Nonstationary dynamic models with finite dependence

A cidiacono, Pe e ; Mille , Robe Allen A icle Nons a iona y dynamic models wi h ini e dependence Quan i a i e Economics P o ided in Coope a ion wi h: The Econome ic Socie y Sugges ed Ci a ion: A cidiacono, Pe e ; Mille , Robe Allen (2019) : Nons a iona y dynamic models wi h ini e dependence, Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 10, Iss. 3, pp. 853-890, h ps://doi.o g/10.3982/QE626 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217158 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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Mille Teppe School o Business, Ca negie Mellon Uni e si y The es ima ion o nons a iona y dynamic disc e e choice models ypically e- qui es making assump ions a beyond he leng h o he da a. We ex end he class o dynamic disc e e choice models ha equi e only a ew-pe iod-ahead condi- ional choice p obabili ies, and de elop algo i hms o calcula e he ini e depen- dence pa hs. We do his bo h in single agen and games se ings, esul ing in ex- p essions o he alue unc ions ha allow o much weake assump ions ega d- ing he ime ho izon and he ansi ions o he s a e a iables beyond he sample pe iod. Keywo ds. Dynamic disc e e choice, ini e dependence, condi ional choice p obabili ies. JEL classi ica ion. C33, C35. 1. In oduc ion Es ima ion o dynamic disc e e choice models is complica ed by he calcula ion o ex- pec ed u u e payo s. These complica ions a e pa icula ly p onounced in games whe e he equilib ium ac ions and u u e s a es o he o he playe s mus be ma gined ou o de i e a playe ’s bes esponse. O igina ing wi h Ho z and Mille (1993), wo-s ep me h- ods p o ide a compu a ionally cheap way o es ima ing s uc u al payo pa ame e s in bo h single-agen and mul iagen se ings. These wo-s ep es ima o s i s es ima e con- di ional choice p obabili ies (CCPs) and hen cha ac e ize u u e payo s as a unc ion o he CCPs when es ima ing he s uc u al payo pa ame e s.1 CCP es ima o s all in o wo classes: hose ha exploi ini e dependence, and hose ha do no .2The o me en ails exp essing he u u e alue e m o i s di e ence ac oss wo al e na i es as a unc ion o jus a ew-pe iod ahead condi ional choice p obabili ies Pe e A cidiacono: [email p o ec ed] Robe A. Mille : [email p o ec ed] We hank he e e ees, Vic o Agui egabi ia, Shakeeb Khan, Jean-Ma c Robin, and semina pa icipan s a Duke, Sciences Po, Toulouse, and To on o o help ul commen s. We acknowledge suppo om Na ional Science Founda ion G an Awa ds SES0721059 and SES0721098. 1See A cidiacono and Ellickson (2011) o a e iew. 2CCP es ima o s ha do no ely on ini e dependence include hose o Ho z, Mille , Sande s, and Smi h (1994), Agui egabi ia and Mi a (2002,2007), Baja i, Benka d, and Le in (2007), and Pesendo e and Schmid -Dengle (2008). ©2019 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0. A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE626 854 A cidiacono and Mille Quan i a i e Economics 10 (2019) and low payo s.3In ui i ely, ρpe iod ini e dependence holds when he e exis wo sequences o choices ha lead o om di e en ini ial choices bu gene a e he same dis ibu ion o s a e a iables ρ+1pe iods la e . The sequences o choices need no be op imal and may in ol e mixing ac oss choices wi hin a pe iod. When