Can we ue eaonally adjued varable n dynamc facor model? Maxmo Camacho + Unverdad de Murca mcamacho@um.e Yulya ovcha Unverdad Rovra--Vrgl yulya.lovcha@gmal.com Gabrel Perez Quro Banco de Epaña and CEPR gabrel.perez@bde.e Abrac We examne he hor-erm performance of wo alernave approache of forecang from dynamc facor model. The fr approach exrac he eaonal componen of he ndvdual varable before emang he model, whle he alernave ue he non eaonally adjued daa n a model ha endogenouly accoun for eaonal adjumen. Our Mone Carlo analy reveal ha he performance of he former alway comparable o or even beer han ha of he laer n all he mulaed cenaro. Our reul have mporan mplcaon for he facor model leraure becaue hey how he ha he common pracce of ung eaonally adjued daa n h ype of model very accurae n erm of forecang ably. Ung fve concden ndcaor, we llurae h reul for US daa. Keyword: Dynamc facor model, eaonal adjumen, hor-erm forecang JE Clafcaon: E3, C, E7. Maxmo Camacho hank CICYT for uppor hrough gran ECO-983 and ECO3-45698. The vew n h paper are hoe of he auhor and do no repreen he vew of he Bank of Span or he Euroyem. + Correpondng Auhor: Unverdad de Murca, Faculad de Economa y Emprea, Deparameno de Meodo Cuanavo para la Economa y la Emprea, 3, Murca, Span. E-mal: mcamacho@um.e.
. Inroducon The lae- receon, omeme referred o a he Grea Receon, magnfed he nere of economc agen n havng effcen hor-erm forecang model ha help monor ongong economc developmen. Th could explan he recen reurgence of dynamc facor model, fr developed by Sock and Waon (99), whch have proven o be ueful n growh and nflaon forecang. Among oher, recen example are Aruoba, Debold and Sco (9), Aruoba and Debold () and Camacho and Perez-Quro (). To our knowledge, all of he forecang analye developed n h relaed leraure ue eaonally adjued daa, where he eaonal componen are exraced ndvdually from each varable eher by he offcal acal offce ha publh he daa or by he analy (when eaonally adjued daa are no avalable) before emang he model. Therefore, only one common facor and everal doyncrac componen are emaed n hee dynamc facor model. We wll call h approach radonal, becaue he andard procedure n he leraure. Th radonal approach ha ome lmaon. Fr, behnd he ndvdual eaonal adjumen here ex he mplc aumpon ha he eaonal componen for each varable necearly doyncrac (no common). Second, removng he eaonal componen from he ndvdual varable before emang he model may lead o loe of nformaon abou he eaonal componen ha could poenally be ueful for forecang. A an alernave o h radonal approach, he rucural dynamc facor model have he advanage of beng formulaed n erm of common componen, uch a rend, eaonal componen and cycle ha have a drec nerpreaon. Modellng hee feaure nde he model could be of grea benef nce hey could be ealy projeced no he fuure, leadng o poenal forecang mprovemen. Th paper am o evaluae he performance of radonal veru rucural facor model. We ue a Mone Carlo exerce o how ha when he daa generang proce exhb doyncrac eaonal componen he radonal dynamc facor model ha ue eaonally adjued daa (he oucome of TRAMO-SEATS) ouperform he rucural dynamc facor model, epecally when he doyncrac eaonal componen are erroneouly Noe ha he leraure on large-cale dynamc facor model, whch nclude a va number of ndcaor, alo ue eaonally adjued daa. Alhough our reul can be exended o large-cale model, we focu on mallcale model for he ake of mplcy. In addon, Bovn and Ng (6), Poncela and Ruz () and Banbura and Modugno (4) recenly how ha large pecfcaon could perform wore han mall pecfcaon due o dffcule n exracng a relevan gnal n he preence of ndcaor of dfferen qualy.
