Dscusson Paper No. 113 Mulvarae Tme Seres Model wh Herarchcal Srucure for Over-dspersed Dscree Oucomes Nobuhko Teru and Masaaka Ban Augus, 213 January, 213 (Frs verson) TOHOKU MANAGEMENT & ACCOUNTING RESEARCH GROUP GRADUATE SCHOOL OF ECONOMICS AND MANAGEMENT TOHOKU UNIVERSITY KAWAUCHI, AOBA-KU, SENDAI 98-8576 JAPAN
Mulvarae Tme Seres Model wh Herarchcal Srucure for Over-dspersed Dscree Oucomes Nobuhko Teru * and Masaaka Ban ** Augus, 213 January, 213 *Graduae School of Economcs and Managemen, Tohoku Unversy, Senda 98-8576, Japan **College of Economcs, Nhon Unversy, Chyoda-ku, Tokyo 12-836, Japan Correspondng Auhor: Nobuhko Teru, eru@econ.ohoku.ac.jp The auhors acknowledge useful commens from wo referees for revsng hs manuscrp. Teru also acknowledges he fnancal suppor of he Japanese Mnsry of Educaon Scenfc Research Grans (A)212433.
Mulvarae Tme Seres Model wh Herarchcal Srucure for Over-dspersed Dscree Oucomes Absrac In hs paper, we propose a mulvarae me seres model for over-dspersed dscree daa o explore he marke srucure based on sales coun dynamcs. We frs dscuss he mcrosrucure o show ha over-dsperson s nheren n he modelng of marke srucure based on sales coun daa. The model s bul on he lkelhood funcon nduced by decomposng sales coun response varables accordng o producs compeveness and condonng on her sum of varables, and augmens hem o hgher levels by usng Posson-Mulnomal relaonshp n a herarchcal way, represened as a ree srucure for he marke defnon. Sae space prors are appled o he srucured lkelhood o develop dynamc generalzed lnear models for dscree oucomes. For over-dsperson problem, Gamma compound Posson varables for produc sales couns and Drchle compound mulnomal varables for her shares are conneced n a herarchcal fashon. Insead of he densy funcon of compound dsrbuons, we propose a daa augmenaon approach for more effcen poseror compuaons n erms of he generaed augmened varables parcularly for generang forecass and predcve densy. We presen he emprcal applcaon usng weekly produc sales me seres n a sore o compare he proposed models accommodang over-dsperson wh alernave no over-dspersed models by several model selecon crera, ncludng n-sample f, ou-of-sample forecasng errors, and nformaon creron. The emprcal resuls show ha he proposed modelng works well for he over-dspersed models based on compound Posson varables and hey provde mproved resuls han models wh no consderaon of over-dsperson. Key words: Compound Posson, Compound Mulnomal, Dscree Oucomes, Dynamc Generalzed Lnear Model, Herarchcal Marke Srucure, MCMC, Over-dsperson
1. Inroducon A full Bayesan analyss on he dynamcs of dscree responses such as couns has been faclaed by he Markov chan Mone Carlo (MCMC) mehods for more han 2 years. The scope s beyond he earler works wh maxmum lkelhood mehod by Harvey and Fernandes (1989) and Ord, Fernandes, and Harvey (1993). In parcular, Wes e al. (1985) and Cargnon e al. (1997) developed me-seres models for varables followng a mulnomal dsrbuon by nroducng dynamc lnear models usng he Bayesan approach. The former proposed a dynamc model wh a mulnomal dsrbuon and he laer deal wh several ses of mulnomal dsrbuons, boh of whch assumed he oal number of varables o be consan. In conras, he sochasc models for a dscree response exhb an neresng dsrbuonal propery: he reproducon of Posson varables and he condonal dsrbuon of hese varables on her sum follow a mulnomal dsrbuon. Teru e al. (21) used hs Posson mulnomal relaonshp n a dynamc generalzed lnear model o propose a mulvarae me-seres model wh a herarchcal srucure beween varables for specfyng marke srucure on he bass of a produc s sales dynamcs. We use he erm marke srucure o refer o how we classfy producs no several groups called submarkes or caegores, so ha he producs are compeve nsde a submarke bu no ousde. Ther model was a macro model for drec aggregae sales, whou consderaon of mcrosrucure. On he oher hand, he Posson varable has a lmed propery of havng dencal frs wo momens, and herefore he over-dsperson has been dscussed as mporan ssues n he leraure parcularly n economercs, as s fully dscussed n Wnkelmann (28). In hs paper, we exend he model of Teru e al. (21) such ha he dscree response varables have over-dspersons. We frs ncorporae he model of an ndvdual consumer s purchase and hen aggregae hose up o a produc sale, afer whch we fnd a mcrosrucure o generae over-dspersons for our applcaon. We prove ha over-dsperson s nheren 1
