MODELING TIME-VARYING TRADING-DAY EFFECTS IN MONTHLY TIME SERIES

MODELING TIME-VARYING TRADING-DAY EFFECTS IN MONTHLY TIME SERIES Wllam R. Bell, Census Bureau and Donald E. K. Marn, Howard Unversy and Census Bureau Donald E. K. Marn, Howard Unversy, Washngon DC 0059 and Census Bureau, Washngon DC 033 KEY WORDS : ARIMA models, unobserved componens, me-varyng regresson. INTRODUCTION Tradng-day effecs reflec varaons n monhly me seres due o he changng composon of monhs wh respec o he numbers of mes each day of he week occurs n he monh. A relevan queson regardng radng-day effecs s wheher hey reman consan over me? Ths s especally pernen for real sales daa n whch radng-day effecs presumably depend on consumers shoppng paerns and on hours ha real sores are open, wo hngs ha have changed over me n he U.S. Seasonal adjusmen praconers somemes deal wh hs ssue by resrcng he lengh of he seres o whch he radng-day model s f. However, hs can provde only a crude approxmaon o radng-day effecs ha vary hrough me. In hs paper we explore some alernave models for me-varyng radng-day effecs and nvesgae possble me varaon n radng-day effecs n some Census Bureau monhly me seres. Monsell (983), Dagum, Quennevlle and Suradhar (99, hereafer DQS), and Dagum and Quennevlle (993, hereafer DQ) consdered sochasc models for me-varyng radng-day coeffcens. Monsell used random walk models for he coeffcens; DQS and DQ consdered a more general model n whch applyng some order of dfferencng (no jus frs order) o he radng-day coeffcens yelds whe nose. A lmaon o he analyss of Monsell and of DQS s ha hey consdered jus radng-day plus whe nose rregular models. Monsell appled hs model o smulaed daa, DQS o daa flered by X- rregular flers o remove rend and seasonaly. DQ consdered a more general model ncludng seasonal, rend and rregular componens n addon o me-varyng radng-day, hough for he example presened hey chose a model wh fxed radng-day coeffcens. Harvey (989) and Bell (004) consdered models for me-varyng radng-day coeffcens n general Dsclamer: Ths repor s released o nform neresed pares of ongong research and o encourage dscusson of work n progress. The vews expressed on sascal, mehodologcal, echncal, or operaonal ssues are hose of he auhors and no necessarly hose of he U.S. Census Bureau. conexs, and we dscuss her models n more deal n laer secons. In he nex secon we revew a model for fxed radng-day effecs and hen dscuss some alernave models ha allow for sochascally me-varyng radng-day coeffcens. Secon 3 dscusses aspecs of fng he models. Secon 4 gves resuls of fng he models o some Census Bureau me seres. The fnal secon gves conclusons and also dscusses drecons for fuure research.. MODELS FOR TRADING-DAY EFFECTS Monhly me seres ha are accumulaons of daly values (flow seres) are ofen affeced by he day-ofweek composon of he monh,.e., by whch days occur fve mes and whch days occur four mes n he monh. Le y be he observed value for monh (or s logarhm) n a monhly flow me seres. A basc model ncorporang radng-day effecs s y = TD + Z, () (Bell and Hllmer 983, hereafer BH) where TD s he radng-day effec, and he remander seres Z follows some me seres model such as an ARIMA (auoregressve-negraed-movng average) model (Box and Jenkns 9) or an ARIMA componens model (Harvey 989). We wll use lnear regresson models for TD. We frs consder a basc model wh fxed coeffcens, and hen dscuss alernave ways ha hs model can be generalzed o allow for sochascally me-varyng radng-day coeffcens. We could nclude addonal regresson effecs (oher han for radng day) n he model (), such as Easer holday effecs as n BH, bu we gnore ha possbly n hs secon o focus aenon on alernave models for TD. In he examples of Secon 4 we brng n some addonal regresson effecs and he me seres model for Z.. A model for fxed radng-day effecs Le he average effec of day on he monhly value of he seres be α, =,,...,, so α s he average effec on he seres of Monday, α he average effec of Tuesday,, and α he average effec of Sunday. For monh, le be he number of Mondays n he monh,, and D D D he number of Tuesdays, he number of Sundays. Also, le