modelled a f hey were common acro ere. Inerengly, when he daa are generaed wh common eaonal componen, he performance of radonal facor model ll comparable o or even beer han ha of rucural facor model, even n he cae ha he eaonal componen are correcly modelled a common acro he ere. One poenal explanaon ha f eaonaly doyncrac, he common eaonaly model would be clearly mpecfed; f here common eaonaly, exracng doyncrac eaonal erm neffcen and uffer from he cure of dmenonaly. The radonal approach apparenly provde he be of boh world: no makng ncorrec aumpon abou common eaonaly, whle keepng a lmed number of parameer o emae. Thee reul have mporan mplcaon for he leraure on facor model nce hey how he good forecang performance of he andard model ha ue eaonally adjued daa wh repec o alernave model ha handle eaonally adjumen endogenouly. The reul obaned n he Mone Carlo analy are confrmed by ung a e of fve concden US economc ndcaor. Our emprcal reul alo ugge ha he andard raegy of forecang from dynamc facor model ha ue eaonally adjued daa he mo advable way o compue he foreca. The paper rucured a follow. Secon decrbe he man feaure of rucural and radonal dynamc facor model. Secon 3 oulne he Mone Carlo mulaon and dcue he reul. Secon 4 addree he emprcal analy. Secon 5 conclude.. Mehodologcal framework.. Srucural facor decompoon The aonary economc varable are aumed o adm a rucural facor decompoon. 3 Therefore, each of he N aonary varable, y, can be wren a he um of hree ochac componen: a common componen, f, whch repreen he overall bune cycle condon; an doyncrac componen, u, whch refer o he parcular dynamc of he ere; and a eaonal componen,, whch refer o he perodc paern and are allowed o be eher We ue he TRAMO-SEATS veron daed March,, a downloaded from he Bank of Span daabae. Alernave fler a X-, X-, and ARIMA model would lead o qualavely mlar reul. 3 We focu he analy on aonary varable. The updae of Aruoba Debold and Sco (9) howed ha modelng he ochac rend were very dapponng nce he growh rae ranformaon faclae beer handlng of benchmark revon, whch ypcally affec level more han growh rae. 3
doyncrac or common. Accordng o h decompoon, he rucural dynamc facor model can be aed a where =,,N, and he are he loadng facor. y f u, () We aume he followng dynamc pecfcaon for he hree componen. The common componen and he doyncrac componen follow auoregreve procee of order p and p, repecvely: where.. d. N, f ~ f where.. d. N, ~, and f u a, wh =,,N. f... ap f p f, () b u... bpu p, (3) For he purpoe of he paper, he reamen of he eaonal componen deerve pecal commen. In andard applcaon ha ue facor decompoon analye, whch we called radonal model n h paper, he eaonal componen of he ere exraced before emang he model and, herefore, model elecon, emaon and forecang carred on from eaonally adjued ere. The eaonal adjumen echnque are developed eher by he reearcher, uually wh he help of auomac procedure, uch a TRAMO- SEATS or X, or by he acal agence, whch n ome cae publh only he eaonally adjued veron of he me ere. In expreon (), h mple ha, =,,N. Alernavely, he dynamc propere of he eaonal componen could be accouned for whn he rucural dynamc facor model. In lne wh rgonomerc eaonaly model (ee Harvey, 989), we aume ha he eaonal componen can be vewed a he um of / cyclcal componen / j j, (4) where he number of obervaon per year. In h expreon, he cyclcal componen are modelled a rgonomerc erm a he eaonal frequence, model j j, hrough he j j co j n j n j co j j j j, (5) j where j=,,/, =,,N, and are muually uncorrelaed noe wh common j j varance j, and he erm j appear by conrucon o form j. In addon, we ue he andard aumpon ha he error erm exhb he ame varance acro frequence,.e., 4
j for all j=,,/. To complee he acal pecfcaon of he model, we aume ha all he durbance drvng he hree ochac componen are muually and erally uncorrelaed. To faclae mulaon and emaon, we prove n he Appendx ha h eaonal componen can be alernavely expreed by ung a eaonal auoregreve negraed movng average pecfcaon. For quarerly daa, 4 he eaonal componen are 3.387.869 where he backhf operaor, ~.. d. N,, (6) reflec ha he eaonal effec allowed o change over me, and =,,N. 5 To derve h expreon, we ued =4, nce he eaonal behavour of our quarerly varable ofen relaed o he me of a year. 6 e u conder ome denfcaon ue regardng he rucural dynamc facor model decrbed n (). e k be he number of common componen n h model. In he cae of he radonal model and he rucural model wh doyncrac eaonaly, he facor he only common componen, whch mple ha k=. Therefore, he mnmum number of me ere requred o denfy hee model N=k+=3. To acheve denfcaon n he radonal model, we alo aume ha he facor loadng of he fr varable one. To acheve denfcaon n he rucural model wh doyncrac eaonaly, we alo aume ha he marx of facor loadng lower-rangular wh un on he man dagonal. I mean ha he facor loadng of he fr varable one, ha fr varable doe no conan eaonal componen ( ), and ha he eaonal 5 paern are proporonal acro ere, =,3,4,5, wh. In he cae of he rucural model wh common eaonaly, he common componen are he facor and he eaon, whch mple ha k= and he mnmum number of me ere requred o denfy he model N=k+=5. Therefore, we work wh fve varable nce h he mnmum number of varable o enure ha he common facor denfed n all model. I worh ponng ou ha he rucural model ha aume doyncrac eaonaly could agn par of he eaonal varably o he doyncrac componen or par of he 4 For he ake of mplcy, we derve all he expreon for quarerly daa. Alhough he expreon would be larger, all he reul obaned n he paper could ealy be obaned for monhly daa. 5 In h cae, he yearly um of he eaonal effec expeced o be zero, nce he durbance erm ha zero expecaon. A model of deermnc eaonaly ealy obaned by mpong. 6 Alhough we focu on rgonomerc eaonaly a n Harvey (989), here are alernave way of allowng eaonal varable o change over me, a n Hannan e al. (97) or Harron and Seven (976). However, he Hannan e al. (97) eaonal model and he Harvey (989) model wh non-equal varance are he ame model n he Gauan cae, or when nnovaon follow a mxure of normal drbuon a n Bruce and Jurke (99). The Harron and Seven (976) eaonaly wh correlaed durbance model and he Hannan model are he ame model, whch are alo dencal o he model ha we ue n he Gauan cae.
