A Modification of the HP Filter. Aiming at Reducing the EndPoint Bias


 Beryl Merritt
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1 Edenösssche Fnanzveralun, Bundesasse 3, CH33 Bern Admnsraon fédérale des fnances, Bundesasse 3, CH33 Berne Ammnsrazone federale delle fnanze, Bundesasse 3, CH33 Berna Sss Federal Fnance Admnsraon, Bundesasse 3, CH33 Bern Doc. o.: ÖT/3/3 Auhor: P.A. Bruchez A Modfcaon of he Fler Amn a Reducn he EndPon Bas Workn Paper 8 Auus 3 The ork of he FFA roup of economc advsors does no necessarl reflec he offcal poson of he Offce or Federal Deparmen or ha of he Federal Councl. The auhors are responsble for he assumpons and an errors hch ma be conaned n he ork. F:\WEBTEAM EFV\Exern\Modfed fler enlsh.doc
2   Absrac I s ell knon ha he HodrckPresco () fler suffers from an endpon bas. Ths s problemac hen he fler s used recursvel for economc polc (n hs case he endpon s he pon of neres). The usual a o deal h hs problem s o exend he seres h ARIMA forecass. Hoever, he usefulness of hs approach s lmed b he qual of he forecas. Ths s h he presen paper explores an alernave a o deal h he endpon bas hch does no use forecass: he penal funcon ha he fler mnmzes s modfed n order o reduce (hoever, hou elmnan) he dfference n reamen of he endpon compared o oher pons; hs elds a modfed fler. I s shon ha (compared o he usual fler) he modfed fler (recursvel appled) has he follon properes. ) The endpon bas s reduced. ) The busness ccle componen s ber; he modfed fler s n fac approxmael equvalen o anoher modfcaon of he fler: ncreasn he ap beeen rend and daa b a consan (43% hen he smoohn parameer s ). ) The amplude response of he modfed fler s closer o he one of he deal fler, bu here s a phase shf.
3 Inroducon The HodrckPresco () fler s ofen used o separae srucural and cclcal componens, or o smooh a curve (hou smoohn o he pon ha becomes a srah lne). Hoever, hs fler has been crczed on several rounds, n parcular because of he «endpon bas»: he las pon of he seres has an exaeraed mpac on he rend a he end of he seres. If one s onl neresed n he properes of he ccle, hs s no ha bad: one smpl has o om he rend values a he end of he seres. Bu f he rend s used for economc polc, hen he las pon s lkel o be he one hch s parcularl neresn. The usual a o solve hs endpon bas problem s o exend he seres (for example h ARIMA forecass 3 ). Thus he neresn pon s no loner a he end of he seres. The usefulness of hs approach s lmed, hoever, b he qual of he forecas 4. We propose here an alernave hch s robus n he sense ha does no use forecass. I s a smple and naural modfcaon of he fler. Hodrck, R.J. and E.C. Presco (997), «Posar US Busness Ccles: An Emprcal Invesaon», Journal of Mone, Cred and Bankn, 9, 6. Kaser R. & A. Maravall (), «Measurn Busness Ccles n Economc Tme Seres», Lecure oes n Sascs, Sprner. 3 A sophscaed exrapolaon process lke he ARIMA forecas has he advanae of leavn no dscreonar choce (or onl a he bennn). I has hoever he dsadvanae of no akn no accoun unexpeced announcemens (hch have an mpac on he fuure bu are no aken no accoun n pas daa). Ths s h an exper forecas mh be preferred o a pure ARIMA forecas. 4 If he seres s exended b four ears forecass ( o 4), and f here s an error n he level of hese four forecass, hen (assumn ha he error s he same for all four forecass) can be shon ha more han 4% of hs error ll pcall ranslae no error of he rend compued for me (usn h smoohn parameer ). Some addonal problems ma appear f exper forecass are used nsead of ARIMA forecas. Exper forecass of GDP roh rae for example mh be based oard he mean roh rae. Ths mples ha forecas errors mh ssemacall depend on busness ccle condons (ben oo pessmsc n boom mes and oo opmsc n recessons). I mh no be possble o correc hs ssemac error n real me because mh be dffcul o assess he busness ccle condons (or because one s compun he rend precsel n order o assess he busness ccle condons). Anoher problem h exper forecass s ha he leave a lo of freedom abou he choce of a crucal value: ho quckl he econom reurns o normal afer a shock.