whenever consumers n he marke do no behave ndependenly, as s usually assumed n economcs and markeng. Then, we propose o buld a mulvarae me seres model wh herarchcal srucures by usng Gamma compound Posson varables for dscree responses havng over-dsperson. We frs develop he sascal modelng of compound dsrbuons o represen a marke srucure, and hen, propose a daa-augmenaon approach. Tha s, we rean he Posson Mulnomal dsrbuonal relaonshp mplyng he number of oal sales and produc s marke share, where we augmen Posson and Mulnomal parameers, whch are generaed by Gamma and Drchle dsrbuons, respecvely. Then, we oban he jon poseror densy by a full Bayesan MCMC procedure. We dscuss he mcrosrucure for he nheren over-dsperson n our problem n Secon 2. Secon 3 descrbes he properes of he compound Posson and compound mulnomal dsrbuons used as he buldng blocks for our model. The srucure of he model s explaned n Secon 4, and daa augmenaon approach o over-dsperson s proposed n Secon 5. Secon 6 descrbes model specfcaon n he dynamc generalzed lnear model and derves jon poseror densy. Secon 7 deals wh he esmaon and forecasng procedures. Secon 8 repors he emprcal applcaon. Concludng remarks are provded n Secon 9. 2. Mcrosrucure for Generang Over-dspersons Suppose ha here are H poenal consumers n he marke and he number of purchases of produc by consumer h a me follows a Posson dsrbuon wh parameer h as x h Posson, (1) h on he ground ha consumers wll make no purchase or small quany of buyng produc a a specfc me. Thus, he oal number of sales for produc, y, also follows Posson dsrbuon 2
H, (2) * h h h1 * h 1 H y x I hc x where I h C s an ndcaor funcon, akng value 1 when consumer h belongs o a poenal consumer se C,.e., when he/she s ready o buy, and oherwse, and H wh * H I h C h1 h beng he ndex for reorderng of consumers n he se C. In hs crcumsance, he over-dsperson phenomenon s derved by evaluang he mean and varance of y as because holds H H * * * h h, (3) E y E x * * h 1 h H y x * * * h x x h h ' * h hh' * * x * x * h h ' E y Var Var Cov, h * h * ' hh' Cov,, Cov x, x for any par of correlaed Posson varables. Tha s, equaon (4) mples ha here s over-dsperson whenever a leas one par of consumers does no behave ndependenly. Ths s no a srong assumpon, as jusfed by dscussons on he exsence of a reference group n socey and s decson makng, gong back o Hyman (1942), and s applcaon o socal psychology on he bass of he consumer behavor heory, such as Park and Lessg (1977) and Bearden and Ezel (1982). In fac, markeng models have been developed based on a commonaly across consumers when hey represen heerogeney n he random effec models. Ths s well explaned n he ex book Ross e al.(25). On he oher hand, Gamma compound Posson varables wh posve parameersa,, denoed as y~ Compound Posson, a, are suable for over-dsperson as hey conan (4) he frs wo momens 3
a ya Ey E y Var 1. (5) In he nex secon, we assume ha he number of produc sales, y, by aggregang over an ndvdual consumer s purchase, as shown n equaon (2), follows he compound Posson dsrbuon. 3. Gamma Compound Posson and Drchle Compound Mulnomal Dsrbuons A Gamma compound Posson varable s defned by he mxure of he Posson varable havng parameer n he Gamma dsrbuon wha,, and s densy funcon s evaluaed as a negave bnomal dsrbuon., a, p y a a a y 1 f y f a, d. y! 1 1 y a (6) Hoadley (1969) dscussed he reproducve propery of Gamma compound Posson varables and he condonal dsrbuon for a se of hese varables when he sum of varables s gven. Tha s, le y1, y2,..., y I be muually ndependen random varables havng a Gamma compound Posson dsrbuon wh he second parameer common across subjecs,.e., Then, he sum n y1 y2... yi follows y ~ Compound Posson a,. (7) n ~ Compound Posson, Furhermore, he condonal dsrbuon of y I a 1. (8) y1 y2 y I,,..., ' when n s gven s shown as y n~ Drchle Compound Mulnomal y n, a, (9) 4