doyncrac varably could be modelled a eaonal. Therefore, he noe can be ranmed from he eaonal componen,, o he doyncrac componen, u, and vce vera, whch may nfluence he n-ample fng performance of he model adverely. 7.. Sae-pace repreenaon To emae model parameer and o nfer unoberved componen by ung he Kalman fler, convenen o rewre he equaon ha decrbe he model dynamc n a ae-pace repreenaon. In he cae of N economc varable, whch are colleced n he vecor Y, he approprae ae-pace form of he model requre he pecfcaon of boh he meauremen equaon, Y Hh e, wh e ~.. d. N, R, and he predcon equaon h, wh ~.. d. N,. h For h purpoe, worh ponng ou ha he eaonal componen.387.869 3 (7) can be wren a.387.869 (8), wh var, =,,5. 3 where The pecfc form of hee wo equaon depend on he aumpon abou he eaonal componen. Ung he aumpon ha N=5, p=p=, when eaonal componen are common acro he dfferen varable, he ae pace repreenaon of he model become: f u y u y.387.869 u3 y 3 3 3.3873.869 3 u, (9) 4 y4 4 4.3874.8694 u5 y 5 5 5.3875.869 5 7 See Geweke (977) and Geweke and Sngleon (98) for a general dcuon of denfcaon n dynamc facor model. 6
and f a f f u u b u b u u3 b3 u3 3 u 4 b4 u 4 4 u5 b5 u5 5 3 where R=, and a dagonal marx wh man dagonal () dag,,,,,,, '. () f, 3 4 5 The ae-pace form of radonal dynamc facor model ha ue eher he offcal eaonally adjued daa e or he eaonally adjued oucome from TRAMO-SEATS can ealy be derved from hee expreon. In parcular, obaned by mpong, and. 3 However, when he eaonal componen aumed o be doyncrac for each economc varable, he ae-pace repreenaon of he model f u u y 3 3 3 3 u3 y A 3 3 3 u 4 y 3 3 3 A 3 3, () u5 y4 4 3 3 A 3 X y 5 5 3 3 3 A X X 3 X 4 and 7
f u u u u u X X X X 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 5 5 3 3 3 3 3 4 5 a b b b b b B B B 3 3 3 3 3 3 33 33 33 B 3 3 3 3 3 3 33 33 33 3 3 3 3 3 3 33 33 33 3 3 3 3 3 3 33 33 33 where A,.387,.869 ', X,, ', Z,, ' f f u u u3 3 u 4 4, (3) u5 5 X Z X 3 Z X 4 Z 3 X 5 Z 4, and =,..,5. B, (4) R, and a dagonal marx wh man dagonal dag,,,,,,,,,,,,,,,,, ' f. (5) 3 4 5 3 4 5 3. Mone Carlo mulaon In h econ, we degn a Mone Carlo expermen o udy ome of he fne-ample propere of rucural dynamc facor model ha accoun for common or doyncrac eaonal componen agan radonal dynamc facor model ha manage eaonally adjued daa. A hown n Table, he expermen conduced wh a comprehenve e of coeffcen n order o capure a wde range of pecfcaon, allowng for dfferen degree of common facor correlaon, dfferen perence of doyncrac componen, and doyncrac componen ha are heerogeneou. To cover a large varey of combnaon, Table repor ha he loadng facor of he fr varable e o uny n order o acheve denfcaon, whle oher facor loadng are eher pove for varable and 3 (.7 and., repecvely) or negave for varable 4 and 5 (-.8 and -.5, repecvely). We generae wo alernave cenaro for he eaonal componen. 8 The fr cenaro, called M, re o mmc he emprcal forecang exerce where eaonal componen are doyncrac. 9 The econd cenaro, called M, re o mmc 8 A menoned before, for denfcaon purpoe, he eaonal componen doe no affec he fr varable. 9 In lne wh our emprcal reul, we e j, j. In parcular, we e around.. 8
he cae of common eaonal componen, where he eaonal facor loadng are eher pove (.9 for varable 3) or negave (-.8 and -.7 for varable 4 and 5, repecvely). The common non-eaonal facor, f, and he ndvdual componen, u, are generaed a fr order auoregreve procee. Accordng o Table, n mulaon S, S, and S3 we replcae uaon where he economc varable hare a rong peren non-eaonal common componen (auoregreve parameer of.9). However, n mulaon S4, S5 and S6 he perence of he facor weak (auoregreve parameer of.) whle moderae n mulaon S7, S8 and S9 (auoregreve parameer of.5). In addon, hee poenal emprcal cae are combned wh everal degree of auoregreve parameer of he doyncrac componen. The perence rong (beween. and.4) n mulaon labelled S, S4 and S7, weak (beween.6 and.9) n mulaon labelled a S, S5 and S8 and mxed (beween. and.9) n mulaon S3, S6 and S9. In lne wh he emprcal reul of Secon 4, he mulaon labelled a S ue moderae perence of he non-eaonal common componen mxed (pove and negave, weak and