysis ha follows focuses on high-pass raher han band-pass versions of filers for wo principal reasons. Firs, comparisons across filers are clearer if we focus on he effecs of a single applicaion of he high-pass filer raher han he wo applicaions implici in he band-pass. Second, he signal exracion filers of Secion 3 below do no have an explici band-pass form. They may be neverheless inerpreed as approximae high-pass filers, as is common in he lieraure. Thus, comparison of boh frequency exracion and signal exracion filers is beer carried ou by focusing on high-pass formas..6 Graphical comparisons We illusrae he relaive effecs of he filers discussed so far by presening he gain of each filer, ha is, he muliplier ha affecs he variabiliy of each frequency componen in he inerval [0, π ]. Figure shows he gains of he high-pass FD and BK filers for cycles up o 8 3 See, e.g., Engle (974) and Sargen (987). 5

9 years in lengh ( ω0 = π 3), assuming for simpliciy ha he sample is infinie. 4 ES and HP filers are discussed in Secion 3.. Figure. Gain of high-pass filers FD BK ES HP Cycles per period When examined in isolaion from he processes o which hey are applied, he filers seem relaively similar. The BK filer is leas accurae in he neighborhood of he sep a ω0 = π 3, bu i ends o dampen frequencies below ha level and is close o uniy for higher frequencies. Secion 4 will show ha looking a he filers in isolaion from he process o which hey are applied may underesimae he poenial for inaccuracy. Figure shows he gain for he band-pass filers corresponding o he frequency exracion problem, wih an upper frequency bound of ω = π 4 or four-quarer cycles. The qualiaive feaures are similar o Figure, wih he cavea ha he BK approximaion has o deal wih wo seps here insead of one. 4 Wih an infinie sample, he CF filer coincides wih he FD, as noed earlier, and is no shown separaely in he figure. 6

10 Figure. Gain of band-pass filers..0 FDBP BKBP Cycles per period 3. The signal exracion problem 3. General principles Le y be a ime series, =,,T, such ha y = g + c (3) where ( ) d L g = A( L) ε (4) Where ε and η are muually independen whie noise series, c = A( L) η (5) AL ( ) al al = is a lag polynomial wih j= a j <, and d is a posiive ineger. Noe ha he saionary series on he righ hand sides of (4) and (5) share he same lag polynomial, hough hey are driven by orhogonal whie noise processes. 7

11 In he frequency exracion problem, y was defined as an I(d) process. In he signal exracion problem, y has an I(d) componen g defined in he same way as before, bu here y has an addiional saionary componen c, which is he objec of invesigaion. is While (983, Secion 5.) shows ha he leas-squares esimae of he rend componen ˆ = y (6) + g ( ) d λ L ( L ) d and ha he corresponding esimae of he cyclical componen is hus cˆ d d ( L) ( L ) = d d λ + ( L) ( L ) y (7) where λ = σ σ. 5 As in he frequency exracion problem, he esimae c minimizes he mean η ε ˆ squared error, hough he characerisics of he benchmark c are differen here. 3. Exponenial smoohing filer (ES). When applying he filer d =, we obain from (7) an esimae cˆ = S( L) y of he cyclical componen by ( L)( L ) + L L ) SL ( ) = λ ( )( (8) which corresponds o exponenial smoohing (ES). The ES filer may also be obained by minimizing a funcion of he form ( y ˆ ) ( ˆ ) g + λ Δg (9) which penalizes boh deviaions of he esimaed rend from he observed value and changes in he rend. The second erm produces smoohness in he rend. See King and Rebelo (993). 3.3 Hodrick-Presco filer (HP). When applying he filer d =, we obain from (7) an esimae cˆ = H( L) y of he cyclical componen by 5 Similar derivaions, a various levels of generaliy, are provided in King and Rebelo (993), Harvey and Jaeger (993), and Harvey and Trimbur (003). 8