where a a I, 1,..., and s densy s derved by n 1 n! a y a py n, a f y n, f ad. (1) n a y! a 4. Models for Defnng Marke Srucure The producs are more or less compeve n her marke. These producs are grouped accordng o he degree of compeveness no several segmens, and furher caegorzng hese groups o hgher levels o make subgroups leads o a herarchcal srucure for he marke defnon. A ree srucure s used o represen he herarchcal naure of compeve relaonshps among producs n a graphcal form, as shown n Fgure 1. Fgure 1(a) ndcaes he marke wh no specfc srucure and each produc sale s drecly accumulaed o marke sale. On he oher hand, Fgure 1(b) shows he suaon ha some groups of produc sales are respecvely accumulaed o sub-markes frs and hen sub marke sales consue marke sale. Fgure 1 (a), (b) Marke Srucures Basc Srucure Le us assume ha here are I producs n he marke and ha y s he number of sales for he produc a me ( 1,..., T), whch follows he Gamma compound Posson dsrbuon ndependenly wh a me-varyng parameer a, defned by (7) for 1,..., I when here s no compeve relaonshp wh each oher. Then, we oban he Gamma compound Posson dsrbuon wh a, for marke sales, defned as he aggregae of produc sales, n I y 1, under he assumpon of no specfc srucure among producs n he marke, as shown n Fgure 1(a). Tha s, we have margnal dsrbuons for 5
produc and marke sales by * y ~ Compound Posson a,, n Compound Posson a,, (11) where a * I a. Furhermore, afer 1 n s gven, he condonal dsrbuon of produc follows sales y y, 1,..., I where a a a y n~ Drchle Compound Mulnomal y n, a, (12),..., ' 1. The sequenal use of equaons (11) and (12) produces a jon I dsrbuon for marke and produc sales, *,,,, p n y a p n a p y n a. (13) We noe ha he condonng se a *,, a has equvalen nformaon wh a, f we ake a as a full-dmensonal vecor wh a nondegeneraed dsrbuon. In conras, Teru e al. (21) proposed a dynamc generalzed lnear model based on Posson varables whou over-dspersons. Ths represens a macromodel for aggregae sales drecly whou consderng he mcrosrucure. They used he reproducve propery of Posson varables and condonal mulnomal dsrbuon when he sum of varables s gven, and proposed a mulvarae me-seres model wh a herarchcal srucure based on he dscree oucomes. Tha s, we have margnal dsrbuons y Posson and n * Posson and condonal dsrbuon y n Mulnomal y n,, where * I I and j j, j 1,..., I 1 1. The lkelhood a me s defned by 6
* where he nduced parameers and and marke shares for each produc. Hgher Order Srucure * *,,, p n y p n p y n, (14), respecvely, represen he expeced oal sales Ths model s exended o a hgher order herarchcal srucure, as developed by Teru e al. (21). Nex, we fully explan he model specfcaons, ncludng he desgn marx of sae space prors, as hs model s appled o acual me-seres daa n he emprcal analyss. Exendng he Posson mulnomal relaonshp n he above-descrbed manner, we decompose he marke srucure no L submarkes [ k ], ' k M k such ha y L k1 y [ k ], where y y M represens N k dmensonal vecor of he producs ha are grouped n M, k 1,..., L. Gven aggregaed submarke sales k m [ k ] Mk y, he condonal [ k] [ k] dsrbuon y m, k 1,..., L, follows ndependen N k -dmensonal mulnomal [ k ] dsrbuon snce l for [ l] [ l] [ k] [ k] y s are orhogonal o each oher,.e., y m y m k by he defnon of a submarke.,..., ' denoe an L-dmensonal vecor of submarke sales. Then, [1] [ L] Nex, le m m m m n follows a mulnomal dsrbuon condonal on he sum of submarke sales,.e., marke sales n We noe ha L m k 1 [ k ]. In bref, we have a hree-layer herarchcal marke srucure model. y and n are ndependen, condonal on m. Then, he jon densy funcon of I produc sales (boom layer), L submarke sales (mddle layer), and marke sales (op layer) are decomposed no,, p n m y p n p m n p y m L [ k] [ k] [ k] p n p m n p y m. k 1 (15) 7