a ini e dependence ep esen a ion exis s, i is possible o elax some o he assump ions abou ime ha a e commonly made when es ima ing dynamic disc e e choice models. Nons a iona y in ini e ho izon models can be es ima ed when ini e de- pendence holds. In ini e ho izon models, assump ions abou he leng h o he ime ho izon and he e olu ion o he s a e a iables beyond he sample pe iod, can be e- laxed. Fo example, a dynamic model o schooling equi es making assump ions ega d- ing he age o e i emen , and also he unc ional o m o u ili ies o olde wo ke s, al- hough he da a a ailable o esea che s migh only ack indi iduals in o hei wen ies o hi ies. Fu he mo e, es ima ion is as because condi ional choice p obabili ies need only be compu ed o a ew pe iods ahead o he cu en choices. Many pape s ha e used he ini e dependence p ope y in es ima ion, o en em- ploying ei he a e minal o enewal ac ion.4Mo e gene al o ms o ini e dependence, whe he a ea u e o he da a o imposed by he au ho s, ha e been applied in models o e ili y and emale labo supply Al ug and Mille (1998), Gayle and Golan (2012), Gayle, Hincapie, and Mille (2018), mig a ion (Bishop (2012), Coa e (2016), Ma ( o hcoming), Ransom (2018)), pa icipa ion in he s ock ma ke Kho unzhina (2013), ag icul u al land use Sco (2013), smoking Ma sumo o (2014), educa ion A cidiacono, Aucejo, Mau el, and Ransom (2016), occupa ional choice James (2014), and housing choices Kho un- zhina and Mille (2016). These pape s demons a e he ad an age o exploi ing ini e dependence in es ima ion: i is no necessa y o sol e he alue unc ion wi hin a nes ed ixed-poin algo i hm, no in e ma ices he size o he s a e space.5 The cu en me hod o de e mining whe he ini e dependence holds o no is o guess and e i y. The main con ibu ion o his pape is o p o ide a sys ema ic way o de e mining whe he ini e dependence holds when he e a e a (la ge bu ) ini e numbe o s a es. To accomplish his, we sligh ly gene alize he de ini ion o ini e dependence gi en in A cidiacono and Mille (2011). Key o he gene aliza ion is ecognizing ha he ex an e alue unc ion can be exp essed as a weigh ed a e age o he condi ional alue unc ions o all he al e na i es plus a unc ion o he condi ional choice p obabili ies, whe e all he weigh s sum o one bu some may be nega i e o g ea e han one. As one o ou examples shows, his sligh gene aliza ion enla ges he class o models ha can be 3See Ho z and Mille (1993), Al ug and Mille (1998), A cidiacono and Mille (2011), Agui egabi ia and Magesan (2013,2017), and Gayle (2017). 4See, o example, Ho z and Mille (1993), Joensen (2009), Sco (2013), A cidiacono, Baye , Ble ins, and Ellickson (2016), Decle q and Ve bo en (2018), Mazu (2017), and Beauchamp (2015). The las h ee exploi one-pe iod ini e dependence o es ima e dynamic games. 5The ini e dependence p ope y has also been di ec ly imposed on he decision making p ocess in mod- els o economize on he s a e space. See, o example, Bishop (2012) and Ma ( o hcoming). Assuming play- e s do no use all he in o ma ion a hei disposal educes he s a e space playe s use o sol e hei op i- miza ion p oblems. This app oach p o ides a pa simonious way o modeling bounded a ionali y when he s a e space is high dimensional. