Trading-Day Adjustment as a Practical Problem

Transcription

1 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).3 Tradng-Day Adusn as a Praa Prob Aa-Toubr, Ky IS, Déarn ds sasqus d our r 8, Bouvard Adoh Pnard 55 PARIS D 4, FRA -a: [email protected] Ladray, Donqu IS, Déarn ds sasqus d our r 8, Bouvard Adoh Pnard 55 PARIS D 4, FRA -a: [email protected] Inroduon Tradng day ffs ar rsn n any ono srs. Thy our whn h nnsy of avy s no h sa aordng o h day of h w. In ha as onhy or quarry srs an onan fuuaons nd o h andar sruur. On or Saurday n a onh for xa drasay as h ra rad urnovr n uroan ounrs. In h wdy usd sasona adusn sofwars Trao-Sas and -ARIMA, radng-day adusn s don usng Rg-ARIMA odng. For h rgrsson ar of h od, h usr an sfy r-dfnd rgrssors and/or usr-dfnd rgrssors. In hs ar, w roos a way o bud a usr-dfnd s of radng-day rgrssors ang no aoun sfs of h aona andar and h sor of avy undr rvw.. Why shoud w a no aoun aona andars n radng-day adusn? Fgur rrsns h avrag quarry nubr of worng days aross uroan ounrs. Worng days ar hr dfnd as h nubr of Mondays, Tusdays, Wdnsdays, Thursdays, Frdays whh ar no ub hodays. W an no bg dffrns bwn ounrs, n aruar n frs and sond quarr. Thrfor hr s an oran ona a on h ono srs. Tha s why urosa ass uroan ounrs o do hr own radng-day orron bfor rforng a dr sasona adusn. Fgur : Avrag quarry nubr of worng days aross uroan ounrs Q Q Q3 Q4 AT B BU Y Z D DK S FI FR GR HU I IT LT LU LV MT L PL PT RO S SI SK UK

2 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).33 vn for a sf ounry, h nubr of worng days for a gvn quarr an vary a o fro a yar o anohr. In Fran, h sond quarr of 0 w hav hr worng days or han h sa quarr n 05 (s Tab ). Ths ay ad o fuuaons n srs ha ar no orrd by sasona adusn. Tab : ubr of worng days n Fran Q Q Q3 Q Why shoud w a no aoun h sor of avy undr rvw? Tradng day ffs ar vry dffrn dndng on h sor of avy. Th foowng grahs (Fgur ) show h offns of radng-day rgrssors for Frnh urnovr ndxs n wo ra rad sors: Mnars and Dsan sang. Fgur : offns of radng-day rgrssors for Mnars and Dsan sang urnovr ndxs Dsan sang Mnars 0,03 0,05 0,0 0,05 0,0 0, ,005-0,0 0,0 0,0 0,008 0,00 0,004 0,00 0-0,00-0,004 Monday Tusday Wdnsday Thursday Frday Saurday Monday Tusday Wdnsday Thursday Frday Saurday For Mnars, Monday s offn s ngav, baus n Fran any shos ar osd on Monday. On h oos Saurday s offn s vry hgh as any o do shong on Saurday. Ohr offns ar osv, bu sar. For Dsan sang, offns fro Monday o Frday ar qu sar, whn Saurday s offn s ngav. Prdfnd rgrssors n Trao-Sas and ARIMA onsdr ony wo ass: h frs on s ha ah day of h w has a dsn ff, hn sx radng-day rgrssors (Monday,, Saurday), ar nrodud n h Rg-ARIMA od, h sond as s foundd on h hyohss ha wdays, fro Monday o Frday, ar qu sar, whras Saurday s sar o Sunday, whh ads o on onras rgrssor. Dsan sang orrsonds o h sond as. Mnars dos no orrsond o any of h wo ass as h dd w offns ar qu sar.

3 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).34 Tha s why oud b usfu o roos radng-day rgrssors whh r o onsdr or varous suaons han hos wo ass. Th ass ods. Th bas od Tradng-day ffs ar nd o h ooson n days of h onh (quarr). Th bas od for fxd radng-day ffs of fow srs s hrfor: α ε () Whr s h raw srs and dnos h nubr of Mondays (), Tusdays () a da and ε foows an ARIMA od: d D ( B ) ( B ) ( B) φ( B ) ε ϑ( B) Θ( B ) a ϕ for a onhy srs. d and D ar h dffrnng ordrs, ϕ, Φ, ϑ, Θ ar oynoas, a s a wh nos. α s h avrag ff of an -y day on varab. Th od rsns a as wo robs (s B, Hr (983) and B (984 and 995)): - α s sas nd o b hghy orrad - s rgrssors ar sasona. For ur radng-day ff saon, rgrssors shoud b sasonay adusd. L us no α and α α for,,. α Mod () an b rwrn as α s h avrag ff of any day; α ε s h sf ff of a -y day and 0 ; and s h ngh of onh. Usng h onsran 0, w an wr: Rang n od () w oban: ) α ε ( (), whr: onras rgrssors ( ) ar no sasona baus for ah -y day, h sasona ar of, whh s s ong-r onhy an, s aos qua o s on. As s sasona, s rad by s dvaon o s ong-r onhy an (La Yar varab). Thn h od bos: Wh : ) αly ε LY ( (3) f Fbruary and La Yar f Fbruary and no La Yar Ohrws