rong) auoregreve parameer for he doyncrac componen. For all of hee daa-generang procee, he gnal-o-noe rao or proporon of he varance arbued o he common facor o he varance of he doyncrac componen a bune cycle frequence (3 o 8 year) alo ncluded n he able. In he mulaon, we ry o cover varou cenaro, accordng o he conrbuon of he common facor o he varaon of he ere. In mulaon labeled a S and S4, he common facor and he doyncrac componen accoun for abou he ame poron of he varance (gnal-o-noe rao cloe o ). In mulaon S, S3 and S7, he varance of he common facor larger han ha of he doyncrac componen (gnal-o-noe rao hgher han ), whle much maller (gnal-o-noe rao lower han ) n mulaon S5, S6 and S8. Fnally, mulaon S9 and S accoun for mxed cenaro. For each of hee cae, we generae a oal of M= e of me ere of lengh T= obervaon. We ue hem o mmc hree dfferen emprcal forecang cenaro. The fr cenaro, called EId, mmc he cae n whch an analy f a rucural dynamc facor model o he non eaonally adjued daa, whoe eaonal componen are reaed a doyncrac. The econd cenaro, called ECo, refer o a mlar cae bu where he eaonal componen common o he la four me ere. The hrd cenaro, called EaTS, mmc he cae n whch he analy ue eaonally adjued daa before emang he andard dynamc facor model,.e., he radonal approach. In our analy, he eaonal componen are exraced from he generaed me ere ung TRAMO-SEATS. The lengh of he generaed me ere nce would refer o 3 year of quarerly obervaon. 9
In each replcaon, m, we emae he wo rucural facor model and he radonal facor model ha ue eaonally adjued daa. We examne he performance of hee model n Table and 3. In each of hee able, he fgure n bracke analyze he ably of he model o nfer he facor whle he re of he fgure refer o he accuracy of he model o nfer he me ere. In Table, we examne he n-ample f of he model by compung he averaged quared dfference acro he T obervaon beween he generaed and he emaed me ere (Mean Squared Error, MSE), whch are alo averaged acro he M replcaon. In Table 3, we compare he ou-of-ample forecang accuracy by compung he error n forecang (one-ep-ahead) he generaed arge ere. For each m-h replcaon, he oneep-ahead foreca are obaned by emang he model wh daa from = o =T-, and by compung he foreca for T. To faclae nerpreaon, he able repor fracon of MSE, where he denomnaor he MSE from forecang ung he pecfcaon ha agree wh he daa-generang parameer. Tha, he fgure compare he MSE of all model agan he uaon where an oracle ha gven he reearcher he correc model o foreca from. Therefore, fgure below one ndcae ha he forecang model doe he job beer han he oracle. The man reul of he Mone Carlo expermen are he followng. Fr, here are wo man poenal ource of eaonal mpecfcaon n rucural dynamc facor model: when he daa are generaed wh doyncrac eaon bu he model ncorrecly aume common eaon (column labelled a M) and when he daa are generaed wh common eaon bu he model ue he erroneou aumpon ha he eaon are doyncrac (column labelled a M). Accordng o he magnude of he fgure repored n he able, he econd ource of mpecfcaon eem o be much le damagng han he fr. Second, when he eaonal componen generaed doyncracally acro he me ere, he radonal approach of dynamc facor model ha ue eaonally adjued daa unequvocally acheve he be performance. The fgure repored n he hrd column of he panel labelled a M how ha h raegy ouperform he rucural facor model ha aume doyncrac eaon. The poenal explanaon ha he rucural facor model may uffer from an denfcaon problem nce hard o denfy eparaely he varance of he ndvdual componen from hoe of he eaonal componen when hey are doyncrac. Anoher explanaon would be ha he greaer number of parameer o be emaed whn he rucural approach generae larger uncerany and noe n he emaon. Thrd, he rucural model ha correcly rea he eaonal componen a common when hey are acually generaed a common (ffh column of he able) uually exhb he