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 855 cheaply es ima ed by exploi ing his mo e inclusi e de ini ion o he ini e dependence p ope y. De e mining whe he ini e dependence holds o a pai o ini ial choices is a nonlin- ea p oblem, ye he algo i hm we p opose o dynamic op imiza ion p oblems only has a ini e numbe o s eps. We pa i ion candida e pa hs o demons a ing ini e depen- dence in say ρpe iods; pa hs ha each he same se o s a es eached wi h a nonze o weigh a e collec ed oge he . Pa i ioning by whe he a weigh is ze o o no , a he han he alue o he weigh , educes an uncoun able in ini y o pa hs o a ini e se . Each el- emen in he pa i ion maps in o a linea sys em o equa ions, and we check he ank o he sys em, also a ini e numbe o ope a ions. The size o he linea sys em is based on he numbe o s a es a ainable in ρ−1pe iods om he ini ial s a e, no he o al numbe o s a es in he model. The algo i hm p oceeds i e a i ely, by checking he de- e minan s o selec ed elemen s in he pa i ion. I one (o mo e) o he elemen s has a nonze o de e minan , hen he pai o choices exhibi s ρpe iod ini e dependence; o h- e wise i does no . Once ini e dependence is es ablished, ano he linea ope a ion (on a ini e numbe o equa ions) yields a se o weigh s ha can be used in any CCP es ima o ha exploi s ini e dependence. In game se ings, ini e dependence is applicable o each playe indi idually. He e, ini e dependence ela es o ansi ion ma ices o he s a e a iables when a desig- na ed playe places a bi a y weigh on each o he possible u u e decisions (so long as he weigh s sum o one wi hin a pe iod) and he o he playe s ollow hei equilib- ium s a egies. Consequen ly, ini e dependence in games canno be asce ained om he ansi ion p imi i es alone (as in he indi idual op imiza ion case). Indeed, whe he o no ini e dependence holds migh also hinge on which equilib ium is played, no a pa adoxical esul , because di e en equilib ia o he same game some imes e eal di e en in o ma ion abou he p imi i es, so na u ally equi e di e en es ima ion ap- p oaches. Up un il now, esea ch on ini e dependence in games has been es ic ed o models wi h a e minal ac ion ( ha ends he p ocess go e ning he s a e a iables o indi id- ual playe s). O he wise one-pe iod ini e dependence ypically ails o hold, because he equilib ium ac ions o he o he playe s depend on wha he designa ed agen has al- eady done. Hence he dis ibu ion o he s a e a iables, which he o he playe s pa ly de e mine, depends on he ac ions o he designa ed playe wo pe iods ea lie . These s ochas ic connec ions, a i al ea u e o many s a egic in e ac ions, has limi ed em- pi ical esea ch in es ima ing games wi h nons a iona i ies. We de elop an algo i hm o sol e o ini e dependence in a b oade class o games han hose cha ac e ized by e minal and enewal ac ions. In he gene al case, a bilinea sys em o equa ions mus be sol ed, whe e he numbe o equa ions is dic a ed by he possible s a es ha can be eached a ew pe iods ahead, bu in some specializa ions, including bu no limi ed o e minal and enewal ac ions, ou algo i hm educes o sol ing a linea sys em o equa- ions. The es o he pape p oceeds as ollows. Sec ion 2lays ou ou amewo k o ana- lyzing ini e dependence in disc e e choice dynamic op imiza ion p oblems and nonco- ope a i e equilib ium games. In Sec ion 3, we de ine ini e dependence, and show how 856 A cidiacono