12 H( L) = ( L) ( L ) λ + ( L) ( L ) (0) which corresponds o he HP filer of Hodrick and Presco (997). The HP filer may also be obained by minimizing a funcion of he form ( ˆ ) ( ˆ ) y g + λ Δ g () which penalizes boh deviaions of he esimaed rend from he observed value and he second difference in he rend. The second erm produces more smoohness in he rend han he corresponding erm for he ES filer. See King and Rebelo (993), Hodrick and Presco (997) and Ehlgen (998). Noe ha he cyclical componen is defined here in a fundamenally differen way han in he frequency exracion problem, hough he language used in connecion wih boh ypes of filer may be similar. The objecive of he ES and HP filers is o exrac a saionary variable ha may conain a large amoun of low frequency variaion. For insance, when c AL ( ) =, he ideal resul is whie noise, which conains all frequencies wih equal weighs. Thus, in ha case, he filer would ideally give he same weigh o all frequencies, including low frequencies ha would be eliminaed as rends in he frequency exracion problem. In pracice, he ES and HP filers dampen low frequencies, bu no o he same exen as he ideal high-pass filer. Refer once again o Figure for illusraions of he gain of he ES and HP filers. 4. Effecs of filering on specific processes 4. Sraegy for comparing filers I was noed earlier ha comparison of he gains of he individual filers in isolaion may be misleading because he ineracion wih he specral characerisics of some processes may play a large role in he final resul. In his secion, he filers are applied o a range of specific processes ha vary as o wo aspecs: he lag polynomial A( L ) of he Wold represenaions of he saionary componens of he series and he degree of inegraion of he series. 4.. Wold represenaions For he Wold represenaions, we follow Ehlgen (998) in selecing he following four processes, which conain a reasonable degree of specral diversiy. 9

13 Whie noise, WN: AL ( ) = Moving average, MA(): A( L) = +.5L Auoregressive process, AR(,.5): ( ) A( L) =.5 L Persisen auoregressive process, AR(,.9): ( ) A( L) =.9 L 4.. Degree of inegraion By consrucion, he BK, ES and HP filers are applicable o I() processes, so we allow he degree of inegraion o be as high as. 6 In he conex of he frequency exracion problem, his calls for I(0), I() and I() cases. The variance of he innovaion is aken o be uniy: σ = in all cases. In he conex of he signal exracion process, here is also a saionary componen, and we denoe he corresponding processes as I()+I(0) and I()+I(0). The variance of he innovaion in he inegraed componen is aken o be uniy: σ = in all cases. The signal exracion seup also requires he specificaion of he parameer λ, which represens he raio of he variances of he wo noise processes. For he I()+I(0) case, we use λ = 600, which is he figure proposed by Hodrick and Presco (997) for quarerly daa. For he I()+I(0) case, we use ( ( )) λ = 300 cos π 6, which is he ES filer parameer value suggesed by King and Rebelo (993) so ha he gain a ω = π 3 (eigh year cycle) is he same as ha of he HP filer wih λ = 600. These five cases of degree of inegraion are combined wih each of he four Wold processes defined above o produce a oal of weny combinaions. The four filers FD, BK, ES and HP are hen applied o each of he combinaions. We firs illusrae a few cases graphically and hen examine all cases numerically o gauge he accuracy of he filers in various condiions. ε ε 4. Graphical illusraions In he graphical illusraions, we focus on he whie noise and AR(,.9) processes, which bes represen he range of resuls. The MA() and AR(,.5) specificaions are qualiaively 6 The HP filer may be applied o inegraed processes up o I(4) as well. 0

14 similar o he whie noise case in erms of heir low frequency performance, and all he filers end o perform fairly well a high frequencies. In general, he mos ineresing es case urns ou o be he persisen AR(,.9) process. In he illusraions based on he frequency exracion problem, he benchmark of performance is he ideal FD filer, which is shown in each figure as a solid line. Figure 3. Frequency exracion problem wih I(0) process Whie noise Cycles per period BK ES HP Benchmark AR(,.9) Cycles per period BK ES HP Benchmark Since he specrum of whie noise is consan, he op panel of Figure 3 is essenially he same as Figure, excep ha he verical axis is rescaled by a facor of π. The BK, ES and HP filer overesimae he low frequency componens below he 3 period frequency and end o underesimae above ha frequency. In general, however, he performance for all filers seems reasonably good. The characerisics of he AR(,.9) case in he lower panel are similar, bu he low frequencies play a larger role here and he overesimaion of low frequency rends seems o be more of a problem in relaive erms.

15 Figure 4. Frequency exracion problem wih I() process Whie noise Cycles per period BK ES HP Benchmark AR(,.9) BK ES HP Benchmark Cycles per period The I() processes in Figure 4 show he same low frequency problem as he AR case in he previous figure. The whie noise case, a random walk, is similar o he saionary bu persisen AR case. The AR case here is even worse han before, paricularly for he ES filer.