be performance. However, worh emphazng ha he accuracy of h model comparable o ha of he radonal facor model ha ue eaonally adjued daa, whch n many cae exhb he lowe MSE. Fourh, he perence of he doyncrac and he common componen ncreae he ze of he dfference acro pecfcaon bu doe no aler he naure of he reul. Regardle of wheher he eaonal componen are common or doyncrac, he radonal facor model acheve relavely beer forecang performance n he cae of hgh perence, whch movae he ue of h approach n cae of doub abou he naure of he eaonal componen. Ffh, he concluon obaned by analyng he MSE acheved by he model on nferrng he facor and hoe acheved by he model on fng he varable are of he ame naure,.e., good facor emaon mple good fng of he daa. In addon, he reul of he ou-of-ample analy (Table 3) are qualavely mlar o hoe of he n-ample performance, alhough a lle weaker. The nuon ha here more noe n he ou-ofample analy, whch generae hgher uncerany acro he model and make dffcul o exrac concluon from he analy. Summng up, hee reul agree wh he general raegy followed by analy ha rounely apply radonal dynamc facor model o me ere ha exhb common, doyncrac, and eaonal componen. Th approach con, pror o fng he facor model, of removng he eaonal componen. When eaonaly doyncrac, h raegy lead o he be reul. When he eaonaly common acro ere, lead o very good reul, whch are comparable o he reul of emang he rucural facor model aocaed wh he daa generang proce. Noably, he radonal approach exhb he advanage of elmnang he poenal damage of ung rucural facor model ha aume common eaonaly when acually doyncrac. 4. Emprcal analy 4.. In-ample analy The fve quarerly ndcaor ued n he emprcal analy are he Unvery of Mchgan conumer enmen ndex, new paenger car and ruck ale, medan uual weekly earnng n conan dollar, oal houng ar (new prvaely owned houng un ared), and
employee on nonagrculural payroll from 978. o 7.4. Accordng o our prelmnary analy of un roo, we fnd ha all of hem conan un roo; herefore all varable are ued n growh rae. The Unvery of Mchgan conumer enmen ndex a conumer confdence ndex publhed by he Unvery of Mchgan and Thomon Reuer. The ndex normalzed o have a value of n December 964 and baed on a lea 5 elephone nervew, whch are conduced each monh n a Uned Sae ample o ae near-me conumer aude on he bune clmae, peronal fnance, and pendng. The ndex doe no conan eaonaly. New paenger car and ruck ale and houng ar were obaned from he Deparmen of Commerce Bureau of Economc Analy (BEA), he medan uual weekly earnng and nonagrculural payroll were obaned from he Bureau of abor Sac. Thee economc ndcaor exhb a key advanage for our udy: hey are avalable a boh non-eaonally adjued and eaonally adjued. In addon, he elecon of hee ndcaor follow he lne uggeed by he nfluenal paper of Sock and Waon (99). We ar he analy wh a e of ndcaor ha nclude an ndcaor from he upply de of he economy (houng ar), an ndcaor from he demand de (car and ruck ale), an ndcaor from he ncome de (weekly earnng), and an ndcaor of he labor marke (employee on nonagrculural payroll). Then, we enlarge he nal e of ndcaor wh he Unvery of Mchgan conumer enmen n order o ncorporae a non-eaonal ere whch agree wh he evoluon of he bune cycle. Table 4 dplay he maxmum lkelhood emae of he rucural dynamc facor model ha accoun for eaonal adjumen and he radonal facor model ha ue eaonally adjued daa where he eaonal componen are exraced before emaon. The fgure ha appear n bracke refer o her andard devaon. 3 The choce of model pecfcaon alway baed on he Schwarz creron. The able alo how he log- kelhood acheved by hee model and her gnal-o-noe rao. There are everal noeworhy feaure from he emae repored n Table 4. Fr, he emaed common facor moderae (or even weak) nce he emae for fr order auocorrelaon range from.6 o.5, whch refer o cae S4 o S n he Mone Carlo expermen. Second, he perence of he ndvdual componen of he fr four varable mxed nce ome auoregreve parameer are mall whle ome oher are large, whch The Grea Receon wa no ncluded o overcome he problem aocaed o he large break of he me ere n hee year. To manage boh eaonally and non-eaonally adjued ere, we ubue manufacurng and rade ale, orgnally ued n Sock and Waon (99) for car and ruck ale. The ame apple o he ubuon of real peronal ncome le ranfer by weekly earnng. 3 Alhough no ncluded o ave pace, we obaned mlar reul when he radonal model con of ung he offcal eaonally adjued veron of he varable.