and Mille Quan i a i e Economics 10 (2019) his p ope y can be used in es ima ion, gene alizing exis ing es ima o s ha exploi i- ni e dependence o o de o accommoda e he many new applica ions ou algo i hm on ini e dependence e eals. The ou h sec ion p o ides a new ep esen a ion o his p ope y, and uses he ep esen a ion o demons a e how o eco e ini e dependence pa hs in single agen op imiza ion p oblems. Sec ion 5ex ends he app oach o mul- iagen equilib ium se ings. New examples wi h ini e dependence, de i ed using he algo i hm, a e p o ided in Sec ion 6, while Sec ion 7concludes. 2. F amewo k This sec ion i s lays ou a gene al class o dynamic disc e e choice models. D awing upon ou p e ious wo k (A cidiacono and Mille (2011)), we ex end ou ep esen a ion o he condi ional alue unc ions which plays an o e a ching ole in ou analysis, and hen modi y ou amewo k o accommoda e games wi h p i a e in o ma ion. 2.1 Dynamic op imiza ion disc e e choice In each pe iod ∈{1T}un il T≤∞, an indi idual chooses among Jmu ually exclu- si e ac ions. Le dj equal one i ac ion j∈{1J}is aken a ime and ze o o he wise. The cu en pe iod payo o ac ion ja ime depends on he s a e x ∈X, a ini e se .6 I ac ion jis aken a ime , he p obabili y o x +1occu ing in pe iod +1is deno ed by j (x +1|x ). The indi idual’s cu en pe iod payo om choosing ja ime is also a ec ed by a choice-speci ic shock, j , which is e ealed o he indi idual a he beginning o he pe- iod . We assume he ec o  ≡(1 J )has con inuous suppo , is d awn om a p obabili y dis ibu ion ha is independen ly and iden ically dis ibu ed o e ime wi h densi y unc ion g( ), and sa is ies E[max{1 J }] ≤ <∞. The indi idual’s cu - en pe iod payo o ac ion ja ime is modeled as uj (x )+j . The indi idual akes in o accoun bo h he cu en pe iod payo as well as how his decision oday will a ec he u u e. Deno ing he discoun ac o by β∈(01), he in- di idual chooses he ec o d ≡(d1 dJ ) o sequen ially maximize he discoun ed sum o payo s: ET  =1 J  j=1 β −1dj uj (x )+j (2.1) whe e a each pe iod he expec a ion is aken o e he u u e alues o x +1xTand  +1T. Exp ession (2.1) is maximized by a Ma ko decision ule which gi es he op imal ac ion condi ional on ,x ,and . We deno e he op imal decision ule a as do (x  ),wi hj h elemen do j (x  ). The p obabili y o choosing ja ime condi ional on x ,pj (x ), is ound by aking do j (x  )and in eg a ing o e  : pj (x )≡do j (x  )g( )d (2.2) 6Ou analysis is based on he assump ion ha x belongs o a ini e se , an assump ion ha is o en made in his li e a u e; see Agui egabi ia and Mi a (2002) o example. Howe e , i is wo h men ioning ha ini e dependence can be applied wi hou making ha assump ion; see Al ug and Mille (1998) o example. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 857 We hen de ine p (x )≡(p1 (x )pJ (x )) as he ec o o condi ional choice p oba- bili ies (CCPs). Deno e V (x ), he ex an e alue unc ion in pe iod , as he discoun ed sum o ex- pec ed u u e payo s jus be o e  is e ealed and condi ional on beha ing acco ding o he op imal decision ule: V (x )≡ET  τ= J  j=1 βτ− do jτ(xττ)ujτ(xτ)+jτ Gi en s a e a iables x and choice jin pe iod , he expec ed alue unc ion in pe- iod +1, discoun ed one pe iod in o he u u e, is βX x +1=1V +1(x +1) j (x +1|x ).Un- de s anda d condi ions, Bellman’s