N = D denoe he lengh of monh (8, 9, 30, or 3), and le α (/ ) α = denoe he average daly effec. To model fxed radng-day effecs, BH sar wh α D = αn + ( α α) D = =. () In equaon () he erm α N s a lengh-of-monh effec ha we do no consder as par of he radng-day effec, and n fac drop from he model for reasons dscussed below. Thus, TD s gven by he second erm on he rgh hand sde of (). Snce = ( α α) = 0, TD may be wren as, (3) = = TD = ( α α)( D D ) = T where = α α and T = D D, =,,...,. The parameers measure he dfferences beween he Monday,, Saurday effecs and he average daly effec, α. The dfference beween he Sunday effec and he average daly effec s gven by α α =. In equaon (3) we use Sunday as he day of = reference, however, clearly any of he oher sx days could be so used. Noce ha f α, =,,..., are equal, he radng-day effec s zero. Also, for a nonleap year February, D = 4, =,,...,, and he radng-day effec s zero. Raher han nclude he erm α N of () n our models, we shall nsead dvde he orgnal seres (before akng logarhms) by N. Ths s conssen wh he defaul opon n he X--ARIMA program (Fndley, Bell, Monsell, Oo and Chen 998). We do hs for he followng reasons: (a) Bell (984) noed ha he lengh-of-monh effec, α N, can be decomposed no a level effec, a fxed seasonal effec, and a leap- February effec. The frs wo wll be accouned for by dfferencng n he models. Thus, he erm α N serves essenally o model leap-february effecs. (b) When y represens logarhms of he orgnal seres, we would expec α o be approxmaely log(9/8) = 0.035, n whch case can be shown ha ncludng α N n he model s approxmaely equvalen o dvdng he orgnal me seres (before akng logs) by N. (c) Includng α N n he model and esmang α somemes yelds mplausble values,.e., values ha dffer subsanally from 0.035.. Models for sochascally me-varyng radngday coeffcens To adap our radng-day model o allow for sochascally me-varyng coeffcens, we le α, =,,..., be he effec of day n monh, and α α, =,,..., be he dfference beween he effec for day n monh and he average weekly effec for he monh, α (/ ) α. By defnon j j j= j= = = ( α α ) = 0, (4) and hus one of he coeffcens can be compued as a funcon of he oher sx. Wh hese me-varyng coeffcens, (3) becomes TD = T = T, (5) = where and T are column vecors wh componens = (,,, ) and T = ( T, T,, T ). To complee he specfcaon we need o gve a model for, or specfy a jon model for he α and derve he mpled model for. We shall use a random walk model o allow for me varaon n he radng-day effecs. We choose he random walk model because s boh smple, and nonsaonary. I also ncludes a fxed coeffcen as a specal case when he random walk nnovaon varance s zero. A smple model seems o be requred because we are no lkely o be able o esmae well an nvolved model for he sochasc radng-day coeffcens. We desre a nonsaonary model because use of a saonary model, such as a saonary auoregressve model of order one (AR()), would mply errac me varaon n he radng-day coeffcens. In fac, unless he AR parameer n he saonary AR() model s exremely close o one, he radng-day coeffcens would show errac varaon around he fxed means, and he model also would no allow for conssen movemen up or down over long perods. Hannan (94) made an analogous observaon n he conex of usng saonary AR models for a me-varyng seasonal componen. The random walk model s more appealng, as wll allow he coeffcens o change more smoohly over me and wll no e hem o fxed means. Bell (004) made a sraghforward generalzaon of (3) by assumng ha he follow ndependen random walk models: ( B) = η, =,,,, () where denoes he backshf operaor Bx =, B ( ) x and he η are muually ndependen whe nose seres wh varances σ. As before, he reference day n () s Sunday wh deermned from (4) as