16 Figure 5. Frequency exracion problem wih I() process 5 Whie noise BK ES HP Benchmark Cycles per period 50 AR(,.9) BK ES HP Benchmark Cycles per period When he process is I(), Figure 5 shows ha all he approximae filers fail dramaically a low frequencies, even hough he are designed o exrac implicily up o wo uni roos. The BK and ES filers produce finie bu very large values a ω = 0. The HP filer, which can exrac up o four uni roos, sill produces a zero value a ω = 0, bu overesimaes subsanially oher low frequency values of he specrum up o he 3-period hreshold. 7 For cases based on he signal exracion problem, recall ha he objecive is o produce a cyclical componen corresponding o he paricular A( L ) or Wold represenaion, which in general conains non-zero low frequency componens. This benchmark process appears in he Figures 6 and 7 as a solid line. 7 Ehlgen (998) has examined a differen bu relaed aspec of he ineracion beween he HP filer and he various ime series processes. 3

17 Figure 6. Signal exracion problem wih I()+I(0) process 0.5 Whie noise FD BK ES HP Benchmark Cycles per period AR(,.9) Cycles per period FD BK ES HP Benchmark Wih an I()+I(0) process, all filers underesimae he low-frequency componens of he ( ) specrum. In Figure 6, he numerical illusraions use 300 cos( 6) λ = π, as in he ES filer. The low-frequency values produced by he BK, ES and HP filers, which were oo high in he frequency exracion problem are in his case no high enough. The FD filer, which assigns no weigh o low frequencies, fails dramaically here in ha range. We see ha he ES filer produces he bes low-frequency resuls, as expeced from heory. Resuls for he I()+I(0) processes are presened in Figure 7, wih λ = 600. The BK and ES filers have finie non-zero values a he zero frequency. For low frequencies more generally, he BK filer ends o approximae he benchmark, while he ES filer ends o overshoo on he posiive side. I is no alogeher clear from he graphics ha he HP filer is bes in his case, as heory suggess. The one-dimensional represenaion of he gain fails o capure he abiliy of he HP filer o srike a balance in is effecs on he wo componens of he I()+I(0) process. The appropriae relaionships emerge clearly when exac mean squared errors are calculaed numerically in he nex secion. 4

18 Figure 7. Signal exracion problem wih I()+I(0) process Whie noise Cycles per period FD BK ES HP Benchmark AR(,.9) Cycles per period FD BK ES HP Benchmark 4.3 Frequency domain errors (RMSEs) Visual represenaions are helpful, bu a quaniaive measure of goodness of fi allows for more precise comparisons of he accuracy of he esimaes produced by he filers. In each case, we use he roo mean square error of he esimae of he cyclical componen, which is equivalen o he opimizaion crieria proposed by Chrisiano and Fizgerald (003) and While (983). Le u ˆ = c c be he esimaion error, where he cyclical componen is esimaed by cˆ ˆ = F( L) y and ˆ ˆ i f( ω) = F( e ω ). The frequency domain represenaion of he RMSE is in he frequency exracion problem and π σ ( ) ˆ u = f ω f( ω) syy( ω) dω π () { ˆ( ) ( ) ˆ( ) ( )} π σu = f ω scc ω + f ω sgg ω d π ω. (3) 5

19 in he signal exracion problem. In he ables ha follow, RMSEs are scaled by π ( ) s ω d π σ ( ) c = cc ω, he volailiy of he benchmark for each problem. Filer rankings for a given process are no affeced, bu he inerpreaion of resuls expressed his way is scaleindependen. Table. Frequency exracion problem: Frequency domain roo-mean-squared errors I(0) I() I() WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) FD BK ES HP Resuls for he frequency exracion problem framework, presened in Table, exhibi some clear general paerns. The benchmark in he frequency domain problem is he ideal FD filer and, by definiion, i has he bes performance in Table. However, he able helps ascerain wheher he oher filers provide good approximaions and, if so, under wha circumsances. In paricular, wha is he second bes filer under each of he various condiions? Several paerns are manifes in he able. Firs, he degree of inegraion plays an imporan role in he accuracy of he approximae filers. The resuls for I() processes are markedly worse han in he saionary cases, and he I() figures are worse by an order of magniude. Clearly, he fac ha he approximae filers annihilae wo o four uni roos is no comfor as far as he accuracy of resuls in he frequency exracion problem is concerned. Among he four Wold specificaions, resuls for he approximae filers in he AR(,.9) case are clearly inferior o he ohers. Even when he process is I(0), he RMSEs are abou four imes as large as for WN. Since many economic series exhibi subsanial auocorrelaion, his paern suggess cauion when applying approximae filers o economic variables. As o he second bes filer, he resuls of he BK and HP filers are very similar for I(0) and I() processes, bu he BK has a sligh edge in hese cases. In he I() cases, he HP filer is clearly aided by is abiliy o deal wih up o four uni roos and is beer han he oher approximaions. The differences from he ideal filer, however, are sill quie large. Barring some 6