agree wh mulaon S4, S9 and S. Fnally, ome of hee auoregreve parameer are negave, whch agree wh mulaon S. Thrd, employmen exhb large pove auoregreve coeffcen for he doyncrac componen, whch lead o he low gnal-onoe rao n EId and EaTS model. 4.. Ou-of-ample analy In h econ, we develop an ou-of-ample forecang analy. For h purpoe, we aume ha he me ere of nere o be forecaed are he eaonally adjued oucome of TRAMO-SEATS of he fve economc ndcaor decrbed n he prevou econ. 4 The h- perod-ahead foreca were compued recurvely and he analy wa conduced o mulae real-me forecang. The fr foreca obaned by emang he model wh daa from = o =, and by compung he foreca +h. Then, he model are re-emaed wh daa from = o =+, and he foreca obaned for +h+. Th proce repeaed unl =Th. The fr mulaed ou-of-ample foreca wa made n 998. and we conder foreca horzon of h= and h=4 perod. 5 For each me ere, he averaged quared dfference beween he foreca and he argeed varable are compued. The reul, whch are repored n Table 5, ugge ome concluon ha are n lne wh he fndng obaned n he Mone Carlo analy. Regardle of he argeed ere and he foreca horzon, he foreca compued from he radonal dynamc facor model ha ue eaonally adjued ere exhb he lowe MSE. 6 In addon, he able how he p-value of Clark and We (7), whch compare he accuracy of he radonal approach wh he wo veron of he rucural approach and he eaonal ARIMA foreca. Overall, he able how ha he beer performance of he radonal approach acally gnfcan a 5% level a -perod foreca horzon. Therefore, cleanng up he economc ndcaor from eaonaly eher by ung he eaonally adjued ere or by ung auomac unvarae procedure before ung he varable n he dynamc facor model eem o be a reaonably raegy o follow. 5. Concluon 4 The reul obaned from he offcal eaonally adjued me ere are que mlar o he eaonally adjued oucome of TRAMO-SEATS. Accordngly, f he former were he ere of nere, he reul would be qualavely mlar o hoe preened n he paper. 5 Each quarer, we updaed he daabae a f all he varable had been oberved n ha quarer. Therefore, we dd no develop a peudo real-me analy nce daa revon or publcaon delay are no reaed. For a careful analy of hee forecang problem, ee for example Camacho, Perez-Quro and Poncela (). 6 There only one excepon: he one-ep-ahead foreca of car. Noably, he gan wh repec o he radonal facor model no large. 3
Depe he effor of recen ude o evaluae he emprcal hor-erm forecang performance of dynamc facor model, ll reman an open queon wheher beer o ue eaonally adjued ndcaor before emang he model or o accoun for he eaonal componen of he raw daa whn a facor model. The fr raegy mplcly aume ha he eaonal componen are doyncrac and he laer raegy could lead o unneceary complexy, epecally for praconer ha are no famlar wh eaonal analy. We ue Mone Carlo expermen o analyze he exen o whch hee wo alernave exhb relave forecang performance gan. Our mulaon reul ugge ha when he daa are generaed under he aumpon ha he eaonal componen are doyncrac he dynamc facor model ha ue eaonally adjued ndcaor exhb he be forecang performance. Inerengly, when he eaonal componen are common o all he me ere, forecang deeroraon wh repec o a dynamc facor model ha accoun for he common eaonaly uually neglgble n our expermen. Noably, he former mprove on he laer n everal cae. In emprcal applcaon, dffcul o decde a pror f he eaonaly common or doyncrac acro ere. Gven ha he deeroraon of he n-ample fng and ou-ofample forecang performance of he dynamc facor model ha ue eaonally adjued ndcaor very mall, whle he performance of he common eaonal componen model very poor n he cae of doyncrac eaonal facor, we rongly recommend he ue of eaonally adjued ere n facor model. We llurae hee reul by ung US daa from 978. o. of he Unvery of Mchgan conumer enmen ndex, new paenger car and ruck ale, medan uual weekly earnng, houng ar, and employee on nonagrculural. In lne wh he mulaon reul, he forecang performance of a dynamc facor model ha ue he eaonally adjued veron of hee ere beer han hoe of dynamc facor model ha aume common or doyncrac eaonal componen. 4