4 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).35. A or arsonous od In od (3), sx radng-day rgrssors offns hav o b sad, as w as ARIMA offns and ossby ours or ohr ys of rgrssors. To g a or robus saon of h offns, woud b br o fnd a or arsonous od. Trao and -ARIMA roos a sfd for of h od, nrodung a wday rgrssor. Th undryng hyohss s ha n so ono avs, wdays (Monday, Tusday,, Frday) ar sar and n h ohr hand ha Saurday s sar o Sunday. You hn hav: and and h fna od s: ( ) 3 Gnra odng 3. Inroduon of aona andars αly ε ow w roos a or gnra for of h od n ordr o nrodu aona andar. Inuvy, h ff of a Monday on h avy s no h sa f hs Monday s ordnary or f s a ub Hoday. Thn, nsad of a aron n svn days, w wr a aron n fourn ys of days. 4 αly ε (4) o ndxs dno Mondays n, Tusdays n,, Frdays n and 8 o 4, Mondays off, Tusdays off,, Frdays off. 4 α α and α α for,, 4. 4 As n h ass od, h nubr of rgrssors an b rdud undr so hyohss. For xa, f w assu ha a Sunday off s quvan o a Sunday n and ha any day off s quvan o a Sunday, hn w hav: and h od bos: 4 αly ε 8 o ha, hr, usng a onras dos no rov h sasona oonn anyor as Sundays days off do no hav h sa ong-r onhy ans as h ohr days. W nd o rov sasona ar of hos rgrssors by rang h by hr dvaon o hr ong-r onhy an. 3. Th vry o od I s ossb o gnraz h rdng od by wrng a aron of h onh n ys of days: α ε L us no (5) α α and α α for,,.

5 As bfor, h od s hangd no: LY ε α and as w hav,, h od an b wrn wh onras rgrssors, n ordr o sov onary robs: ε α ) (, W hn ry o forua hyohss n ordr o sfy h od. As s rfrab o hav onras rgrssors, w dfn wo ys of hyohss: quay of offns of onras varabs quay of offns of ohr day-varabs Th onras varab s a obnaon of dffrn days ha ar suosd o hav h sa bhavour (and hn h sa offn): { } { },,,,,, wh Th nubr of ns of s s nod. L,,, L b ss of «quvan day-varabs»: { } { } { } { } { } { } # and,,,,,, # and,,,,,, # and,,,,,, L M L L o ha and,,, L a a aron of h days of h onhs. W us h sa hod as bfor o ransfor h od: { } { } And { } Th fna for of h vry o od s hrfor: { } ε α () Fnay, on us no forg o rov sasonay fro hos rgrssors. In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).3

6 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).3 4 How o dfn ss of rgrssors rvan wh sor of avy? W hos o ay h 5-varab od (s son 3. od 4) o h Frnh urnovr ndxs a A 5-dg v, wh -ARIMA. Aong h hgh nubr of ossb hyohss o oban a or arsonous od, how o hoos h or rvan ons, aordng o h sor of avy? Frs of a, w arbrary dd h hyohss ha Sunday n, Sunday off and a day off ar sar, whh ads o sx radng-day rgrssors n onrass o SundaysDays off. Thos rgrssors wr nrodud as usr-dfnd varabs n Dra and h offns wr oud wh -ARIMA. Thn h da was o xor h offns of hos radng day-rgrssors usng an aggorang hrarha usr anayss. Th foowng grah (n ba ns) shows h usr r n ra rad sor (5 srs) (Fgur3): Fgur 3: usr r of radng-day s offns of Frnh urnovr ndxs n ra rad ung h branhs a dffrn vs (s vra dod ns) ads o ran a son of rvan ss of rgrssors. In ra rad, usng h 4 h rd n fro h f rs o dfn a s of hr rgrssors: Monday aon, Tusday un Frday oghr, and Saurday aon. I orrsonds o h as of Mnars, whh s frqun n Frnh ra rad. Fgur 4 shows h usr r orrsondng o h ndusry urnovr ndxs. Afr hs xoraon, w fnay ddd o ran fv ss of radng-day rgrssors : S0: La Yar aon (hyohss ha a days of h w ar sar) S: varabs: Mon. n, Tu. n, Wd. n, Thu. n, Fr. n, Sa. n; onras: Sun. Days off S: varab: (Mon. Tu. Wd. Thu. Fr.) n; onras: Sa.Sun. Days off S3: 5 varabs: Mon. n, Tu. n, Wd. n, Thu. n, Fr. n; onras : Sa.Sun. Days off S4: 3 varabs: Mon. n, (Tu. Wd. Thu. Fr.) n, Sa. n; onras : Sun. Days off La Yar s aso rsn n ah od wh S, S, S3, S4. S0 orrsonds o sasonay adusd ngh of onh ff. W wand o h f oud b rvan for so avs ha hav onnuous roduon ross ran ha ndusrs. S4 was rand for ra rad, whosa rad and srvs bu no for ndusry and onsruon. Th nx s s o dfn a hod o s for ah sr h bs s aong hos fv.