4  4  Secon explans h he fler has an endpon bas, Secon 3 proposes a modfcaon of he fler, Secon 4 descrbes some properes of he modfed fler, and lasl Secon 5 concludes.. Wh does he fler have an endpon bas? The fler defnes he rend such as o mnmze he follon penal funcon: T T ( ) [( T λ T ) ( )] () here s he value a me of he varable for hch e an o compue he rend. values of hs varable are knon up o pon T n me. If here ere onl he frs erm, he soluon ould be a all me and here ould be no cclcal componen. Bu he second erm mposes a penal on chanes n he rend s slope. If λ ere nfne, no chane n he rend s slope ould be alloed, and he rend ould be a srah lne (he same as he ordnar leas squares reresson lne, here me ould be he ndependen varable). The ber λ s, he smooher he rend. Ths s h λ s called he «smoohn parameer». I can be seen a hs pon ha he fler suffers from an endpon bas. The second erm of he penal funcon feaures a sum from T o T and no from T o T as n he frs erm (f he sum n he second erm ere aken from T o T here ould be more unknon han daa ). The consequence s ha T, T, and do no appear n he second erm T T of () as ofen as he oher : and appear onl one me, T T T and T appear o mes, hle all oher appear hree mes. Thus, for T, T, T and T he penal for a rend knk s loer han ould be f all ere reaed equall. When choosn T, T, T and T o mnmze he penal funcon, larer rend knks ll be alloed a he ends of he seres
5  5  han ould be he case hou ha bas, resuln n daa a he ends of he seres havn an exaeraed mpac on he rend. 3. Proposon o modf he fler The prevous commens sues an eas a o modf he fler n order o reduce he endpon bas: all should appear n he same measure n he second erm of he penal funcon hch penalzes chanes n he rend s slope. Ths can be done b mnmzn he follon penal funcon nsead of (): T T ( ~ ) T λ T [( ~ ~ ) ( ~ ~ )] () here λ λ for T3 o T λ λ 3/ λ λ 3 for T and T for T and T The dea s o compensae for he fac ha for ceran values of, ~ appears less ofen n he second erm of he penal funcon, b ncreasn he correspondn value of λ (for example ~ T appears onl once nsead of hree mes, hus s λ T s hree mes he usual λ). Ths modfcaon s no enouh o make all ~ ener smmercall no he penal funcon, bu makes more smmercal n he sense ha a chane n he rend s slope alas coss hree penal erms. Thus does no compleel solve he endpon bas, bu reduces. Ths modfcaon s que naural and does no lead o a suaon n hch he λs near he end of he seres could be se a dscreon. Bu he proof of he puddn s n he ean, and he man arumen n favor of he modfed fler s s properes.
6 Some properes of he modfed fler Appled recursvel, he modfed fler has he hree follon advanaes compared o he usual fler: he endpon bas s reduced ( 4.), he busness ccle componen (dfference beeen acual daa and rend) s ber ( 4.), ccles a busness ccles frequences are elmnaed from he rend o a reaer exen ( 4.3). 4.. The endpon bas s reduced The rend can be expressed as a lnear combnaon of he daa of he nal seres: (3) T The follon raph shos he ehs (vercal axs) for all (horzonal axs) from o  (for λ and ). Weh of acual daa a me T for compuaon of he rend a me T (lambda, nerval lenh ) modfed We can see ha a revson of he daa a me T ll have less effec on he rend a me T h he modfed fler han h he usual fler. Ths
7  7  reduces he endpon bas 5. The mpac s he same for he daa a T, and hen reaer for several ears 6. The fac ha he modfed fler ves more eh han he fler o he daa of he pas s he counerpar o he fac ha he las pon has less mpac on he rend. A monoone curve for he ehs of he modfed fler mh have been more sasfn han hs shape 7 h s maxmum for daa a . Hoever, hs maxmum has he desrable proper of ben belo he maxmum of he ehs. The prevous raph s made for recursve applcaon of he flers on an nerval of lenh, bu hen λ he lenh of hs nerval s no mporan so lon as s reaer han. For he modfed fler hs can be seen on he follon raph (he usual fler has he same proper), hch shos he ehs for he modfed fler appled o nervals of varous lenhs. 