. () = ( ) + + Though he reference day s usually chosen o be Sunday, n prncple we could change he model () o allow any gven day o serve as he reference day. The model () has he dsadvanage ha he correspondng o he reference day has dfferen sascal properes han he coeffcens correspondng o he oher sx days. To see hs wh Sunday as he reference day, noe ha applyng he dfference operaor B o boh sdes of () gves η ( B) = ( η + + η ). (8) Thus, alhough η,, η are assumed ndependen of one anoher, η wll be correlaed wh all of hem (excep f Var( η ) = 0 for some, n whch case η = 0 almos surely). Analogous resuls obvously hold f we change he reference day. How much effec hs ssue has on acual esmaes wll be examned for he examples n Secon 4. Harvey (989, pp. 43-44) nsead generalzed (3) by specfyng a model for,,, ha mplcly assumed ha he α follow ndependen random walk models ( B) α = ε, =,,...,, (9) where he ε are muually ndependen whe nose seres wh common varance σ ε. If we sll defne η = ( B) as n () and (8), bu now whou assumng ha he η are ndependen of one anoher, hen we see from he defnon of he and from (9) ha η = ( B) = ( B)( α α ) = ε ε, =,,...,, (0) where ε (/ ) ε. From (0) follows ha he η, = =,,..., have mean zero, varance ( j ) ε ( / ) σ ε and Cov η, η = σ /,, j =,,,; j. Thus, unlke wh Bell s model, he seven nnovaons η, η,, η are correlaed wh one anoher, bu all have he same sascal properes. However, he assumpon of a common varance for he ε, mplyng he common varance ( / ) σ ε for he η, s more resrcve han he assumpon of sx dfferen nnovaon varances n (). Ths rases a queson as o wheher hs resrcon s approprae n suaons where he effecs of one day may be changng faser over me han hose of oher days. We also noe ha (0) mples a mulvarae random walk model for he snce he nnovaons η are cross-correlaed. In hs form, Harvey s model canno be handled by he sofware we use for model fng because he sofware requres ndependen unobserved componens. We deal wh hs ssue n he nex secon. Noce ha he varance-covarance marx of η,, η for boh Bell s and Harvey s models s ( ) sngular snce he consran () mples ha Var η,, η = 0, where s a column vecor ( ) of seven ones. 3. FITTING THE MODELS TO DATA In hs secon we gve deals on fng me seres models ha nclude eher fxed or sochascally mevaryng radng-day effecs usng he REGCMPNT program (Bell 004). We frs gve background nformaon on he RegComponen model, and hen dscuss he seps necessary o conver Harvey s model o hs form so ha can be f by he REGCMPNT program. 3. The RegComponen model The general form of he RegComponen me seres model s y x δ h z = + = k () where x = ( x,, xr ) s a row vecor of known regresson varables a me and δ s he correspondng column vecor of fxed regresson parameers. The h, =,, k are seres of known consans ha we call scale facors and z, =,, k are seres of ndependen unobserved componen seres followng ARIMA models. The REGCMPNT program mplemens lkelhood evaluaon and maxmzaon, forecasng, and sgnal exracon for RegComponen models. The program pus he model n sae space form and uses he Kalman fler wh a suable smooher o do he calculaons. See Bell (004) for more deals. Fxed radng-day effecs are handled n REGCMPNT by ncorporang he radng-day varables and regresson coeffcens n (3) as par of he regresson effecs x δ. Bell s model () for mevaryng radng-day coeffcens s also easly handled n REGCMPNT by denfyng sx of he ARIMA componens n () wh he me-varyng z coeffcens, =,,...,, and seng he correspondng scale facors h o he radng-day varables T,, T. An addonal ARIMA componen s necessary for he resdual seres Z n (). An mporan pon s ha he RegComponen model, and hence he REGCMPNT program, requres ha he nnovaons n he models for he n () be z