20 powerful reason o avoid frequency domain calculaions, he able suggess ha he FD filer is o be preferred. Table. Signal exracion problem: Frequency domain roo-mean-squared errors I()+I(0) I()+I(0) WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) FD BK ES HP In he signal exracion problem, he goal is o obain he saionary series corresponding o he given A( L ) specificaion. Thus, all four filers are approximaions and have posiive RMSEs. For each process, he mos accurae filer is known by consrucion, since he ES filer is derived o be opimal in he I()+I(0) cases, whereas he HP filer is opimal in he I()+I(0) cases. In conras o he frequency exracion problem, Table shows ha he FD filer is ouperformed here by all he ohers in he I()+I(0) case, and by all bu ES in he I()+I(0) case. Anoher conras wih he previous able is ha he resuls in Table are no very sensiive o he degree of inegraion of he series. Alhough all he cases examined conain an inegraed componen, hey all conain also an I(0) componen whose innovaion is more variable. This saionary componen is clearly very influenial in he comparaive resuls. As in Table, he AR(,.9) resuls are clearly worse han in he oher cases. 4.4 Volailiy disorion Anoher measure of he error involved in esimaing he cyclical componen is a possible disorion of he overall cyclical variabiliy of he process. Thus, we can look a he variance raio σ σ as an indicaor of his overall disorion in variabiliy, where is he benchmark cyclical ĉ c c process and cˆ is a paricular esimae. Resuls are shown in Tables 3 and 4 for he frequency and signal exracion problems, respecively. There are essenially wo ypes of siuaions, no unrelaed, in which he overall volailiy ends o be disored. One is in he frequency exracion problem when he filers are applied o 7

21 I() processes, as seen in he las four columns of Table 3. The resuls of applying he ES filer o he auoregressive I() processes are similar bu less pronounced. The oher siuaion is in he signal exracion problem when he Wold represenaion is AR(,.9). In hese cases, he esimaes fail o capure he large rend-like componens of he persisen AR(,.9) process and, wih one excepion, hey underesimae he volailiy by more han 30%. Table 3. Frequency exracion problem: Volailiy relaive o benchmark I(0) I() I() WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) FD BK ES HP Table 4. Signal exracion problem: Volailiy relaive o benchmark I()+I(0) I()+I(0) WN MA() AR(,.5) AR(,.9) WN MA() AR(,.5) AR(,.9) FD BK ES HP Differencing, over-differencing, and deerminisic rends 5. Pre-differencing inegraed processes As before, le y be a ime series, =,,T, such ha ( ) d L y = A( L) ε (4) Frequency exracion approaches generally work bes when he observable series is saionary. In principle, he FD filer annihilaes he specrum for frequencies in a neighborhood of zero. Similarly, he CF filer may be applied o I() processes, he BK and ES filers may be used wih I() processes, and he HP filer may even be applied o I(4) processes. However, he damping effecs of hese filers for low frequencies may be limied in pracical applicaions, and he reained low frequency componens may conain subsanial rend-like properies. Thus, a 8

22 sandard firs sep, paricularly if he filer is applied in he frequency domain, is o exrac all uni roos from y and o focus on he saionary variable ( ) d L y (5) whose specrum is finie as ω 0. The difference operaor Δ d ( L) = ( L) d (6) has he frequency domain represenaion and is gain is given by This gain is ploed in Figure 8 for d = and. i ( e ω ) d δ ( ω) = (7) d [ ] δ ( ω) = ( cos ω) d (8) d Figure 8. Effecs of firs and second differencing (gain) Firs Second Cycles per period 9

23 Noe ha he gain implies ha he filer dampens variaion in frequencies 0 ω < π 3, paricularly frequencies close o zero, and ha i amplifies variaion for π 3 < ω π. The frequency π 3 corresponds o 6 periods per cycle. Wih quarerly daa, 6 quarers per cycle is ofen aken as he high frequency bound of he business cycle specrum (see, eg, Baxer and King (999), Sock and Wason (999)). Wih monhly daa, 6 monhs is well ouside he normal range of business cycle frequencies. Hence, he operaor Δ d ( L) ends o amplify only frequencies ha are generally considered o correspond o shor-erm noise and ha are ofen censored in frequency domain analyses of business cycle flucuaions. applicaion of Clearly, appropriae applicaion of he differencing filer o variables wih uni roos (e.g., Δ d ( L) o he process y in (4)) produces saionary variables amenable o he applicaion of frequency exracion echniques. Overdifferencing (he applicaion of Δ d+ j ( L) wih j > 0 o y in (4)) in general produces saionary series, bu may lead o undesirable resuls. I may excessively dampen low frequencies and amplify high frequencies if he laer are reained. 5. Tesing for overdifferencing Granger and Haanaka (964) and Granger (966) idenify a ypical specral shape for economic variables. They refer o he fac ha he specrum of an economic series ends o be decreasing as a funcion of he frequency from 0 o π. The ypical specral shape is consisen wih he presence in many series of auoregressive feaures, wih real roos greaer han and possibly uni or near-uni roos. For example, for he AR() series x defined by wih 0< ρ < and ε whie noise, he specrum is given by In his case, s ( ω) < 0 for ω from 0 o π. xx ( ρlx ) = ε (9) ω = σ ρ ω+ρ (30) sxx ( ) ε ( cos ) However, if he firs difference operaor specrum is reversed. The specrum is hen Δ x is applied o his series, he slope of he 0