5 The endpon problem s reduced n he sense ha he excess eh of he las daa s reduced. Hoever, does no follo from hs ha he dfference beeen he expos and recursve rend s necessarl smaller h he modfed fler han h he usual. I am ndebed o Yvan Lenler ho made me aare of hs pon (he shoed an example h US daa n hch he dfference beeen recursve and expos rend s larer h he modfed fler). 6 The sum of he ehs s equal o, bu some ehs are neave. Ths s rue for he fler as ell as for he modfed fler. Ths ould also be rue for an ordnar leas square reresson of he daa on me. The smples a o undersand h some ehs are neave s o consder an ordnar leas square reresson (h me as ndependen varable) n a case n hch he slope of he reresson lne s posve and he scaer conans man pons. An ncrease (small enouh such ha he cener of rav of he scaer, hrouh hch he reresson lne mus pass, be approxmael consan) of he daa near he end of he me reresson nerval ll ncrease he rend a he end of he nerval, hle an ncrease of he daa near he bennn of he nerval ll decrease he rend a he end of he nerval (because ll reduce he slope of he reresson lne). Thus, s usfed ha some ehs are neave. The fac ha he neave ehs are more neave h he modfed should no a pror be seen as an undesrable feaure of hs fler. 7 Hoever, hs shape mh be apprecaed for praccal reasons no dscussed here: old daa should have lle eh because he have lle mpac on curren daa, and ver recen daa should have less eh because he mh be revsed; hs leads o a shape feaurn a maxmum.
8  8  Wehs for compuaon of he rend a he end of he seres, modfed () appled o an nerval of lenh We can see ha hese ehs are almos dencal for and (n fac he o curves are so smlar ha he canno be dsnushed over he area n hch he are boh defned). 4.. The busness ccle componen s ber The follon raph shos an example of daa (acuall hs daa s a real GDP seres) and her rend compued recursvel h he fler and he modfed fler (he rend a me T s compued b appln he fler o an nerval of lenh fnshn a T).
9  9  acual daa () () modfed T Flers recursvel appled on an nerval of lenh We can see ha accordn o he fler he rend s (almos) alas suaed beeen he acual daa curve and he rend accordn o he modfed fler. Ths means ha compared o he fler, he modfed fler arbues a reaer poron of he flucuaons o he busness ccle. The fac ha he daa, he rend accordn o he fler, and he rend accordn o he modfed fler nersec a he same pon ma appear surprsn. In fac, he do no cross precsel a he same pon, bu nearl. The nuve explanaon of hs phenomenon s smple. Suppose ha a a ceran dae he rend accordn o he modfed fler s equal o he acual daa. Ths mples ha n he penal funcon he value of /λ T s no relevan because s mulpled b ( T T ) hch s null. The fac ha he value of λ a me T has been rpled (n comparson o he usual fler) has no effec. Furhermore, T T mples ha [( T T ) ( T T )]. Indeed, f [( modf T n order o reduce he value of hs expresson, even f ha means T T ) ( T T )] ere dfferen from, ould be preferable o ncreasn ( T T ) : he dervave of ( T T ) h respec o T ben null
10   hen ), ould no cos much o ncrease hs value slhl. Bu f ( T T [( T T ) ( T T )], ha means ha havn mulpled λ b 3/ a me T had relavel lle mpac. Indeed, f one exracs he erms of he penal funcon conann T, one obans (afer havn mulpled hem b λ T ): {[( ) ( )] [( ) ( } ( T T ) λ T T T T T T T T 3)] The fac ha b T [( T T ) ( T T )] makes he erm hch s mulpled λ dmnsh oard zero, and hus reduces he mpac of λ T, and hereb he effec of havn mulpled hs coeffcen b 3/. Concluson: he modfed fler dffers from he usual fler b he values of λ a T and a T, bu f T T hen he value of λ a pon T has no effec and he value of λ a pon T has less effec han usual. So, f a a ven dae he rend accordn o he modfed fler s equal o he acual daa, s normal ha he rend accordn o he usual fler also be approxmael (bu no exacl) equal o he acual daa. I s shon n he appendx ha s possble o be more precse han us san ha he busness ccle componen s larer h he modfed fler: s 43% larer (for λ) Busness ccle flucuaons are aenuaed o a reaer exen (bu here s a phase shf) The follon raph shos he rend of an arfcal seres, sn(), compued h he fler as ell as he rend obaned h he modfed fler. The rend compued h he fler appled expos o he enre seres s also shon (hs rend s no dran for he las four ears of he seres).