, * Then, we oban margnal and condonal daa dsrbuons: n Posson m n Mulnomal n, y m Mulnomal m,, where k L [ k ] [ k ] [ k ] [ k ] N k [ k] [ k] [ k] / j j1 N j L Nk [ j] [ k] j / 1 k1 1 j 1,..., L 1 [ k] [ k 1,...,, ], M k,, 1,..., 1, and [ k ] s he parameer of he Posson varable N k classfed o he submarke M k. The srucure of hs model s llusraed n Fgure 1(b). 5. Daa Augmenaon Approach o Over-dsperson Daa Augmenaon Exendng he model for accommodang over-dsperson, nsead of drec use of denses (6) and (1), we ake a daa augmenaon approach o keep he orgnal parameers,.e., mplyng expeced sales and marke shares for each produc n he modelng, and use he * ' generaed sample of augmened varable z lkelhood for he parameersa, :, ' n he MCMC process o defne he *, =Gamma, Drchle p z a a a. (16) By usng he relaon of poseror densy * *,,,,,, * * * *,, p a n y p a n y d d (17) p n p a d p y n p a d, we evaluae hese negrals by augmenng he gamma and Drchle parameers n erms of generang he s-h samples *( s ) *( ) *( ) ( ) from gamma p s s, s a and ( s) from Drchle p ( s ) ( s ) ( s) *( s) ( s) n, a n MCMC eraons. Then, condonal on z oban he Posson Mulnomal lkelhood funcon, ' ', we 8
*( s) ( s) pn p y n,, (18) and hs forms a buldng block o consue a herarchcal srucure. In case of wo-layer models and hree submarke (L=3), s exended as follows: for * ' [1] [2] [3] z,, ', ', ' ',,,,, ;,, * * * * pn p a, d p a n m y p a z n m y dz L p m n p a d p y m p a d [ k] [ k] [ k] [ k] [ k] [ k], m,, k 1 (19) where am a, k 1,..., L Mk [ k ] and, a a M. k We evaluae hese negrals by augmenng he gamma and Drchle parameers erms of generaed samples ( s ( ) Drchle ) s p n, am *( s ) *( ) *( ) ( ) from Gamma p s s, s a and ( ) ( ) [ k ] s [ k ] s p a ( s) *( s) '( s) [1]( s) [2]( s) [3]( s), ( s) and [ k]( s) z n from, respecvely. Then, condonal on z,, ', ', ' ', we oban he Posson Mulnomal lkelhood funcon: L *( s ) ( s ) [ k ]( s ) [ k, ]( s ), [ k ]( s ) p n p m n p y m. (2) k 1 6. Model Specfcaon and Jon Poseror Densy Dynamc Generalzed Lnear Model Usng he expecaon of produc sales leads o E y a. (21) Thus, we nerpre ha he expeced sale s decomposed no an ndvdual mean a and a common mean across producs. In urn, we model he mean funcon as a f x, 9
connecng wh markeng mx varables x and sochasc error. We specfy he srucure n more deal as log a a log x ', 1,..., I. (22) Tha s, he ndvdual mean has a produc-specfc mean a and a common me rend log, * and s assumed o be mosly explaned by markeng mx varables. We se a log a a * and denoe log, and hen condonal on a, we have he dynamc equaon a * * Ths consues he srucural equaon F v * = 1, 2,..., I, ' x '. (23) * *, where a ai, F s he marx defned by he srucure on 1,..., ' and n equaon (24), and v s he error vecor comprsng of. As for he dynamc sae vecor used n he * applcaon, we specfy he second-order local common rend for and he frs-order local rend model for he response parameer,.e., w ; w 1w3, 1,... I, * * 1 1 1 1 2 (24) and hs specfes he sysem equaon H 1 w. Coupled wh he daa dsrbuon (15), we defne he dynamc generalzed lnear model wh he sae space pror: F v, v ~ N(, V) H 1 w, w ~ N(, W). (25) Jon Poseror Densy Under he usual assumpon ha he pror densy for he covarance marx of srucural and sysem equaons pvw, pv pw, we can express he pror dsrbuon as 1
,,, p V W a T 1,,, 1,, p X V a p W a p V p W. (26) Nex, we se he pror dsrbuon pa of he ndvdual mean parameer a by a N a a N a b IG n s. (27) 2 2 2 2, / ;, ; /2, /2 By arrangng he erm n (22) as c a, where c log a log x ', for 1,..., T, we consue he lkelhood for he mean a condonal on he error varance and derve condonal poseror dsrbuon p a V W a 2 closed form as,,,,, n a 2 a Tc v, (28) T 2 2 2 N, 2 2 T a v where T c c / T and 1 a means he se of ak, k 1,..., I excludng a. Fnally, we oban he jon poseror densy wh daa augmenaon of z, ', ', ', ' ' by equaon (29): * [1] [2] [3] 2 p,, V, W, a, ; z daa * * 3 [ k ] [ k,,,, ] [ k ], T k 1 1 p n p p m n p p y m p I 1,,, 1, p X V a p W p V p W 2 2 p a,, V, W, a, p a p. In (29), daa means he observed daa y, x. (29) Our model wh over-dspersons s characerzed as he dynamc generalzed lnear models 11