24 and s ( ω) > 0 for ω from 0 o π. ΔΔ x x s ΔΔ x x( ω) = σε ( cos ω) ( ρcos ω+ρ ) (3) Noe ha a decreasing specrum is no a feaure of every possible ARIMA specificaion. For insance, he specrum of x in (9) is increasing in ω if < ρ < 0. However, many sandard empirical specificaions share he ypical shape. Anoher example is an MA() process wih x = ( + al) ε (3) a > 0, which also has a monoonically decreasing specrum. This case is a bi differen from he AR() case in ha he specrum of Δ x is no monoonically increasing. Recall, however, ha he effec of Δ swiches from dampening o amplifying a ω = π 3. For he firs-differenced MA() series, he specrum is increasing on average, in he sense ha he average specrum for ω < π 3 is less han he average specrum for ω > π 3. More precisely, π 3 ( ) < ( ) π s 0 ΔΔ x x d s 3 x x ) d π ΔΔ (33) π 3 ( ω) ω π 3 ( ω ω Figure 9 illusraes he specra of he level and firs difference of he AR() and MA() processes wih ρ = a =.5.

25 Figure 9. Specrum of AR() and MA() processes and heir firs differences AR(,.5) Level Difference Cycles per period MA() Level Difference Cycles per period The foregoing paerns sugges a sraegy for esing for overdifferencing of economic series. If he series x has a bounded specrum ha is higher on average for 0 ω < π 3 han for π 3 < ω π, bu he relaive magniudes are reversed for Δ x, he shape of he resuling specrum is dominaed by he difference operaor raher han he original series, a sign of overdifferencing. This hypohesis may be esed empirically using specral mehods. Operaing in he frequency domain, we ake advanage of saisical sampling resuls available for specral funcions. Suppose he specrum of x, =,,T, is esimaed using he periodogram I xx j πt = iλ j T ( λ ) = xe, (34) where λ = π jt, j=,,t. Brillinger (98, Secion 5.) shows ha asympoically (as j T ) EI ( λ ) = s ( λ ) (35) xx j xx j

26 Cov I Var I ( λ ) = s ( λ ) (36) xx j xx j { xx j I xx k } Moreover, Brillinger (98, Theorem 5.6.3) implies ha ( λ ), ( λ ) = 0 for j k. (37) n j0 + n I xx( λ j) (38) j= j0 is asympoically normal when n as T. Le n be he ineger such ha λ n π 3 and λ > n + π 3. Then n S = Ixx( λ j) and n j= S T = Ixx( λ j) T (39) n j= n + are asympoically normal and independen wih By (35)-(37), we have also π 3 = ( π 3 ) ( ω) ω and ( ) ES s d 0 xx E S = π 3 s xx ( ω) dω (40) π π 3 n T = xx λ j + xx λ j n j= ( T n ) j= n + ( ) Var S S s ( ) s ( ) (4) Hence, he hypohesis 0 π 3 ( ) < ( ) π 0 xx π 3 xx (4) H : π 3 f ( ω) dω π 3 f ( ω) dω is rejeced wih confidence level α if S z = Var S S ( S ) > z (43) where z is asympoically sandard normal and N( z > z α ) = α (N is he sandard normal cdf). α 5.3 Deerminisic rends So far we have assumed ha here are no consan erms in he equaions for he observable non-observable variables, bu such consans are likely o appear in mos empirical applicaions. For insance, consider a simple form of equaion () in which An empirical model would include a consan α such ha ( ) L y d = and AL ( ) =. = α + ε (44) 3