11 acual daa: snus h perodc 6.8 () appled recursvel () appled expos () modfed recursvel appled We can see ha he amplude and he perodc of he oscllaons are approxmael he same for he fler and for he modfed fler. The boh aenuae hese oscllaons, as desred snce a ccle over a sxear perod can be consdered o be a busness ccle (remember ha hese flers compue a rend hch should no conan an busness ccle flucuaons). The rend accordn o he modfed fler s slhl ou of phase h respec o he fler (he dfference s small compared o he perodc). Inuvel, hs reflecs he fac ha he modfed fler ves less eh o he endpon of he seres (ha as he oal because hs eh s exaeraed n he fler) and herefore more eh o he pas daa. Some nera resuls from hs 8. Wha happens f e ake a snus for anoher perod? The follon raph provdes an example compued h a snus h a loner perodc. 8 There s a radeoff beeen connun o reduce a snus of a ven perod b he same amoun, avodn phase shf and elmnan he endpon bas. See Schps Bernard, «Ene Anmerkunen zur Sasonberenun von Zerehen», KOF Konunkur Berch /3.
12 acual daa: snus h perodc () appled recursvel () appled expos () modfed recursvel appled Here he rend accordn o he modfed fler s sll ou of phase relave o he fler, and he ampludes are smlar. The flucuaons are no aenuaed. Ths s correc snce a ccle over a enear perod s no a busness ccle. The ampludes can dffer o a reaer exen f he perodc s eaker, as he follon raph shos:..5. acual daa: snus h perodc 3 () appled recursvel () appled expos () modfed recursvel appled Therefore, becomes clear ha he modfed fler has he advanae of aenuan hh frequenc flucuaons o a reaer exen han he usual fler. The amplude accordn o he modfed fler becomes closer o he one ha ould have been obaned b appln he usual fler expos. Ths
13  3  hoever as no he case for oscllaons of lare perodc. Thus he usual fler appled recursvel can be closer o he expos fler han he modfed fler (also appled recursvel) f he daa conan enouh oscllaons of lare perodc, hch can be he case for nonsaonar daa. If s possble o make accurae forecass, s preferable o exend he seres h hese forecass raher han usn he modfed fler: n hs a e e closer o rend accordn o he usual fler expos. Hoever, he modfed fler has he advanae of ben robus n he sense ha does no depend upon forecass and herefore does no depend upon her qual. Follon he specral analss approach e can compue analcall he mpac of he o flers on sn(*π/τ), a snus of perod τ and frequenc π/ τ. Le be a lnear fler here are he ehs (remember ha boh he usual and he modfed are flers of hs pe). Appln hs fler o he snus ll ve he follon rend: sn[(  ) * π/ τ ]. Usn he den sn(θϕ)sn(θ)cos(ϕ)cos(θ)sn(ϕ), hs can be ren: sn( * π / τ )[ cos(* π/ τ )] cos( * π/ τ )[ sn(* π/ τ )] (4) sn( * π / τ Ω) [ cos(* π/ τ )] [ sn(* π/ τ )] sn(* π/ τ ) here Ω arc( ) cos(* π/ τ )] Ths s a snusod of amplude [ cos(* π / τ )] [ sn(* π/ τ )] and phase shf Ω.