of he Posson Mulnomal dsrbuon perurbed by he Gamma Drchle dsrbuons. 7. Esmaon and Forecasng Samplng he lnk funcon for MCMC In addon o he sandard Bayesan nference on sae space modelng by dynamc lnear models (DLMs) by Wes and Harrson (1997), we use he MCMC approach o esmae he model by usng Meropols Hasngs samplng specfcally for he condonal poseror densy of lnk funcons based on he daa-augmened represenaon,,,daa,,,, p F V p n m y z p z p F V dz, (3) * [1] [2] [3] where z, ', ', ', ' '. Once he values of for equaon (3) are gven, he srucural equaons coupled wh he sysem equaons n her sae space prors n equaon (25) consue he convenonal Gaussan sae space models. The mul-move sampler by Carer and Kohn (1994) and Fruhwrh-Schnaer (1994) s used o sample he sae vecor. We assume ha he nal values of he sae vecor follow a mulvarae normal dsrbuon N, d I. The mean vecor was se as he esmae of he coeffcen on sac regresson,.e., he regresson wh me-nvaran coeffcen, and we se d =.1 for he emprcal applcaon. Predcve Densy Nex, one-sep-ahead predcve densy p y daa 1 s evaluaed by 12
y 1,daa 1 1, 1 1 1 1,,,daa T T T T T T T p T 1 T, V, W,daa pv, Wd TdVdW, p z p z p V W (31) where p,,,daa T 1 T 1 V W s he condonal predcve densy of lnk parameers when he predced sae vecor, and srucural and sysem error covarance marces are gven. The MCMC mehod s appled o evaluae hs predcve densy by he augmenng procedure. To evaluae hs densy, we frs exend equaon (16) o defne he predcve lkelhood of condonal on T 1 by T 1,,,,daa Gamma, p z V W a d * * T1 T 1 T1 T1 T1 T1 T1 T1 3 Drchle a Drchle a d d [ k] [ k] [ k] T1 m, T1 T1 T1 T1 T1 k 1 (32) The deals of he samplng scheme for MCMC are descrbed n he appendx. 8. Emprcal Applcaon Daa and Varables We use he sore level scanner, pon of sales (POS), me seres n he curry roux caegory ha was appled o our prevous model n Teru e al. (21) for comparson wh he model wh over-dsperson. The weekly seres comprses hree manufacurers ha produce hree producs each, for a oal of nne producs durng 11 weeks. Table 1 Summary of Daa Fgure 2 Hsogram of Weekly Produc Sales Daa Table 1 descrbes he summary sascs for sales, prce dsplay, and feaure daa. In parcular, sales daa conans varance o gve an evdence of he presence of over-dsperson 13
n he daa. Fgure 2 show he hsogram of brand sales. The frs 1 weeks are used for esmaon and he las 1 weeks are reserved for valdaon of forecasng. The daa conan he amoun of produc sales for y, and prces, dsplay (n-sore promoon), and feaures (adversng n newspaper) for markeng mx varables x. x conans no only varables of her own, bu also hose wh oher producs. The dsplay and feaures are bnary daa akng a value of 1 when was on and when was off. The logs of prce daa are used. Model Comparson Each of he hree makers, A, B, and C, produces hree caegores of producs accordng o he level of spcness o accommodae he dfference n consumer ases (1: No spcy, 2: Medum spcy, 3: Spcy). Followng he dscusson of Teru (211), we assume hree possble marke srucures: (1) produc caegory, (2) makers, and (3) usage,.e., ordnary or luxury usage, as shown n Fgure 3, and compare hese models wh over-dsperson and whou over-dsperson. Tha s, we have sx models o be compared. Fgure 3 Comparave Models The op of Table 2-1 shows he log of margnal lkelhood (LML) as an n-sample f creron and wo ypes of predcve measures,.e., he devance nformaon crera (DIC) by Spegelhaler e al. (22), and he roo mean squared errors (RMSE) of 1-sep-ahead forecass of hold-ou samples as ou-of-sample crera. There are hree levels for forecasng he RMSE: marke, submarke, and produc. These errors, ncludng he null model of no srucure, are repored n he lower panel of he able 2-2. We apply wo ypes of measures: sum1 and sum2. sum1 s he sum of all errors 14