ndependen. Snce hs s no rue for he nnovaons assocaed wh he random walk model for he n Harvey s model (9), o f hs model specal reamen s needed. We dscuss hs nex. 3. Expressng Harvey s model as a RegComponen model For noaonal purposes, le I m denoe an deny marx of order m, a column vecor of m ones, and m 0m a column vecor of m zeroes. To conver Harvey s model (9) o an equvalen model n RegComponen form, we make a lnear ransformaon of α o a mulple of α and sx seres γ, γ,, γ followng ndependen random walks. We do hs usng a marx of he form G C G =. (/ ) Any marx C wh rows ha are orhogonal o each oher and o he consan vecor wll work. A convenen choce s generaed by he rgonomerc funcons wh perod seven. Usng hese rgonomerc funcons, he sx rows of he marx C ha we used have enres (for columns j =,,, ) = ( π j ) c j = /sn( π j/) = ( π j ) c4 j = /sn( 4 π j/) = ( π j ) c ( π j ) c /cos / j c3 /cos 4 / j c5 /cos / j = /sn /. The facor / s used o normalze he rows of C o have lengh one. The marx G s hus orhonormal,.e. GG = GG = I. Defne he new varables γ = ( γ, γ,, γ ) by γ Gα, and noe hen ha G γ = α. We have, ( B) γ = ( B) Gα = Gε ξ. () From (), Var ( ξ) = GVar( ε) G = σ ε GG = σ ε I. Thus he vecors ξ have he same scalar varancecovarance marx as ε. We recover from γ n he followng manner. Defne he paroned marx H by [ ] H I 0 (/ ). Then, f α ( α,, α ), = Hα = HG γ =Γ γ (3) where Γ HG s he marx gven by Γ= ( C 0 ), (4) wh C beng he marx obaned by akng he frs sx rows of he marx C. From (3) and (4) we j oban as a lnear combnaon of he sx coeffcens (,,, ) γ γ γ γ usng he equaon = C γ. (5) Subsung (5) no (5) gves TD = T = T C γ = ( C T) γ. Ths shows ha by ransformng he radng-day varables each monh from T o r = CT, we can wre TD = rγ n erms of he vecor γ ha has componens ha follow sx ndependen random walks wh common varances σ. Havng esmaed γ, γ,, γ by sgnal exracon wh he fed model, we conver hese resuls o esmaes of usng (5). Tha s, leng y ( y, y,, y ) denoe he avalable daa, E( y) = CE ( γ y), wh sgnal exracon varance-covarance marx Var( y) = CVar ( γ y) C, where E( γ y) and Var( γ y) are respecvely he sgnal exracon mean and covarance marx of γ. (These were compued here by he REGCMPNT program, hough a modfcaon was needed o prn he full covarance marces. Ordnarly he program only prns ou he dagonal elemens, whch are he componen sgnal exracon varances Var( γ y).) 4. RESULTS FOR EXAMPLE SERIES To nvesgae possble me varaon of radngday effecs and o compare resuls usng he alernave models dscussed n Secon, we used REGCMPNT o f models o hree U.S. real sales me seres publshed by he Census Bureau: sales of deparmen sores (excludng leased deparmens), sales of women s clohng sores, and sales of shoe sores. All hree me seres are for he 384-monh perod from January 9 o December 998. (Noe: resuls presened here for deparmen sore sales dffer some from hose presened n Bell (004) because he me frame of he seres used s dfferen, and because benchmark revsons are regularly made o he daa over me.) Le y denoe he monhly seres obaned by akng logarhms of he specfc real sales seres n queson, afer dvdng by lengh of monh. In he model for y, n addon o radng-day effecs, we nclude an Easer effec wh a 0-day wndow (BH) and an arlne model for he remander: y = TD + ωe + Z, ( B)( B ) Z = ( θ B)( ΘB ) a, () ε n

where E s he Easer effec varable and assocaed parameer, and ω he s whe nose wh varance σ a. We f () wh TD represenng eher fxed or sochascally me-varyng radng-day effecs, usng he models dscussed n he las wo secons. Fgures and are plos of he sgnal exracon esmaes of he me-varyng radng-day coeffcens, =,,, over he 384 monhs for real sales of deparmen sores and of women s clohng sores, respecvely. The esmaes for shoe sore sales are no shown due o space lmaons. The esmaes ˆ assumng fxed radng-day effecs (3) are ncluded on he plos as a dashed sragh lne for comparson purposes. Egh dfferen plos of esmaed sochascally me-varyng coeffcens are ncluded on each graph. Seven of hese come from Bell s model (), wh each of he days of he week servng as he reference day n one of hem. The plo for whch he