7 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).38 Fgur 4: usr r of radng-day s offns of Frnh urnovr ndxs n ndusry and onsruon 5 An agorh for son of h bs s of rgrssors 5. Dsron of h agorh For ah srs, a Rg-ARIMA od was oud wh -ARIMA usng h or o s of rgrssors (S) as «usr-dfnd» rgrssors. Auoaay dnfd ARIMA od and ours (wh ra vau5) wr hn usd o ou radng day saon wh ah ohr s of rgrssors. Thn, h son bwn h dffrn ss was don aordng o an agorh whh obns wo rra: Fshr s of onsrans on offns: f h hyohss of quay of svra offns s no rd, s ossb o u h orrsondng days n h sa grou (for xa Monday, Thusday,, Frday oghr). AI rron gvs nforaon abou h quay of h Rg-ARIMA od wh ah s. Fgur 5 dsrbs h o agorh of son, for ah srs of ndusry and onsruon, of h bs s of rgrssors aong S0, S, S, S3. 5. Rsus Agorhs of son wr ad for ah urnovr ndxs: h on of Fgur 5 for ndusryonsruon, and a sar on, bu nudng S4 s, for ohr sors. W gv n Tab h rsus of hs son ross. Tab : Dsrbuon (n %) of h radng-day ss sd n ah sor of avy o TD ff S0 (LY aon) S S S3 S4 Ra Trad (5) 5 4 Whosa Trad (89) Srvs (38) Indusry () o sd

8 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).39 Fgur 5: Agorh of son, for ah ndusry and onsruon srs, of h bs s of rgrssors H0 : 0 5% v o rd Rd H0 : α 0 (La Yar s offn),5% v H0 : 5 and 5% v o rd Fna ho o TD rgrssor Rd Fna ho S0 (LY aon) o rd and AI(S) < AI(S) and AI(S) < AI(S3) Fna ho S Rd H0 : 5% v o rd And AI(S3) < AI(S) Rd Fna ho S3 Fna ho S W an no ha n ra rad, for a arg aory of srs, S4 s was sd as h bs on, whh onfrs usrng rsus. In whosa rad, S4 s aso h or frquny sd. In srvs and n ndusry-onsruon, S s or ofn sd. x for ra rad, h rnag of srs whou radng-day ffs s gra. Aos no srs sd La Yar aon (S0). Fnay, w an as f radng-day adusn wh hos nw ss of rgrssors s good. Th dagnos hosn for quay h s h rsdua radng-day sra as dagnos rodud n Dra. For ah s of rgrssors, w aud h rnag of srs wh bad dagnos. W aso auad hs rnag n as h srs wr adusd wh h s sd aordng o our agorh. Rsus for ra rad sor ar rsnd n Tab 3. Tab 3 Prnag of ra rad urnovr ndxs wh rsdua radng-day sra as o TD rgrssor S0 S S S3 S4 Sd s % 5% 9% 3% 4% % % As oud b xd, hs rooron s sar wh h os o s (S), and hghr wh or arsonous ss. Bu h rooron oband wh h ss sd aordng o our agorh s bary grar han h on wh h o s S.. onuson In hs ar, w dsrbd a way o s u rvan ss of radng-day rgrssors, n ordr o a no aoun of boh sfs of aona andar and sor of avy. W aso roosd an agorh for hoosng, for ah srs, h bs on, ang no aoun arsony rron bu aso quay of h Rg- ARIMA od. An aaon on Frnh urnovr ndxs d o vry sasfyng rsus.