nduced by he model n whch we have no specfc preference on he levels o be predced. sum2 s defned as he sum of marke and produc errors by consderng ha he numbers of submarkes dffer beween null and oher srucures. Accordng o he hree crera, he proposed models based on compound Posson varables accommodang over-dsperson mprove he models wh no over-dspersons. In parcular, he compound Posson varable model under he marke srucure (3) shows he bes performance. Table 2-1 Model Comparson: Overall Table 2-2 Model Comparson: Decomposon of RMSE Table 3 Esmaes of Produc Inercep Fgure 4 In-sample Performance and Forecasng Marke, Submarke, and Produc Fgure 5 Esmaes of Srucural Parameer The lef panels of Fgure 4 show he predced f of n-sample daa for marke, submarke of usage, and produc levels, where each observaon s denoed by a do and he esmaes are conneced by sragh lnes. We observe ha he model fs he marke sales well over he observaonal perod. The rgh panel of each elemen depcs s 1-sep-ahead forecasng for hese sales, where he mean values of he predced densy a each predcon sep are conneced by a connuous lne, and he 2.5% and 97.5% quanles of he densy a each sep are conneced by dashed lnes. The hold-ou samples are denoed by dos n he fgure. Ths shows ha he marke wll gradually expand over he nex 1 weeks, and hese forecass are conssen wh he movemen of hold-ou samples. We generae he forecass keepng he las observaon x for he predcon seps. T 15
Fgure 5(a), (b) show he rajecory of esmaed parameers appears ha and 1,..., 9 a a. I s are flucuang downward for he frs perod and hen urnng upward wh local rends around a mean level of.9998. a1,..., a 9 move more heerogeneously wh large flucuaons, whch should be proporonal o he observed dscree oucomes. Fgure 5(c) ndcaes rends for produc A2 and B3 sales, whch belong o a dfferen caegory; shows he oppose rends. Fgure 5(d) depcs he me-varyng prce coeffcen esmaes n response o prce and promoons ( End dsplay and Adversng ). We confrm he compeve relaonshp beween submarkes, and more neresngly, we fnd ha producs B1 and C2 are no hosle o A1 n he sense of prcng sraegy, as hey have he same coeffcen sgn as ha of A1. 9. Concludng Remarks In hs sudy, we proposed a mulvarae me seres model for over-dspersed dscree daa. Theren, we exended he model wh he herarchcal srucure by Teru e al. (21) o accommodae he over-dsperson problem nheren n he modelng of marke srucure based on sales coun dynamcs. We frs dscussed he mechansm of he mcrosrucure for generang over-dsperson n a number of dscree sales daa. The Gamma compound Posson varable for produc sales coun responses and Drchle compound mulnomal varables for produc share are conneced n a herarchcal fashon as a ree srucure for depcng a marke. The model s based on he lkelhood generaed by decomposng sales coun response varables accordng o he degree of compeveness among producs and condonng on her sum, and bulds hem up o hgher levels, represened as a ree srucure. Sae space prors are appled o he lkelhood generaed by he compound dsrbuons o develop dynamc generalzed lnear models for dscree responses wh a 16
herarchcal srucure. We frs showed ha over-dsperson s nheren o he problems where consumers are more or less dependen. Then, we model he compound dsrbuons for accommodang over-dspersons. However, nsead of he drec use of he densy funcon of compound dsrbuons, we augmen varables o make he numercal negraons for mxng easer, and provde more effcen algorhms compared o he mehod ha makes drec use of compound dsrbuons. The emprcal analyss by weekly produc sales daa n a sore showed ha our modelng worked well and he models wh over-dsperson, whch s consruced by compound Poson varables, performs beer han he models whou over-dsperson. There are a few problems for fuure research. One s he exenson of he heorecal sudy. The zero-nflaed Posson (ZIP) model by, for example, Lamber (1992) could be also appled o our modelng when he daa conan many zeros. In parcular, hs could be mporan f we furher ncorporae modelng of ndvdual consumer behavor n he analyss. Ths could be accommodaed by mxure dsrbuons hrough herarchcal models, as used n hs sudy. However, hs addonal mxure modelng demands more complcaed compuaon procedures. The expeced gans from hs exenson could no be subsanal, compared wh he developmen of a new model, and hus, we would lke o leave hs modfcaon of he model for fuure research. 17