reference day corresponds o he day of he esmaed coeffcen s ploed wh long dark dashes, o dfferenae from he plos for he oher sx reference days ha are ploed as doed lnes. These sx ploed curves end o look smlar, and n many cases, some of hem approxmaely concde, so ha he plos for less han sx of he reference days are vsble. The eghh lne n each graph, correspondng o me-varyng coeffcen esmaes from Harvey s model (9), s ploed as a sold lne. For each of he hree seres, he fxed-effecs esmaes ndcae large posve effecs for Frday and Saurday, and a large negave effec for Sunday. Mos of he esmaed fxed effecs for he oher days were no sgnfcanly dfferen from zero. For deparmen sore sales, he mos noceable feaure n he graphs of he me-varyng coeffcens s he large ncrease n he effec of Sunday, from an effec on sales of less han % early n he seres, o a nearly neural effec a he seres end. Also, whle he esmaed fxed-effec coeffcen for Tuesday s close o zero, he esmaed me-varyng coeffcens for Tuesday show decreases n he effec hrough me from posve o negave values. The Frday coeffcen esmaes also decrease over me. Plos of he coeffcen esmaes for he oher days of he week vary less over me. For real sales of women s clohng sores, he ncrease n he esmaed me-varyng coeffcen for Sunday was much less han for he deparmen sores seres: a lle more han half a percen over he lengh of he seres. The Frday coeffcen dsplayed a smlarly moderae decrease, bu he esmaed coeffcens for he oher days showed very lle movemen. a The esmaed me-varyng coeffcens for he daa on shoe sore sales dsplay more varaon han for he seres on sales of women s clohng sores. However, he esmaed coeffcens do no move up or down n a conssen fashon. Many of he plos of he coeffcen esmaes from Bell s model for he reference days used n consrucng radng-day varables show somewha errac varaon over me. Ths hghlghs he pon made earler, ha for Bell s model he coeffcen correspondng o he reference day has dfferen sascal properes han he coeffcens for he oher sx days. For deparmen sores and women s clohng sores, he esmaes from Harvey s model behave smlarly o he esmaes from Bell s model wh reference days dfferen from he coeffcen day. For he daa on real sales of shoe sores, however, esmang Harvey s model yelded ˆ σ 0. The resulng esmaes of he ε had vrually no varaon, and n fac nearly concded wh he fxed-effecs esmaes. In conras, he esmaes from Bell s model vary over me for some of he days, as noed above. 5. CONCLUSIONS AND FUTURE RESEARCH In hs paper we consdered he modelng of mevaryng radng-day effecs usng models gven n Harvey (989) and Bell (004). The models are specal cases of RegComponen models (Bell 004), hough n he case of Harvey s model a lnear ransformaon was needed n order o wre he model n RegComponen form. The REGCMPNT program was used o f he models o me seres of real sales of U.S. deparmen sores, women s clohng sores, and shoe sores for he perod January 9 o December 998. Esmaes from he varous models used were mosly n agreemen wh regard o esmaed me varaon of he radng-day coeffcens for he example seres. The man excepon was ha n Bell s model, esmaes of he radng-day coeffcen for he reference day used n consrucng he radng-day varables ofen showed errac me varaon. Resuls hus vared wh he choce of he reference day. Though hs errac behavor was no presen n all cases and was no necessarly mporan relave o he overall me varaon, s noneheless a lmaon o Bell s model. The oher excepon was ha for shoe sore sales, resuls from Harvey s model showed no me varaon n he coeffcens snce he one nnovaon varance for he radng-day componen was esmaed o be approxmaely zero. The radng-day coeffcens for hs example esmaed from Bell s models dd no move n a conssen upward or downward drecon, rasng some concerns abou wheher hs esmaed me varaon s real. On he oher hand, snce Harvey s radng-day model uses only one varance, f