p inciple applies and V (x )can be ecu si ely ex- p essed as V (x )= J  j=1do j (x  )uj (x )+j +β X  x +1=1 V +1(x +1) j (x +1|x )g( )d  We hen de ine he choice-speci ic condi ional alue unc ion, j (x ),as he lowpay- o o ac ionjwi hou j plus he expec ed u u e u ili y condi ional on ollowing he op imal decision ule om pe iod +1on:7 j (x )=uj (x )+β X  x +1=1 V +1(x +1) j (x +1|x ) (2.3) Ou analysis is based on a ep esen a ion o j (x ) ha sligh ly gene alizes Theo- em 1 o A cidiacono and Mille (2011). Bo h esul s a e based on hei Lemma 1, ha o e e y ∈{1T}and p∈J, heJdimensional simplex, he e exis s a eal- alued unc ion ψj(p) such ha ψjp (x)≡V (x) − j (x) (2.4) To in e p e (2.4), no e ha he alue o commi ing o ac ion ja pe iod be o e see- ing  and beha ing op imally he ea e is j (x )+E[j ]. The e o e, he expec ed loss om p ecommi ing o j e sus wai ing un il  is obse ed and only hen making an op- imal choice, V (x ), is he cons an ψj[p (x )]minus E[j ], a composi e unc ion ha only depends on x h ough he condi ional choice p obabili ies. This esul leads o he ollowing heo em, p o ed using an induc ion. Theo em 1. Fo each choice j∈{1J}and τ∈{ +1T},le any ωτ(xτj) de- no e any mapping om he s a e space {1X} o RJsa is ying he cons ain s ha 7Fo ease o exposi ion, we e e o j (x )as he condi ional alue unc ion in he emainde o he pape . 858 A cidiacono and Mille Quan i a i e Economics 10 (2019) |ωkτ(xτj)|<∞and J k=1ωkτ(xτj)=1.Recu si ely,de ine κτ+1(xτ+1|x j)as κτ+1(xτ+1|x j) ≡⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ j (x +1|x ) o τ=  X  xτ=1 J  k=1 ωkτ(xτj) kτ(xτ+1|xτ)κτ(xτ|x j) o τ= +1T (2.5) Then o T<T, j (x )=uj (x )+ T  τ= +1 J  k=1 X  xτ=1 βτ− ukτ(xτ)+ψkpτ(xτ)ωkτ(xτj)κτ(xτ|x j) + X  xT+1 βT+1− VT+1(xT+1)κT+1(xT+1|x j) (2.6) and o T=T, j (x )=uj (x )+ T  τ= +1 J  k=1 X  xτ=1 βτ− ukτ(xτ)+ψkpτ(xτ)ωkτ(xτj)κτ(xτ|x j) (2.7) Fo he pu poses o his wo k, i is con enien o in e p e Tas he inal pe iod in he sample; ypically T<T.A cidiacono and Mille (2011) p o ed he heo em when T=Tand ωkτ(xτj)≥0 o all kand τ.In ha case,κτ+1(xτ+1|x j) is he p obabili y o eaching xτ+1by ollowing he sequence de ined by ωτ(xτj)and he alue unc ion ep esen a ion ex ending o e he whole decision-making ho izon.8 2.2 Ex ension o dynamic games This amewo k ex ends na u ally o dynamic games. In he games se ing, we assume ha he e a e Nplaye s making choices in pe iods ∈{1T}. The sys ema ic pa o payo s o he n h playe no only depends on his own choice in pe iod , deno ed by d(n) ≡(d(n) 1 d(n) J ), and he s a e a iables x , bu also he choices o he o he playe s, which we now deno e by d(∼n) ≡(d(1) d(n−1) d(n+1) d(N) ).Deno eby U(n) j (x d(∼n) )+(n) j he low u ili y o playe nin pe iod ,whe e(n) j is an iden ically and independen ly dis ibu ed andom a iable ha is p i a e in o ma ion o playe n. Al hough he playe s all ace he same obse ed s a e a iables, hese s a e a iables yp- ically a ec playe s in di e en ways. Fo example, adding o he n h playe ’s capi al may inc ease his payo s and educe he payo s o he o he s. Fo his eason, he payo unc ion is supe sc ip ed by n. The playe s make simul aneous choices in each pe iod. We deno e by P (d(∼n) |x ) he join condi ional choice p obabili y ha he playe s aside om ncollec i ely choose 8The ex ension o nega i e weigh s is also no ed in Gayle (2017). Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 859 d(∼n) a ime gi en he s a e a iables x . Since (n) is independen ly dis ibu ed ac oss all he playe s, P (d(∼n) |x )has he p oduc ep esen a ion: P d(∼n) |x = N  n=1 n=n J  j=1 d(n) j p(n) j (x )(2.8) We assume each playe ac s like a Bayesian when o ming his belie s abou he choices o he o he playe s and ha a Ma ko -pe ec equilib ium is played. Hence, he belie s o he playe s ma ch he p obabili ies gi en in equa ion (2.8). Taking he expec a ion o U(n) j (x d(∼n) )o e d(∼n) , we de ine he sys ema ic componen o he cu en u ili y o playe nas a unc ion o he s a e a iables as u(n) j (x )= d(∼n) ∈JN−1 P d(∼n) |x U(n) j x d(∼n) (2.9) Fo u u e e e ence, we call u(n) j (x ) he educed o m payo o playe n om aking ac ion jin pe iod when he s a e is x . The alues o he s a e a iables a pe iod +1a e de e mined by he pe iod choices by all he playe s as well as he alues o he pe iod s a e a iables. We conside a model in which he s a e a iables can be pa i ioned in o hose ha a e a ec ed by only one o he playe s, and hose ha a e exogenous. Fo example, o explain he numbe and size o i ms in an indus y, he s a e a iables o he model migh be indica o s o whe he each po en ial i m is ac i e o no , and a scala o measu e i m capi al o capaci y; each i m con ols hei own s a e a iables, h ough hei en y and exi choices, as well as hei in es men decisions.9The pa i ion can be exp essed as x ≡(x(0) x(1) x(N) ), whe e x(0) deno es he s a es ha a e exogenously de e mined by ansi ion p obabili y 0 (x(0) +1|x(0) ),andx(n) ∈X(n) ≡{1X(n)}is he componen o he s a e con olled o in luenced by playe n.Le (n) j (x(n) +1|x(n) )deno e he p obabili y ha x(n) +1occu s a ime +1when playe nchooses ja ime gi en x(n) . Many models in indus ial o gani- za ion exploi his specialized s uc u e because i p o ides a lexible way o playe s o in e ac while keeping he model simple enough o be empi ically ac able.10 Since he ansi ions o he exogenous a iables do no subs an i ely e ec ou analysis, we igno e hem o he es o he pape o conse e on no a ion. Deno e he s a e a iables associa ed wi h all he playe s aside om nas x(∼n) ≡x(1) x(n−1) x(n+1) x(N)  ∈X(∼n) ≡X(1)××X(n−1)×X(n+1)××X(N) 9The second example in A cidiacono and Mille (2011) also belongs o his class o models. 10All he empi ical applica ions o s uc u al modeling o which we a e awa e ha e his p ope y, includ- ing hose based on E icson and Pakes (1995). Fo example, i ms a ec hei own p oduc quali y h ough hei own in es men decisions, bu do no di ec ly a ec he p oduc quali y o o he playe s. Thus each i m’s decisions a ec he p oduc quali y o o he playe s only h ough he e ec on he decisions o he o he playe s. 860 A cidiacono and Mille Quan i a i e Economics 10 (2019) Unde his speci ica ion, he educed o m ansi ion gene a ed by hei equilib ium choice p obabili ies is de ined as (∼n) x(∼n) +1|x ≡ N  n=1 n=n J  k=1 p(n) k (x ) (n) k x(n) +1|x(n)  As in Sec ion 2.1, conside o all τ∈{ T}any sequence o decision weigh s: ω(n) τ(xτj)≡ω(n) 1τ(xτj)ω(n) Jτ (xτj) subjec o he cons ain s J k=1ω(n) kτ (xτj)=1and s a ing alue ω(n) j (x j)=1.Gi en he equilib ium ac ions o he o he playe s impounded in (∼n) (x(∼n) +1|x ),we ecu - si ely de ine κ(n) τ+1(xτ+1|x j) o he sequence o decision weigh s