Technical Appendix to Modeling Movie Lifecycles and Market Share. All our models were estimated using Markov Chain Monte Carlo simulation (MCMC).

Transcription

1 Technical Appenix o Moeling Movie Lifecycles an Marke Share Deman Moel All our moels were esimae using Markov Chain Mone Carlo simulaion (MCMC). This meho is wiely use in he markeing leraure an is escribe in eail in Gelman Carlin Sern an Rubin (995). Given he vas leraure on he subjec we lim our iscussion o he specificaion of he various isribuions use. In paragraph. we escribe he conional isribuions of ηi γi β i. These isribuions are non-conjugae an hus are hanle hrough a Meropolis sep. In paragraphs.2 o.7 we escribe he isribuions use for he error erms ( an V Suio Genre σ ) an he parameers of he hierarchical regression ( θ an θ ). As hese isribuions are conjugae we hanle hem wh a Gibbs sep. For noaional convenience we efine m as he number of movies in he aase (m = 404) we use i o inex movies o inex ime (62 weeks) an o refer o he eman moel.. Conional isribuion of ηi γi β i (Meropolis Sep) In secion 3 of he paper we efine he eman moel (equaion 7) as: γi / βi ( w)/ βi ηi ε S = w e +. Given his specificaion he likelihoo funcion for movie i is given by: η ˆ ˆ i η i η i η i I 2 2 ( )~ ( exp ( ( ˆ Lη ) /2 )) exp ˆ ˆ i γi βi S S σ γi γi V γi γ i 2 β ˆ ˆ i β β i i β i

2 where: ˆ γ i / β i ( w )/ β i ηi S = w e η = ln( η ) i i γ = γ i i β = ln( β ) i i ε σ an 2 ~ N(0 ) ˆ η i Suio Genre ˆ γi = zi + θi + θ i. ˆ βi.2 Conional isribuion of σ (Gibbs Sep) σ is he sanar eviaion of he error erm in he eman moel. I is isribue Inverse Gamma as follows: where: ( + ( + ) ( )) 2 + σ ~ IG ν n νv ns / ν n ν =3; v = (Priors chosen o be reasonable uninformaive) n is he oal number of movie/week observaion s S. = ( S ˆ ) 2 2 i 2

3 .3 Conional isribuion of (Gibbs Sep) V V is he variance of he vecor of error erms (imension 3) in he hierarchical regression of he parameers of he eman moel. The prior on V is isribue Inverse Wishar ( ν 0 V0). Consequenly he poserior conional on V is isribue Inverse Wishar: ˆ η ˆ i η i V ~ ˆ ˆ IW ν0 + m V0 + γi zi γi zi. i ˆ ˆ β i β i Where m represens he oal number of movies (i.e. 404). We chose ν 0 = 7 V0 = I(3) 7 as a locally iffuse prior (I(3) is a 3x3 ieny marix)..4 Conional isribuion of (Gibbs Sep) Le η β = γ θ θ i β i Genre Suio i i i i be ha componen no aribuable o suio or genre. An le β be a sacke marix of β across all movies. i where: ( ( )) ( ) ( ) Then δ = vec ~ N V ZZ + A ( ) = vec D ( ) ( ˆ β ) D = ZZ + A ZZD+ A D ( ) ˆD = ZZ Z β D= sack( ) is he prior on 3

4 Z is a sacke marix of he iniviual movies z i s. The prior on is iffuse given by: = D N A A vec( ) ~ (0 ) = I() 0.0 Suio.5 Conional isribuion of θ (Gibbs Sep) We use a muli-imensional version of he massively caegorical approach suggese in Seenburgh e al (2003) o allow for an inercep shif ( level. Suio θ ) on each parameer a he suio Le η i Suio Genre β = i γi θi zi β i be ha componen of he coefficiens no aribuable o Suio Suio he coninuous variables or genre. An le be a sacke marix of β across all β i movies. Furher le N Suio Suio s = i ISuio i s i= β β = be he sum of he vecors of hese parial coefficiens over all movies prouce by a paricular suio inexe here by s. Finally le he number of movies represene by suio s be θ m s. Then ~ N(( I(3) m + V ) β ( I(3) m + V ) ) Suio Suio s s Suio s s Suio.6 Conional isribuion of V Suio (Gibbs Sep) The prior on Wishar: is IW ( ν V ). Thus he poserior conional on is Inverse V Suio Suio0 Suio0 V IW m V Suio Suio Suio Suio ( )( θ ) Suio ~ νsuio0 + Suio0 + βi θs i βi si i 4 V Suio

5 Where Suio θ s i is he vecor of coefficiens for he suio ha represens movie i an m once more represens he oal number of movies. We chose ν = 7 V = I(3) 7 as a locally iffuse prior. Suio0 Suio0 Genre.7 Conional isribuion of an (Gibbs Sep) θ V Genre This procees exacly as for he suio coefficiens in.5 an.6 excep ha he inexing is performe across movies ha belong o each genre insea of movies represene by each suio. 2 Esimaion for he Marke Share Moel The esimaion of he marke share moel is similar in naure o he esimaion of he eman moel. The hierarchical srucure on he parameers ηims γims β ims is ienical o he srucure pu on ηi γi β i excep ha we resric η ims o be beween 0 an by using a Log ransform raher han log since can be inerpree in erms of expece marke share. Thus he seps escribe in secions.3 o.7 are ienical an will no be repeae here. Wha is ifferen here is ha we moel marke share raher han eman; we inrouce an ousie goo; an we use a Poisson approximaion for he log specificaion. These hree changes are escribe in he nex hree secions. 2. Conional isribuion of ηims γims β ims (Meropolis Sep) Our choice moel is efine by equaion as: V M e I = e V V O j + e I j j. Given his specificaion he likelihoo funcion for movie i is given by: 5