esmaon error led o hs varance beng ncorrecly esmaed a s boundary value of zero, hs would ncorrecly sugges no me varaon n any of he radng-day coeffcens. The resuls sugges several opcs for fuure research. One s o examne mehods of esng for he presence of me-varyng radng-day coeffcens. DSQ and DQ consdered hs opc, bu he mehods suggesed have sgnfcan lmaons. The work of Harvey and Srebel (998) on esng for deermnsc versus ndeermnsc cycles may be applcable. Anoher opc nvolves generalzng Harvey s model o allow for dfferen nnovaon varances for he dfferen days. We can easly do hs by droppng he common varance assumpon made for he random walk models gven n (9), and followng hrough wh he model dervaon as n Secon. However, f we conver he resulng model o RegComponen form as n Subsecon 3., he ransformaon marx ha wll orhogonalze he ε va () wll depend on he unknown varances. Ths generalzaon canno be accommodaed by our curren sofware. REFERENCES Bell, Wllam R. (984), Seasonal decomposon of deermnsc effecs, Research Repor 84/0, Sascal Research Dvson, U.S. Census Bureau. Bell, Wllam R. (004), On RegComponen Tme Seres Models and Ther Applcaons, n Sae Space and Unobserved Componen Models: Theory and Applcaons, eds. Andrew C. Harvey, Sem Jan Koopman, and Nel Shephard, Cambrdge, UK: Cambrdge Unversy Press, forhcomng. Bell, Wllam R. and Hllmer, Seven C. (983), Modelng me seres wh calendar varaon, Journal of he Amercan Sascal Assocaon, 8, 5-534. G Box, G. E. P. and Jenkns, G. M. (9), Tme Seres Analyss: Forecasng and Conrol (revsed edon), San Francsco: Holden Day. Dagum, E. B. and Quennevlle, B. (993), Dynamc Lnear Models for Tme Seres Componens, Journal of Economercs, 55, 333-35. Dagum, E. B., Quennevlle, B. and Suradhar, B. (99), Tradng-Day Varaons Mulple Regresson Models wh Random Parameers, Inernaonal Sascal Revew, 0, 5-3. Fndley, Davd. F., Monsell, Bran C., Bell, Wllam R., Oo, Mark C., and Chen, Bor-Chung, (998), New Capables and Mehods of he X--ARIMA Seasonal Adjusmen Program (wh dscusson), Journal of Busness and Economc Sascs,, -. Hannan, Edward J. (94), The Esmaon of a Changng Seasonal Paern, Journal of he Amercan Sascal Assocaon, 59, 03-0. Harvey, A. C. (989), Forecasng, Srucural Tme Seres Models and he Kalman Fler, Cambrdge, U. K.: Cambrdge Unversy Press. Harvey, A. and Srebel, M. (998), Tess for deermnsc versus ndeermnsc cycles, Journal of Tme Seres Analyss, 9(5), 505-59. Monsell, B. C. (983), Usng he Kalman Smooher o Adjus for Movng Tradng Day, Research Repor 83/04, Sascal Research Dvson, U.S. Census Bureau. Fgure Capons Fgure. Esmaed coeffcens of he daly radng-day effecs for real sales n U.S. Deparmen sores (excludng leased deparmens) durng he perod January 9 o December 998 Fgure. Esmaed coeffcens of he daly radng-day effecs for real sales n U.S. Women s clohng sores durng he perod January 9 o December 998. Legend for Fgures The esmaes of he sochascally me-varyng coeffcen usng Bell s model are ploed wh a dark dashed lne when he reference day corresponds o he day of he parameer esmae, and as doed lnes for oher reference days. Esmaes from Harvey s model are ploed as a sold lne, and he esmaed coeffcen for he fxed-effec model appears as a dashed sragh lne.

Sunday coeffen Deparmen Sores Monday coeffen Deparmen Sores esmaed coeffcen -0.030-0.00-0.00 0.0 esmaed coeffcen me me Tuesday coeffen Deparmen Sores Wednesday coeffen Deparmen Sores esmaed coeffcen esmaed coeffcen me me Thursday coeffen Deparmen Sores Frday coeffen Deparmen Sores esmaed coeffcen esmaed coeffcen me me Saurday coeffen Deparmen Sores esmaed coeffcen me

Sunday coeffen Women's Clohng Monday coeffen Women's Clohng esmaed coeffcen -0.030-0.00-0.00 0.0 esmaed coeffcen me me Tuesday coeffen Women's Clohng Wednesday coeffen Women's Clohng esmaed coeffcen esmaed coeffcen me me Thursday coeffen Women's Clohng Frday coeffen Women's Clohng esmaed coeffcen esmaed coeffcen me me Saurday coeffen Women's Clohng esmaed coeffcen me