9 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).40 RFRS B, W. R. (984), Sasona Dooson of Drns ffs, Rsarh Ror, Sasa Rsarh Dvson, U.S. Burau of h nsus, Washngon D.., RR84/0. B, W. R. (995), orron o Sasona Dooson of Drns ffs (n RR84/0), Rsarh Ror, Sasa Rsarh Dvson, U.S. Burau of h nsus, Washngon D.., RR95/0. B, W. R., Hr, S.. (983), Modng T srs wh andar Varaon, Journa of h Aran Sasa Assoaon, 383, 8, Bssa, M., Dhfa, R., Ladray, D.,Lassoud, A., Maghrab, B. (009), Ls ffs d andrr dans s sérs unsnns, STATO 03. hn B-., Fndy D. F. (993), Muav radng day adusns: - and RgARIMA oard, Aran Sasa Assoaon Annua Mngs, Axandra. ho P. (003), A o on h Tran of Fbruars n Tradng-Day Adusn, Worng Par, T Srs Rsarh and Anayss nr, Sass anada. vand W. P. (98), andar Adusn of T Srs, Dsusson Par, Sa Suds Son, Board of Govrnors of h Fdra Rsrv Sys, 98. vand W. P., Gru M. R. (983), Modng srs whn andar ffs ar rsn, Ad T Srs Anayss of ono Daa, Znr A. (dor), U.S. Darn of or, U.S. Burau of h nsus, Washngon D.., 5-. Dagu,. B., (980), Th -ARIMA Sasona Adusn Mhod, Sass anada, aaogu - 4 Dagu,. B., Qunnv, B., Suradhar, B. (99), Tradng-day varaons u rgrsson od wh rando arars, Inrnaona Sasa Rvw,, 0, 5-3. Fournr, J-M., Ladray, D. (995), Ls ffs d andrr dans 'anays ononur d a roduon, ono Sasqu, 85-8, 5-. Fndy D. F., Souu R. J. (000), Don and Modng of Tradng Day ffs, Prodngs of h Inrnaona onfrn on sabshn Survys. Góz, V. and Marava, A. (99), Progras TRAMO (T srs Rgrsson wh Ara nos, Mssng obsrvaons, and Ours) and SATS (Sgna xraon n Ara T Srs). Insruons for h Usr, Worng Par 98, Rsarh Darn, Bano d saña. Ladray, D., Qunnv B. (00), Désasonnasr av a Méhod, Mhodooga, nuéro séa 8-, Unvrsé Lbr d Bruxs Lyras, Dra P., Fdaush, Roxann M., and B, Wa R., Drnng Sasonay: A oarson of Dagnoss Fro --ARIMA, U.S. nsus Burau Maard, V. (994), Théor raqu d a orron ds ffs d ours ouvrabs, doun d rava, Dron ds uds Synhèss onoqus, IS, Pars, G9405. Marava, A. (008), os on Progras Trao and Sas, Ban of San Souu R. J., Fndy D. F. (999), On h Sru Dagnoss Usd by --ARIMA o Inda h Prsn of Tradng Day ffs afr Modng or Adusn, ASA Prodngs of h Jon Sasa Mngs (Busnss and ono Sass Son). Tondrng. (005), Frquny Asd Qusons abou andars, Vrson.8, avaab on U.S. nsus Burau (009), --ARIMA Rfrn Manua, Vrson 0.3, T Srs Saff, Sasa Rsarh Dvson, Washngon, D Young A. H. (95), sang radng-day varaons n onhy ono srs, Thna Par, U.S. Darn of or, U.S. Burau of h nsus, Washngon D.. Young A. H. (983), on on «Modng T Srs whn andar ffs ar Prsn», Ad T Srs Anayss of ono Daa, Znr, A. (dor), U.S. Darn of or, U.S. Burau of h nsus, 8-.

10 In. Sasa Ins.: Pro. 58h Word Sasa ongrss, 0, Dubn (Ssson STS030).4 ABSTRAT Th nubr of worng days an xan so shor-r ovns n h srs. On or Saurday n a onh for xa drasay as h ra rad urnovr n uroan ounrs. Aar h day ooson of h onh, ohr andars ffs suh as ub hodays or rgous vns ay aso aff h srs. Ths rod fuuaons, as w as h sasonay, ar usuay dd and nad n ordr o xhb h rrguar or non-rod ovns whh ar robaby of os nrs and oran. In hs rsnaon, w fous on wo oran raa robs: h dsgn of adqua rgrssors ang no aoun h sfs of naona andars and h ho of a orr s of rgrssors, radng off bwn h arsony of h od and h quay of h saon.