27 Summing from = 0 we obain y y0 α ε j j= = + + (45) which clearly conains a deerminisic rend. Since he assumpion in frequency domain mehods is ha he series is purely non-deerminisic, his rend mus be exraced before applying specral mehods. If analysis of an empirical ime series as in equaion (44) suggess ha a firs difference filer should be applied, derending is accomplished simply by subracing he sample mean of ( L) y. In his case, preliminary exracion of he mean is helpful in obaining good empirical esimaes of he frequency domain properies of ime series, paricularly a low frequencies. This is paricularly clear when he specrum is esimaed using he periodogram, as described earlier. For a saionary variable wih mean x, he esimae of he periodogram a 0 ω = is ( T ) x π so ha his poin conains informaion abou he firs momen of he series, raher han he second. If i is imporan o work wih he I() process y direcly, derending may be accomplished by removing a linear rend from he series, ha is, by regressing y on a consan and, and using he residuals as an esimae of j= ε. This procedure is necessary for he j applicaion of he FD and ES filers o series ha exhibi significan linear rends. The BK and HP filers are consruced in ways ha incorporae he exracion of linear rends direcly ino he filer, so preliminary derending is no necessary. 6. Empirical illusraions 6. Applicaion of various filers o GDP and GDP deflaor A few pracical issues arise when applying he ime series filers o acual daa. Firs, i is normally convenien o exrac he mean from he raw series. The mean has a direc effec on he zero frequency componen of he periodogram, and also on oher low frequencies, paricularly if smoohing windows are used. Second, for series ha conain a linear rend, or a componen ha looks in pracice like a deerminisic linear rend, ha componen should be exraced as well. This sep is less imporan wih he BK and HP filers, which incorporae a leas wo levels of differencing and implicily 4

28 exrac linear erms. I is much more imporan when using he FD and ES filers, which do no auomaically pre-difference he daa. Third, should he daa be pre-differenced? This issue may be addressed by he compuaion of he z saisic defined in Secion 5. or by applicaion of uni roo ess in he ime domain, such as hose of Dickey and Fuller (979) or Phillips and Perron (988). The firs row of Table 5 shows he resuls of applying he z saisic es o log levels of GDP and he GDP deflaor. The negaive resuls indicae ha he specrum is generally decreasing, as in he ypical specral shape, wih significance a he 0% level for boh variables. The second row applies he same es o firs differences and shows very similar resuls, suggesing ha his degree of differencing is appropriae. 8 Applicaion of a second difference in he las row leads o large posiive values, suggesing ha he difference operaor dominaes he resuls and ha his sep may consiue over-differencing. 9 Table 5. GDP and GDP deflaor: z ess for over-differencing Quarerly daa, 959Q o 006Q GDP GDP deflaor z saisic p value z saisic p value Log level Firs difference Second difference The foregoing resuls indicae ha we should apply he filers o eiher log levels or firs differences of he variables. Figure 0 presens he applicaion of he filers o log levels of GDP. In he case of he FD and ES filers, a simple linear rend is exraced for reasons given earlier. The figure shows he inverse Fourier ransform afer he applicaion of each filer in he frequency domain. Resuls are fairly consisen across filers, hough he lack of differencing in 8 Using alernaive mehods, also in he frequency-domain, Müller and Wason (006) similarly conclude ha a uni roo model is ofen consisen wih he observed low-frequency variabiliy of weny U.S. macroeconomic and financial ime series. 9 Sandard uni roo ess lead o similar conclusions. Using a -es for log levels wih consan and rend, a uni roo canno be rejeced in eiher variable. Dickey-Fuller (979) p values (0 lags) are.39 for GDP and.000 for he deflaor, whereas and Phillips-Perron (988) p values (4 lags) are.8 and.998, respecively. A es for firs differences wih a consan erm rejecs a second uni roo wih p values of.000 and.06 (Dickey-Fuller) and.000 and.05 (Phillips-Perron). Compuaion of p values is as in MacKinnon (996). 5

29 he FD and ES filers produces some visible discrepancies, paricularly oward he ends of he sample. Figure 0. High-pass filers applied o GDP in log levels FD BK ES HP The same approach is applied o firs (log) differences of GDP in Figure. No prior derending is necessary in his case for he FD and ES filers. The homogeneiy of he resuls suggess ha all he filers produce very reasonable approximaions o he heoreical FD filer if he appropriae level of differencing is firs applied. 6

30 Figure. High-pass filers applied o GDP in firs differences FD BK ES HP Wih he GDP deflaor, he resuls are qualiaively similar, hough he sronger rends in his variable lead o greaer deviaions across filers. Because of he persisence of he series, resuls for FD and ES in Figure are quie clearly ouliers in he absence of any differencing. Once again, however, he applicaion o firs differences leads o fairly homogeneous resuls. Figure. High-pass filers applied o GDP deflaor in log levels FD BK ES HP