ω(n) kτ (xτj) o e pe- iods τ∈{ +1T}in a simila manne o (2.5)as κ(n) τ+1(xτ+1|x j) ≡ 0τx(0) τ+1|x(0) τX  xτ=1 J  k=1 (∼n) τx(∼n) τ+1|xτω(n) kτ (xτj) (n) kτ x(n) τ+1|x(n) τκ(n) τ(xτ|x j) (2.10) wi h ini ializing unc ion: κ(n) +1(x +1|x j)≡ (n) j x(n) +1|x(n)  x(∼n) +1|x  0 x(0) +1|x(0) (2.11) Le ing j x +1|x = 0 x(0) +1|x(0)  (∼n) x(∼n) +1|x  (n) j x(n) +1|x(n) (2.12) and adding nsupe sc ip s o all he o he e ms in (2.7), i now ollows ha Theo em 1 applies o his mul iagen se ing in exac ly he same way as in a single agen se ing. 3. The ini e dependence p ope y Theo em 1shows ha he u u e alue e m can be exp essed ela i e o any weigh ed choice sequence as long as he sum o he weigh s add up o one in each pe iod. Gi en ha many pa hs can be chosen, i may be possible o line up he dis ibu ion o s a es gi en wo di e en ini ial choices a some poin in he u u e, say ρpe iods la e . I his is he case, hen exp essing he u u e alue e ms ela i e o hese sequences esul s in he u u e alue e ms a e ρpe iods cancel ou once di e ences in he condi ional alue unc ion a e aken ac oss he wo choices. Hence any in o ma ion ha would esul in di e ences be ween he wo choices in he u u e is al eady embedded in he condi ional choice p obabili ies. In his sec ion, we o malize he concep o ini e dependence. We hen show how i can be used in es ima ion. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 867 Then one-pe iod dependence holds i and only i he e exis s an (A1 +1+A2 +1) ec o o unknowns deno ed by D +1sol ing K +1=H +1Ω +1(A2 +12)◦K2 +1(A2 +1) Ω +1(A1 +11)◦K1 +1(A1 +1)≡H +1D +1(4.4) No e ha i he weigh s placed on all he s a es in Aj +1bu one a e he same ac oss he wo pa hs hen he weigh s placed on he emaining s a e mus be he same as well. Asolu ion o(4.4) o D +1exis s i and only i he ank o H +1equals he ank o he augmen ed ma ix H∗ +1≡[K +1   H +1] o med by augmen ing H +1wi h he ex a column K +1. Deno e he ank o H +1by R +1and he ank o o H∗ +1by R∗ +1. Clea ly, R +1≤R∗ +1≤ R +1+1and R +1≤min{A +2−1A1 +1+A2 +1}. The e a e wo cases o conside : 1. Suppose R +1=A1 +1+A2 +1. I in addi ion R +1=A +2−1, implying H +1is squa e, we sol e o he weigh s by in e ing H +1and hen elemen -by-elemen di iding bo h sides o (4.4) by he ma ching K ec o s, yielding Ω +1(A2 +12) Ω +1(A1 +11)=H−1 +1K +1◦ K2 +1(A2 +1) K1 +1(A1 +1)(4.5) whe e ◦/ e e s o elemen -by-elemen di ision. I R +1>A +2−1, we successi ely elim- ina e A1 +1+A2 +1−A +2+1linea ly dependen columns o H +1 o o m a squa e ma ix o ank A +2−1. We now emo e he co esponding elemen s in D +1in (4.4)so ha he educed A +2−1dimensional ec o con o ms wi h he squa e ma ix, by dele - ing he elemen s ha would ha e been mul iplied by he columns emo ed om H +1, e ec i ely gi ing ze o weigh o he second ac ion o he emo ed elemen s. Finally, an analogous equa ion o (4.5) is sol ed o he weigh s cha ac e izing ini e dependence.21 2. Al e na i ely, R +1<A 1 +1+A2 +1. Fi s , we successi ely elimina e A1 +1+ A2 +1−R +1linea ly dependen columns o H +1 o o m an (A +2−1)×R +1ma- ix deno ed by H +1. This ope a ion co esponds o educing he ec o leng h o D +1 om A1 +1+A2 +1 o R +1by e ec i ely se ing A1 +1+A2 +1−R +1weigh s o ze o. Deno e he R +1×1 ec o o weigh s no elimina ed by D +1. We now elimina e A +2−R +1−1 ows o H +1 o o m an R +1dimensional