Appendx: MCMC Algorhm A.1 Dynamc Generalzed Lnear Model We summarze he pror and condonal poseror dsrbuon used for our proposed model below. Condonal Poseror Dsrbuon We run 1, MCMC eraons n he model. In all models, we used he las 5, eraons o esmae he poseror dsrbuon of model parameers. When he nal values of he parameers are gven, he condonal poseror densy of he necessary parameers s generaed as follows. () s defned by equaon (23), and we use Meropols Hasngs wh a random walk algorhm,, ( s) ( s1) N 1 I where I s an deny marx wh correspondng dmensons. Accepance probably s defned as ( s) ( s) ( s) p ( ) ( 1) F,, V,daa s s, mn,1. ( s1) ( s1) ( s1) p F,, V,daa Afer obanng he draw of hese parameers, we use Wes and Harrson s (1997) sandard Bayesan nference procedure on sae space modelng by DLM. () The mul-move sampler by Carer and Kohn (1994) and Fruhwrh-Schnaer (1994) s used o sample he sae vecor () * Generae (v) (v) Generae [ k ] Generae (v) a *( s ) * *( s) ( s) from gamma p a, ( s ) ( s) from Drchle p n, a m [ k]( s) [ k [ ] from Drchle ] k ( s p ) a 18
Generae a from ( s ) 2 2 b a ma N,, 2 2 b m b m where m a a / m. We se a, b 1 n he emprcal analyss. 1 (v) 2 Generae m ( )/2, /2. 1 2( s ) from IG n 2 m s a a We se n 2, s m n he emprcal analyss. A.2 Forecasng Dscree Oucomes and Consung Predcve Densy Gven he s-h draw of MCMC ( ) ( ) ( ) s, s, s T V W, ( s ) () oban he forecas p,,,daa T 1 T 1 V W p,,,daa T1 T V W from T 1 by he algorhm of Gaussan sae space model; z T from p 1 zt 1 T 1 ( s) * ( s) ( s) [1] ( s) [2] ( s) [3] ( s) zt 1 1, T T 1 ', T 1 ', T 1 ', T 1 ' ' ; () generae he random number ( s ) * ( s n ) T1 Posson T 1 ( s ) ( s ) (v) gven nt ogeher wh he parameer values 1 T 1 ( s ) () oban he forecas samplng from he mulnomal dsrbuon ( s) ( s ) ( s ) ( s ) T 1 T1 T1 T1 o ge he parameer for he marke sales forecas; ( s), generae by mt 1 m n Mulnomal n, for he submarke sales forecass; [ k]( s) (v) gven [ k]( s) ogeher wh he parameer values T 1 m1 of he mulnomal dsrbuon of submarke M k, generae he respecve produc s forecass by samplng from he mulnomal dsrbuon [ k ]( s ) [ k]( s) [ k]( s) [ k ]( s y ) T 1 mt1 Mulnomal m T1, T 1 for k=1,, L; (v) erae seps () (v) M mes. [ k]( s) Then, he emprcal dsrbuon of y 1, s b,..., M approxmaes he predcve [ k ] densy (31) n z y. We se he burn-n parameer b = 5, and he oal number of T1 1 eraons M = 1, for he emprcal applcaon afer checkng he convergence. Seven hours of compuaon were necessary o mplemen our emprcal analyss. By exendng he 19
above forecasng seps up o H sep ahead, we obaned he MCMC sample pah [ k ]( s ) [ k ]( s ) [ k ]( s ) 1 2 H y, y,..., y for he jon predcve densy. 2
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Bayesan forecasng, Journal of he Amercan Sascal Assocaon, vol. 8, pp. 73-97. Wnkelmann, R. (28), Economerc Analyss of Coun Daa (Ffh edon), Sprnger, Hedelberg. 22
Table 1 Summary of Daa weekly weekly weekly weekly Brand Sales Prce (/1g) Dsplay Feaure Average (Varance) Average Average Average A1 33.74 (157.47) 9.4 2.9.4 B1 74.11 (4287.59) 9.33 2.1.4 C1 51.29 (234.13) 89.59 2.9.4 A2 61.54 (2915.8) 87.29 1.52.23 B2 87. (447.92) 83.71 2.73.23 C2 49.67 (1551.92) 8.62 3.39.2 A3 27.22 (578.82) 11.22.69.19 B3 48.13 (1947.57) 99.31.73.19 C3 24.42 (959.64) 99.3.72.18 Table 2-1 Model Comparson: Overall ML DIC RMSE(sum1) RMSE(sum2) Posson Null 15453-362 792.71 792.71 Produc Caegory 15558-381 648.96 48.6 Maker 15575-388 858.24 515.76 Usage 15593-3995 621.13 449.51 Compound Posson Null 15797-3117 653.99 653.99 Produc Caegory 15827-3122 65.41 467.65 Maker 15829-31235 77.48 49. Usage 15868-3139 612.19 412.85 ML: he log of margnal lkelhood, DIC: Devaon nformaon measure RMSE: he roo mean squared errors of 1 sep ahead forecass 23