31 Figure 3. High-pass filers applied o GDP deflaor in firs differences FD BK ES HP The series in he foregoing figures are subsanially smooher han he raw daa, bu greaer smoohness consisen wih some noions of he business cycle may be achieved by censoring high-frequency movemens in a band-pass filer. As noed, band-pass filers are sraighforward exensions of he FD and BK high-pass filers. Figure 4 compares he FD versions of he band-pass and high-pass filers, boh applied o he firs difference of GDP. 8

32 Figure 4. Band-pass filer compared wih high-pass filer: FD filer ( o 8 years and less han 8 years) applied o GDP in firs differences 5 Band pass High pass Quaniaive measures of he feaures of he foregoing graphical illusraions may be obained by calculaing RMSEs of he resuling series, using he FD filer as a benchmark as in Secion 3. Table 6 confirms ha differences across filers are much larger for levels han for firs differenced series, hough given he visual resuls of Figures 0 and, he use of he FD filer as a benchmark mus be aken wih a grain of sal. The resuls for firs differences are more easily inerpreed. We see ha all he RMSEs are relaively small. The HP filer provides he bes approximaion for GDP bu he BK filer does slighly beer for he GDP deflaor. These differenial resuls are indicaive of he ineracions beween he filers and he process o which hey are applied. Table 6. RMSE relaive o FD filer: GDP and he GDP deflaor Quarerly daa, 959Q o 006Q GDP GDP deflaor Level Firs difference Level Firs difference FD BK* ES HP * Sample for BK is 96Q o 003Q, since K observaions mus be dropped a eiher end. 9

33 6. Correspondence of filered GDP o NBER recessions One possible benchmark for he business cycle componens derived from he various filers is how well hey mach he daing of recessions from he NBER. Figure 5 illusraes how differen filers have differen qualiaive characerisics in relaion o NBER recessions. Consider he applicaion of band-pass FD and BK filers o GDP. The FD filer is applied o firs differences o avoid disorions from he rend, whereas he BK filer is applied o log levels, since Baxer and King (999) show ha i can annihilae up o wo degrees of inegraion. The resuls are visually quie differen. The BK filer, implici differencing nowihsanding, produces a series ha has he flavor of a level, as far as recessions are concerned. Noe ha his series ends o peak before he sar of each shaded recession and fall sharply o a rough afer he end of each recession, suggesing he presence of a subsanial residual low-frequency componen (Cf., BK resuls in Figure 4). In conras, he FD filered series ends o be negaive during he course of each recession. 0 In addiion, volailiy seems oversaed, which Table 3 suggess is a feaure of BK wih highly-persisen processes. Figure 5. BK levels versus FD firs differences for GDP FD Diff. BK Level Murray (003) provides evidence ha he BK filer allows he firs difference of a sochasic rend o pass hrough wih U.S. real GDP. In he erminology of he presen paper, he BK filer applied o he log level is a subopimal soluion o he signal exracion problem. 30

34 Which represenaion is more accurae? One es is o include each filered series in a probi equaion in which he binary dependen variable is he recession indicaor. Table 7 provides he pseudo- R for each such experimen, using he high-pass versions of he four filers, as well as band-pass versions of he FD and BK filers. The filers are applied o boh levels and firs differences, and he unfilered series are included as well. In he unfilered, FD and ES cases, a simple linear rend is exraced from he levels. If he filers are applied o log levels of GDP, he HP filer produces he bes relaive fi, and he unfilered series is a disan las. However, he resuls in general sugges ha firsdifferencing is enirely appropriae, given he much more significan resuls obained. Wih firs differences, all he high-pass filers are inferior o he unfilered series. However, he band-pass versions of boh FD and BK are somewha beer han he unfilered series. The BK band-pass filer produces he bes resuls, hough noe ha he sample period is shorer by six years because of he need o drop observaions a boh ends. Esrella (998) shows ha his pseudo- R, in addiion o a measure of fi, is a monoonic ransformaion of he likelihood raio es saisic for exclusion of he single explanaory variable. If he sample period is held consan, he pseudo- R and he likelihood raio es produce he same rankings of models. 3