14  4  The ornal snus had amplude of, hus hs formula ves he amplude response of he fler. Snce e kno he values of he ehs for he usual and he modfed flers, e can compue he amplude response for varous frequences. For he, he modfed and an deal fler (defned as one hch elmnaes all ccles of perod equal or smaller han 8 ears, and leaves he oher ccles unchaned), he follon raph shos he amplude response on he vercal axs as a funcon of he frequenc (expressed as a fracon of π, hus equal o /τ ) ndcaed on he horzonal axs (e do no consder perods smaller han he me un). Amplude response / Perod modfed Ideal The deal fler keeps onl lo frequenc oscllaons (as ell as ver hh frequenc oscllaons hch are close enouh o he me un ha he are observaonall equvalen o lo frequenc oscllaons). The amplude response of he modfed fler s ofen closer o he deal han he one of he fler (he fac ha he hree curves cross a he same pon s a desrable, bu unexpeced, feaure). The follon raph shos he phase shf 9 (he opmal phase shf s null): 9 I s ell knon ha he raphs for he amplude response and he phase shf feaure a smmer: he amplude response for /τ s he same as for /τ, and he phase shf s he
15  5  Phase shf (as a fracon of p) / Perod modfed Ideal The phase shf s usuall orse for he modfed fler. Luckl, he ncrease of phase shf s he lares for frequences hch are sronl aenuaed: Phase shf modfed Ideal Amplude response 5. Concluson A modfed verson of he fler s proposed hch makes possble o reduce he endpon bas a he cos of a phase shf. The usual approach conssn n exendn he seres h forecas values s preferable hen hese forecass are oppose. To see hs, plu /τ n place of /τ no equaon (4) and noce ha Σ cos(π( /τ)) Σ cos(π/τ) and Σ sn(π(/τ)) Σ sn(π/τ).
16  6  accurae. Hoever, he approach presened here has he advanae of ben more robus n he sense ha s ndependen of he qual of he forecass. I mh be parcularl useful for recursve applcaon on earl GDP daa snce he GDP forecas for several ears ahead s unlkel o be accurae.
17  7  Appendx Equaon (3) can be reren as: ( ) Thus ( ) [ ] ~ here ~ (4) oce ha ~ The las equal follos from he assumpon ha he fler leaves a srah lne ab unchaned. ( ) ( ) ( ) ( ) b b a b b a b a b a Thus
18  8  Ths means ha he dfference beeen he acual daa and he rend can be ren as a lnear combnaon of he ncreases n from one perod o he nex. The follon raph shos he ehs ~ ωlda (for lambda and nerval lenh ) modfed Snce ~ s posve for <4 e can re (for and modfed ): ( ) [ ] ( ) [ ] ( ) ( ) ~ ~ ~ ~ ~ ~ ~ Ths means ha he dfference beeen he acual daa and he rend s proporonal o he dfference beeen a ehed averae (posve ehs, sum equal o ) of he earl (assumn ha he me un s he ear) chanes of he four mos recen daes, and a ehed averae (posve or ver close o, sum) of he prevous earl chanes. Thus, he rend ll be belo he acual daa f he recen earl chanes of he daa are on (ehed) averae larer han chanes n prevous ears.
19  9  These ehs ~ 3 ~ for o 3 and ~ 4 ~ for 4 o  are farl smlar (alhouh no exacl equal) for he fler and he modfed fler: modfed In frs approxmaon, he maor dfference beeen he fler and he modfed fler s he proporonal coeffcen ~ hch s.64 for he modfed and.4 for. Ths means ha n frs approxmaon he man dfference beeen () and modfed () s ha he ap  s 43% larer for he modfed. Ths confrms our resul of secon 4., h he addonal nformaon ha he busness ccle componen s n frs approxmaon alas 43% larer for he modfed. 3 M M The approxmaon W ( ) * ( ) 3 ~ 3 ~ α h α.43 can be used o derve some properes of he modfed fler. For example mples ha α M α
20   Thus he rend accordn o s a ehed averae of he acual daa and he rend accordn o he modfed. Furher properes can be compued. For example, he specral properes of he modfed fler can be compued as a funcon of he specral properes of he fler: If e π/τ hen A e π ϕ τ M αa α e ϕ [ ] ( ) τ τ ϕ α αa e ( α ) e αa e ( α ) ( α ) e π ϕ π ϕ ar( αa e τ ( α )) π e π τ Thus, A M ( ϕ M [ αa cos( ϕ ) ( α )] [ αa sn( ϕ )] ( α ) ( α ) αa cos( ϕ ) ( αa ) αa sn( ϕ ) ) αa cos( ϕ ) ( α ) The follon raph presens M compued h α. 43, and from n he complex space: modfed alpha We see ha he specral properes of he fler compued from correced h α.43 are usuall ver close o he one of he modfed.
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