Table 2-2 Model Comparson: Decomposon of RMSE Posson Null Produc Caegory Maker Usage Marke 222.65 Marke 164.36 Marke 138.73 Marke 171.34 A1 47.52 Caegory A 65.71 Maker 1 133.65 Usage 1 125.81 A2 31.21 A1 45.75 A1 49.74 A1 38.74 A3 76.93 A2 27.64 B1 62.98 B1 75.44 B1 15.19 A3 9.66 C1 37.2 C1 42.16 B2 78.92 Caegory B 67.77 Maker 2 141.38 A2 19.21 B3 83.77 B1 71.43 A2 52.44 B2 33.27 C1 54.63 B2 47.72 B2 59.21 C2 19.28 C2 51.56 B3 35.98 C2 31.8 C3 4.32 Caegory C 35.41 Maker 3 67.46 Usage 2 45.81 C1 44.85 A3 22.5 A3 17.19 C2 19.1 B3 39.22 B3 23.71 C3 13.68 C3 22.65 C3 9.17 (sum1) 792.71 (sum1) 648.96 (sum1) 858.24 (sum1) 621.13 (sum2) 792.71 (sum2) 48.6 (sum2) 515.76 (sum2) 449.51 Null Produc Caegory Compound Posson Maker Usage Marke 35.95 Marke 215.32 Marke 198.35 Marke 23.94 A1 42.34 Caegory A 49.88 Maker 1 124.41 Usage 1 139.2 A2 93.88 A1 2.4 A1 29.99 A1 26.89 A3 55.41 A2 2.69 B1 63.23 B1 3.3 B1 28.2 A3 25.7 C1 35.25 C1 21.26 B2 52.6 Caegory B 81.87 Maker 2 75.66 A2 16.61 B3 15.59 B1 5.56 A2 17.35 B2 29.37 C1 22.97 B2 32.68 B2 47.74 C2 2.5 C2 27.33 B3 33.89 C2 13.26 C3 1.26 Caegory C 51.1 Maker 3 8.41 Usage 2 6.14 C1 25.87 A3 14.96 A3 22.1 C2 19.19 B3 47.4 B3 2.79 C3 24.34 C3 22.47 C3 21.18 (sum1) 653.99 (sum1) 65.41 (sum1) 77.48 (sum1) 612.19 (sum2) 653.99 (sum2) 467.65 (sum2) 49. (sum2) 412.85 24
Table 3 Esmaes of Produc Inercep Caegory Brand Pos Mean Pos S.D. A1.17.6 B1.81.7 Usage 1 C1 -.1.6 A2.69.1 B2.94.7 C2.26.6 A3 -.48.8 Usage 2 B3.33.8 C3-1.3.8 25
Fgure 1 Marke Srucure (a) No Specfc Srucure n y 1 y 2 y I (b)three-layer Herarchcal Marke Srucure n Marke m1 m2 m3 Submarke (Caegory) [1] y 1 [1] y 2 [1] [2] [2] [2] y N 1 y 1 y 2 [3] [3] [3] y y 1 y 2 y N N 3 2 Produc 26
Fgure 2 Hsogram of Weekly Produc Sales Daa 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 A1 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 B1 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 C1 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 A2 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 B2 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 C2 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 A3 5 4 3 2 1 8 23 38 53 68 83 98 113128143158 B3 27
5 4 3 2 1 8 23 38 53 68 83 98 113128143158 C3 Fgure 3 Comparave Models Maker Produc Caegory A B C 1 A1 B1 C1 1 2 A2 B2 C2 3 A3 B3 C3 2 Usage Three makers produce hree caegores of producs and hey are classfed no wo groups by her usages. 28
Fgure 4 In-sample Performance and Forecasng: Marke, Submarke, and Produc (a) In-sample f Marke Level (b) Forecasng Marke (marke sales) Marke (marke sales) 14 12 1 8 6 4 2 1 1 19 28 37 46 55 64 73 82 91 1 12 1 8 6 4 2 11 12 13 14 15 16 17 18 19 11 Caegory Level Usage 1 (Caegory sales) Usage 1 (caegory sales) 1 8 6 4 2 1 1 19 28 37 46 55 64 73 82 91 1 8 7 6 5 4 3 2 1 11 12 13 14 15 16 17 18 19 11 Usage 2 (Caegory sales) Usage 2 (caegory sales) 1 5 8 4 6 3 4 2 2 1 1 1 19 28 37 46 55 64 73 82 91 1 11 12 13 14 15 16 17 18 19 11 Produc Level 18 16 14 12 1 8 6 4 2 Usage 1, A1 (brand sales) 1 1 19 28 37 46 55 64 73 82 91 1 18 16 14 12 1 8 6 4 2 Usage 1, A1 (brand sales) 11 12 13 14 15 16 17 18 19 11 29
Usage 2, B3 (brand sales) Usage 2, B3 (brand sales) 3 3 25 25 2 2 15 15 1 1 5 5 1 1 19 28 37 46 55 64 73 82 91 1 11 12 13 14 15 16 17 18 19 11 Fgure 5 Esmaes of Srucural Parameer (a) Marke, Tau 1..9999.9998.9997.9996.9995 1 1 19 28 37 46 55 64 73 82 91 1 (b) a Usage1 A1 Usage1 B1 3. 3. 25. 25. 2. 2. 15. 15. 1. 1. 5. 5... 1 1 19 28 37 46 55 64 73 82 91 1 1 1 19 28 37 46 55 64 73 82 91 1 Usage1 C1 Usage1 A2 3. 3. 25. 25. 2. 2. 15. 15. 1. 1. 5. 5... 1 1 19 28 37 46 55 64 73 82 91 1 1 1 19 28 37 46 55 64 73 82 91 1 3
Usage1 B2 Usage1 C2 4. 35. 3. 25. 2. 15. 1. 5.. 3. 25. 2. 15. 1. 5.. 1 1 19 28 37 46 55 64 73 82 91 1 1 1 19 28 37 46 55 64 73 82 91 1 Usage2 A3 Usage2 B3 3. 25. 3. 25. 2. 2. 15. 15. 1. 1. 5. 5... 1 1 19 28 37 46 55 64 73 82 91 1 1 1 19 28 37 46 55 64 73 82 91 1 Usage2 C3 3. 25. 2. 15. 1. 5.. 1 1 19 28 37 46 55 64 73 82 91 1 (c) Trend Usage 1, A2, Trend Usage 2, B3, Trend.41.41.41.4.4.4.4.4.39 1 1 19 28 37 46 55 64 73 82 91 1.1.1.1.1.1.1.1.1 1 1 19 28 37 46 55 64 73 82 91 1 31
(d) Response Parameers Usage 1, A1, Prce coef Usage 2, B3, Prce coef.3.2.1. -.1 -.2 -.3 1 1 19 28 37 46 55 64 73 82 91 1. -.1 -.1 -.2 -.2 -.3 1 1 19 28 37 46 55 64 73 82 91 1 A1 B1 C1 A2 B2 C2 A3 B3 C3 Usage 1, C2, End coef Usage 1, C2, Ad coef.1.8.6.4.2. -.2 -.4 -.6 1 1 19 28 37 46 55 64 73 82 91 1.15.1.5. -.5 -.1 1 1 19 28 37 46 55 64 73 82 91 1 A1 B1 C1 A2 B2 C2 A1 B1 C1 A2 B2 C2 32