35 Table 7. Probi equaions for NBER recession indicaor: pseudo- R for filered GDP Quarerly daa, 959Q o 006Q Level Firs difference Unfilered* FD* BK** ES* HP FD band* BK band** * Derended level. ** Sample for BK is 96Q o 003Q, since K observaions mus be dropped a eiher end. 7. Conclusions This paper shows ha he feaures of individual ime-series filers designed o exrac business cycle flucuaions inerac sysemaically wih he characerisics of he processes o which hey are applied. The exac naure of his ineracion may no always be sraighforward and is implicaions may differ dramaically from illusraions based on applicaion o whie noise. In frequency exracion problems, he ideal soluion involves he applicaion of he FD filer o saionary daa. If he daa are in fac saionary, he BK, HP and ES filers also produce good resuls, hough hey are somewha less accurae. If he daa process is inegraed, all filers benefi from preliminary exracion of uni roos, even if he filers produce finie specra wihou differencing. The implici differencing incorporaed in he BK and HP filers helps dampen low frequency componens, bu he effecs of hese componens are no alogeher eliminaed and end o disor resuls when applied o highly persisen processes. Preliminary applicaion of he appropriae level of differencing o inegraed processes, wihou over-differencing, leads o fairly similar resuls across filers. The FD filer emerges as somewha preferable, however, paricularly on heoreical grounds. The FD and BK filers have he addiional advanage ha band-pass versions are easily compued, hough he laer has he drawback ha observaions are los a eiher end of he sample in eiher high- or band-pass versions. 3

36 In signal exracion problems, he ideal soluion differs sysemaically from ha of frequency exracion problems in ha i may include large low-frequency componens. In conras o he frequency exracion problem, he cyclical componen is always esimaed wih error, even asympoically. The ES and HP filers are he bes performers in he cases in which hey are heoreically opimal. In he signal exracion problem wih eiher I()+I(0) or I()+I(0) daa, he appropriae choice of hese wo filers is he bes course of acion. If i is unclear wheher he rend componen is I() or I(), he HP is he safer choice. Errors wih he HP filer in he I()+I(0) case are relaively moderae, whereas errors wih ES in he I()+I(0) case are he larges of any filer, even hose no expressly designed for signal exracion. In general, imporan differences beween he frequency and signal exracion problems and he diverse ineracions beween filers and processes sugges ha filers mus be carefully seleced for any paricular applicaion. No single mehod can accommodae all circumsances well. References Baxer, Marianne and Rober G. King, 999, Measuring business cycles: Approximae band-pass filers for economic ime series, The Review of Economics and Saisics 8: Brillinger, David R., 98, Time series: Daa analysis and heory, San Francisco: Holden-Day. Chrisiano, Lawrence J. and Terry J. Fizgerald, 003, The band pass filer, Inernaional Economic Review 44: Comin, Diego and Mark Gerler, 006, Medium-erm business cycles, American Economic Review 96: Dickey, David A. and Wayne A. Fuller, 979, Disribuion of he Esimaors for Auoregressive Time Series Wih a Uni Roo, Journal of he American Saisical Associaion 74: Ehlgen, Jürgen, 998, Disorionary effecs of he opimal Hodrick-Presco filer, Economics Leers 6: Engle, Rober F., 974, Band specrum regression, Inernaional Economic Review 5: -. Esrella, Aruro, 998, A new measure of fi for equaions wih dichoomous dependen variables, Journal of Economic and Business Saisics 6:

37 Granger, C.W.J., 966, The ypical specral shape of an economic variable, Economerica 34: Granger, C.W.J. and M. Haanaka, 964, Specral analysis of economic ime series, Princeon: Princeon Universiy Press. Harvey, Andrew C. and Thomas M. Trimbur, 003, General model-based filers for exracing cycles and rends in economic ime series, The Review of Economics and Saisics 85: Harvey, Andrew C. and A. Jaeger, 993, Derending, sylized facs and he business cycle, Journal of Applied Economerics 8: Hodrick, Rober J. and Edward C. Presco, 997, Poswar U.S. business cycles: An empirical inesigaion, Journal of Money, Credi, and Banking 9: -6. King, Rober G. and Sergio T. Rebelo, 993, Low frequency filering and real business cycles, Journal of Economic Dynamics and Conrol 7: Koopmans, Lamber H., 998, The specral analysis of ime series, San Diego: Academic Press. MacKinnon, James G., 996, Numerical disribuion funcions for uni roo and coinegraion ess, Journal of Applied Economerics : Müller, Ulrich K. and Mark W. Wason, 006, Tesing models of low-frequency variabiliy, NBER Working Paper No. 67. Murray, Chrisian J., 003, Cyclical properies of Baxer-King filered ime series, The Review of Economics and Saisics 85: Phillips, Peer C.B. and Pierre Perron, 988, Tesing for a Uni Roo in Time Series Regression, Biomerika 75: Sargen, Thomas J., 987, Macroeconomic heory, nd ediion, Boson: Academic Press. Sock, James H. and Mark W. Wason, 999, Business cycle flucuaions in U.S. macroeconomic ime series, in John B. Taylor and Michael Woodford, eds, Handbook of Macroeconomics, vol. A, -64. While, Peer, 983, Predicion and regulaion by linear leas-square mehods, nd ediion, Minneapolis: Universiy of Minnesoa Press. 34