6 I S V η ˆ ˆ ims η ims η ims η ( e ) L( η ˆ ims γims βims )~ exp γ S ims γims Vc γims 2 ˆ γ V V O j e e I β ˆ + j ims β β ims ims ˆ β j where: ims ims ims (C-) γ β β ims / ims ( w)/ ims = ηi ms ; V w e S S S = O + j j= J (we iscuss his erm in more eails in 2.3) ( ) η = log η /( η ) ; ims ims ims γ ims = γims ; β = ln( β ); an ims ims ˆ η ims Suio Genre ˆ γi ms = X i ms + θi ms + θ i ms. ˆ βims As in any log specificaion η ims is ienifiable only o an arbrary consan. We show in he nex secion how we se his consan such ha η ims can be inerpree as he expece marke share for movie i in s opening week. 2.2 Poisson Transform The likelihoo (C-) above can be messy o evaluae irecly an inee one encouners machine precision problems as S is frequenly in he millions. A common ransform use o make esimaion more efficien is o use a Poisson approximaion (Baker (995); see also BUGS Manual p. 5; BUGS Examples Vol. 2 p. 5 which we follow closely in our iscussion). 6

7 To illusrae how his approximaion works le us look a a simplifie likelihoo formulaion where we ignore he prior an he ousie goo (we will re-inrouce he ousie goo in he nex secion). The likelihoo for his simple case is given by: LDaa ( η γ β )~ I V ( e ) ms ms ms S i= = J V T j= IS j ( e I j ) Suppose we assume ha he aa is acually generae by S ~ Poisson( ω ) ω = ωe V Then he likelihoo is given by: LDaa ( η γ β )~ ω e I T I S ω ms ms ms = i= T I I S V = ω exp S V Iexp ω e I = i= i= Le he ω ' s have inepenen gamma (ab) priors; inegraing hem ou gives a marginal likelihoo of V s: T I I S V a bω LDaa ( ηims γims βims ) exp S V I ω exp ω e I ω e ω = i= i= I exp S V I T i= ims ims ims S + a = LDaa ( η γ β ) I V ( e + b) i= We noe ha as ab 0 he likelihoos become he same. The inclusion of an ousie goo an he assumpion ha he oal eman ( S ) is consan allows us o make ω invarian o ime (i.e. ω = ω ). Since his consan is unienifie 7

8 S for our log moel we arbrarily se o ω = e. As an imporan sie effec by aing his furher consrain η i can now be irecly inerpree as he expece of movie i marke share in s opening week. 2.3 The Ousie Goo (Meropolis Sep) Deman for movies is highly seasonal. A useful way o irecly accoun for seasonaly is o imagine non-eman for movies. Tha is in every perio here is a percenage of he oal poenial marke ha chooses no o aen movies in ha perio. Those who chose no o go o he movies are sai o consume he ousie goo. This is analogous o he no-purchase opion in he choice leraure. We efine his eman for he ousie goo as S O. If we assume ha oal eman is consan over ime ( S = S ) an is allocae every week beween he movies in heaer an he ousie goo as a funcion of he qualy of he movies an he qualy of he oher opions available o consumers we can compue S O as: S S S I O i= I = The oal eman can be arbrarily se o a large number. We se such ha oal sales in each perio is.2 oal ickes sol in he highes eman week whin he aase o ensure ha here is posive eman for he ousie goo in each perio. We ese values of. an.4; here were no effecs oher han an inercep shif. Nex o accoun for seasonal change in he eman for movies we moel he eman for he ousie goo as a funcion of he eman for he ousie goo in pas years. Using a simple auo-regressive moel he marke share of he ousie goo is given as: V O S (C-2) K O ( k52) = α + φk ln k = S 8

9 SO 52 k where is ( S -oal eman for movies in week k years previously. In (C-2) we ivie he lagge eman by S so ha his can be hough of as marke share for he ousie goo in week k years previously. Wh his ousie goo specificaion he conional likelihoo for he parameers for each movie becomes: η ˆ η η ˆ η T ims ims ims ims V S / V / S S e ˆ ˆ ims ims ims ims ims c ims ims = 2 β ˆ ˆ ims β β ims ims βims L( η γ β )~ I e e exp γ γ V γ γ an for he ousie goo is: α α φ φ L( αφ.. φ)~ e e exp V k T V S / V / S S O O e O = 2 φ k φ k 6 where he prior on hese parameers is chosen o be iffuse N(0 ) where V = I0. In pracice we foun k=3 o work bes (i.e. we use lagge eman for he ousie goo in he previous 3 years o preic curren year sales). V O O Reference Baker S. G. (995) The mulinomial-poisson ransformaion The Saisician Vol BUGS Manual hp:// version BUGS Examples hp:// Vol. 2 Version

10 Gelman A. J. Carlin H. Sern an D. Rubin (995) Bayesian Daa Analysis Chapman an Hall Lonon. 0