5 Appendx Snce he emprcal daa are quarerly, he eaonal componen of each me ere,, he um of wo cyclcal componen,, whch are evaluaed a he eaonal frequence,, and. Accordng o (4), he dynamc of he fr cyclcal componen co n n co. (A) Ung co and n and rearrangng erm, one can oban, (A) whch mple ha, where. If var var,, hen var. Smlarly, he dynamc of he econd cyclcal componen can be obaned from co n n co, (A3) whch, ung co and n, lead o. (A4) Th expreon mple ha, where and var. e u addonally aume ha. Accordngly, he eaonal componen of each me ere can be expreed a, (A5) or 3. (A6)
Snce he greae polynomal of he wo erm from he rgh-hand de of power wo, he reulng polynomal (he reul of ummaon) of power wo a well 3. (A7) To fnd he unknown coeffcen and, we derve he pecra of rgh-hand de of boh expreon. On he one hand, he pecrum of rgh-hand de of (A6) e e e e co co 6 4co co. (A8) The fr equaly follow from he fac ha e e co,. The la equaon ue and. On he oher hand, he pecrum of rgh-hand de of (A7) e e e e co co. (A9) Snce he wo pecra mu repreen he ame dynamc, one can ue he yem of hree equaon wh hree unknown 6, 4 and 4 3 o oban 4 6 4. The real oluon of h equaon are.387 and. 683, and ung agan he yem of equaon, eay o oban ha hey correpond o value. 869 and 5. 745. The fr par of oluon.387,.869 produce nverble MA polynomal n (A7), oppoe, he econd par of oluon reul n non-nverble (A7). Ung he fr par of real oluon we fnd 5.355 from he la equaon of he yem. In h way, he eaonal componen for he ere gven by: 3.387.869. 6
Reference Aruoba, B., Debold, F., and Sco, C. 9. Real-me meauremen of bune condon. Journal of Bune and Economc Sac 7: 47-47. Aruoba, B., and Debold, F.. Real-me macroeconomc monorng: Real acvy, nflaon, and neracon. Amercan Economc Revew : -4. Banbura, M., and Modugno, M.. Maxmum lkelhood emaon of facor model on daae wh arbrary paern of mng daa. Journal of Appled Economerc 9: 33-6. Bovn, J., and Ng, S. 6. Are more daa alway beer for facor analy? Journal of Economerc 3: 69-94. Bruce, A.G. and Jurke, S.R. 996. Non-Gauan eaonal adjumen: X- ARIMA veru robu rucural model. Journal of Forecang 5: 35-37. Camacho, M., and Perez-Quro, G.. Inroducng he Euro-STING: Shor Term INdcaor of euro area Growh. Journal of Appled Economerc 5: 663-694. Camacho, M, Perez-Quro, and Poncela, P.. Markov-wchng dynamc facor model n real me. CEPR Dcuon Paper no. 8866. Clark, T., and We, K. 7. Approxmaely normal e for equal predcve accuracy n need model. Journal of Economerc 38: 9-3. Geweke, J.F. 977. The dynamc facor analy of economc me ere model, n D. Agner and A. Goldberger (ed.), aen Varable n Socoeconomc Model, aen Varable n Socoeconomc Model. Geweke, J.F. and Sngleon, K.J. 98. Maxmum lkelhood confrmaory facor analy of economc me ere, Inernaonal Economc Revew, 37-54. Hannan, E. J., Terrell, R. D. and Tuckwell, N. E. 97. The eaonal adjumen of economc me ere, Inernaonal Economc Revew, 4-5. Harron, P.J. and Seven, C.F. 976. Bayean Forecang. Journal of he Royal Sacal Socey, Sere B 38: 5:47. Harvey, A. 989. Forecang. Srucural me ere model and Kalman fler, Cambrdge Unvery Pre. Poncela, P., and Ruz, E.. More no alway beer: back o he Kalman fler n dynamc facor model. UC3M Workng paper, Sac and Economerc, /7. Sock, J., and Waon, M. 99. A probably model of he concden economc ndcaor. In Kajal ahr and Geoffrey Moore edor, eadng economc ndcaor, new approache and forecang record. Cambrdge Unvery Pre, Cambrdge. 7
Table. Parameer ued n Mone Carlo mulaon Fxed parameer for all mulaon,. 7, 3., 4. 8, 5. 5 f,. 7,.8, 3. 9, 4, 5. 9 Parameer ha conrol doyncrac veru common eaon M (doyncrac eaon) M (common eaon)., 3. 9,.8, 5.., 3. 9, 4. 8, 5. 7 4 Parameer ha conrol non-eaonal facor and ndvdual componen No eaonal componen S S S3 S4 S5 S6 S7 S8 S9 S Common rong rong rong weak weak weak mod mod. mod mod Idoynchr. weak rong mxed weak rong mxed weak rong mxed p-n a=.9 b =.3 b =. b 3 =.4 b 4 =. b 5 =.3 a=.9 b =.7 b =.8 b 3 =.9 b 4 =.7 b 5 =.6 a=.9 b =.9 b =. b 3 =.5 b 4 =.9 b 5 =.3 a=. b =.3 b =. b 3 =.4 b 4 =. b 5 =.3 a=. b =.7 b =.8 b 3 =.9 b 4 =.7 b 5 =.6 a=. b =.9 b =. b 3 =.5 b 4 =.9 b 5 =.3 a=.5 b =.3 b =. b 3 =.4 b 4 =. b 5 =.3 a=.5 b =.7 b =.8 b 3 =.9 b 4 =.7 b 5 =.6 a=.5 b =.9 b =. b 3 =.5 b 4 =.9 b 5 =.3 a=.5 b =.9 b =-. b 3 =-.4 b 4 =-. b 5 =. SNR.46 4.67 8.9 6..88 3.83.8.89..8.4 4.67 6.7.64.88.6.6.6.79.4.5..5.6..6.6.87.8.4 3.5.3.5.7.53.8.3.53.34.3.57.3.89.8.53.57.8.83.5.67 Noe. Parameer refer o he loadng facor. Parameer f and refer o he varance of noe of he common non-eaonal facor and he doyncrac componen, repecvely. Parameer refer o he varance of he noe of he cyclcal componen. Parameer a and b refer o he auoregreve parameer of he common facor and he doyncrac componen, repecvely. SNR he gnal-o-noe of he varance of he common facor a bune cycle frequence (3 o 8 year) o he varance of he doyncrac componen n he ere. 8
Specfcaon S S S3 S4 S5 S6 S7 S8 S9 S Table. In-ample Mone Carlo reul M (doyncrac eaon) M (common eaon) EId ECo EaTS EId ECo EaTS () (.57) (.9) (.) () (.9) 7.6.46.83.73 6.3.37 3.3.85 9.9.49.4.4 3.6.33 3.38.9 () () () () () () () () () (.5) 6.39 7.56 6.84 4.5 (.34).3 6.5 6. 3.95 (.69) 8. 5.3 7.44.6 (.7) 6.94 6.73 9.45.8 (.73) 6.36 5.5 7..4 (.73) 7.68 5.39 6.7 3.3 (.8) 7.9 5.39 6.98 5.4 (.74) 9.74 6.6 5. 4.57 (.).5..3.33.4 ().44.37.39.3 (.94).48.47.39.4 (.94).6.5.45.45 (.).55.58.5.44 (.98).5.5.47.44 (.93).54.48.46.39 ().5.54.39.4 (.98).54.5.4.39 ()..8.99.99 (.7).87 3.8. 3.36 (.4).95 3.3.84 3.6 (.7).97 3.7.33 3.6 (.9).88 3.7. 3.3 (.5). 4.7.95 3.8 (.5).89 3.35.9 3.45 (.8).87 3.53.4 3.59 (.4).3 3.4.8 3.55 (.)..74 () () () () () () () () () (.98).65.9.9.9 (.94).77.88.84.5 (.98).69.6.99. (.).69.58.7.4 (.).8.9.4.6 (.95).7.97.4.99 (.99).63.6.9 (.98).85.8.97.94 (.98).4.97.99.97.98 Noe. Expreon S o S are decrbed n Table. Fgure n parenhee refer o he MSE of he common facor whle oher fgure refer o he MSE of ere o 5. Column labelled a M and M refer o daa generaed procee wh doyncrac and common eaonal componen, repecvely. EId, Eco, and EaTS refer o model wh doyncrac eaon, common eaon, and model whoe ndcaor are eaonally adjued (TRAMO-SEATS) before emaon, repecvely. owe MSE are hghlghed n bold. 9
Table 3. One-perod-ahead Mone Carlo reul Specfcaon M (doyncrac eaon) M (common eaon) EId ECo EaTS EId Eco EaTS S () (.).97.43.9.84 3.48 (.9).67.77.5.7 (.)...3.. () (.94).97.96.97..98 S () (.8)..9.47.39.83 (.79).99.7.63.67.3 (.97).99.98.97 () (.96).99.93.93.94.9 S3 () (.).73.4.3.38 (.93).58.7.76.4 (.)...5. () (.9).99.96.97.99.98 S4 () ().99.3.3.3. (.99)..75.79.75.48 (.)..... () (.)....4 S5 () (.)..5.7..3 (.98)..75.7.84.43 (.).96.. () (.99)..96.95.97.99 S6 () (.98).9.65.4..37 (.).94.37.94.78.4 ()..4. () ().98.98.93.97 S7 () (.97).98.8.7.5.89 (.).84.84.8.34 (.99).3.. () (.97).97.97.98.97 S8 () (.).5.8.5.45 3.3 (.88).96.76.74.66.3 (.98)..4..3 () (.99)..93.99.99 S9 () (.3)..9.3.7.93 (.88).99.77.74.7.33 (.97).99.98.. () (.99)..97.93.96 S () (.).5..3.33.4 (.98).97.8.63.74.49 (.)..74 () (.98).4.97.99.97.98 Noe. See noe of Table.
Table 4. Maxmum lkelhood emae Emaed model EId Eco EaTS.59 (.4).84 (.33). (.4) 3.37 (.7).4 (.5).7 (.37) 4 -.8 (.5) -.39 (.) -.39 (.39) 5. (.3).38 (.5).9 (.97) a.5 (.9).6 (.3).5 (.) b -.4 (.9) -.5 (.9) -.3 (.9) b -.5 (.9) -.3 (.9) -.45 (.9) b 3 -.6 (.) -.95 (.4) -.6 (.9) b 4.8 (.5) -.58 (.8).84 (.5) b 5.5 (.).6 (.5) -.7 (.3) f.3 (.8).33 (.).6 (.9).94 (.6).94 (.6).94 (.6).48 (.4).68 (.7).83 (.6) 3.5 (.3). (.7).9 (.6) 4.8 (.).67 (.4).53 (.3) 5. (.8).6 (.3).5 (.6).6 (.4).9 (.7) - 3.8 (.3) - - 4. (.) - - 5.5 (.) - - 3 - -.94 (.8) - 4 -.7 (.3) - 5 -.88 (.67) - log -9.7 53.97 6.6 SNR.4.56.3. 99.5.33.36..3.33.89.3. 5.36 Noe. See noe of Table. SNR he gnal-o-noe rao of he varance of he common facor a bune cycle frequence (3 o 8 year) o he varance of he doyncrac componen n he adjued ere.
EId Eco EaTS TSW Table 5. Emprcal forecang analy Senm Car Wage Employ Houng -perod-ahead foreca: Relave MSE.8.39.39.3.3.78.53 6.6.6.57.7.3.3...74.7 4.37.6.5 -perod-ahead foreca: Equal accuracy e EaTS/Ed EaTS/Eco EaTS/TSW.3.39..56.3.6.3.....3 EId Eco EaTS TSW EaTS/Ed EaTS/Eco EaTS/TSW 4-perod-ahead foreca: Relave MSE.7.7.8.64.74 6.43.63.7.8.65.7.88.4.6.38.66 4-perod-ahead foreca: Equal accuracy e.77.338.48.8.38.9..8.34.84.96.7.9.8.4.33.33.33.37.354.38.473 Noe. See noe of Table. To compare he reul acro me ere ealy, he fgure how he MSE dvded by he n-ample andard devaon of each me ere.