squa e ma ix wi h ank R +1deno ed by H +1. S ic ly o no a ional pu poses, so wi hou loss o gene ali y, we eo de he equa ions de ining (4.4) so ha he linea ly independen equa ions a e he bo om ones. This allows us o pa i ion H +1≡[H +1   H +1]and K +1≡[K +1   K +1], whe e H +1is (A +2−1−R +1)×R +1, while K +1is (A +2−1−R +1)×1and K +1 is R +1×1. In e ing H +1,weob ainD +1=H−1 +1K +1. Thus a solu ion o (4.4) a ains in his kni e edged case i and only i D +1sol es A +2−R +1−1addi ional equa ions K +1=H +1H−1 +1K +1. 21The se o weigh s gene a ed by his p ocedu e depends on which linea ly dependen columns a e emo ed. The e o e, he weigh ec o s sa is ying ini e dependence a e no unique. 868 A cidiacono and Mille Quan i a i e Economics 10 (2019) To illus a e he algo i hm in he enewal and e minal s a e models men ioned abo e, le X≡{12X}, and suppose he i s choice deno es he e minal o enewal choice which e u ns he s a e a iable x o he alue one, while he second inc eases x by one uni o all x<Xand e u ns Xwhen x=X.22 Because he ansi ions a e de e - minis ic, A1 +1=A2 +1=1,wi hA1 +1={1}and A2 +1={x +1}.AlsoA +2=3,wi h A +2={12x +2}. I now ollows ha in his example: F1 +1(A1 +1)=F1 +1(A2 +1)=10  F2 +1(A1 +1)=01 F2 +1(A2 +1)=00  H +1=−11 0−1o H−1 +1=−1−1 0−1 Subs i u ing hese exp essions in o (4.5), and no ing ha Ω +1(Aj +1j)=ω +1(x j) be- cause K1 (A1 +1)=K2 (A1 +1)=1, demons a es ha ze o weigh is placed on he non- enewal/non e minal ac ion o achie e one-pe iod dependence: ω +1(x2) ω +1(x1)=−1−1 0−11−1 00 1 1◦ 1 1=0 0 The limi a ions o he guess and e i y app oach become e iden when such a widely used class o models in empi ical analysis is e ealed o ha e such a simple s uc u e. The class o models exhibi ing e en one-pe iod ini e dependence is much la ge han e minal and enewal models, and he me hod de eloped he e p o ides a sys ema ic way o disco e ing hem. 4.2 Sol ing nonlinea sys ems o a ain ρ-pe iod dependence Analyzing he exis ence o ini e dependence o ρ>1in oduces nonlinea i y in o he sys em. Fo con enience, we elabel he wo ini ial choices iand jin equa ion (3.1)as1 and 2, and he ini ial s a e as x . Analogous o he one-pe iod ini e dependence case, o any τ∈{ +1 +ρ−1}we say xτ∈{1X}is a ainable by a sequence o decision weigh s om ini ial choice j∈{12}i he weigh on xτis nonze o.23 Le Ajτ ∈{1X} deno e he numbe o a ainable s a es, and Ajτ ⊆X he se o a ainable s a es o he sequence beginning wi h choice j.24 De ine Kjτ(Ajτ)as an Ajτ ec o con aining he weigh s o ansi ioning o each o he Ajτ a ainable s a es gi en he choice sequence beginning wi h jand s a e x . Simila ly, le Aτ+1∈{1X}deno e he numbe o s a es ha a e a ainable by a leas one o he sequences beginning ei he wi h choice 1o 2, and deno e by Aτ+1⊆X he co esponding se . Gi en an ini ial s a e and choice, we 22Mo e o mally, 1 +1(1|x )=1, o all and x , while o all , 2 +1(x +1|x )=1i x <X and 2 +1(X|X) =1. 23Fo example, suppose X≡{123}and x =3.Alsoassume 1 +1(1|3)=3/4, and 1 +1(2|3)=1/4.Then he he i s wo s a es a e a ainable in +1 om aking he i s choice bu he hi d is no . 24In ou simple example, A1 +1=2and A1 +1={12}. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 869 deno e by Fkτ(Ajτ) he i s Aτ+1−1columns o he Ajτ ×Aτ+1 ansi ion ma ix om Ajτ o Aτ+1when kis chosen a pe iod τ,wi h Fkτ(Ajτ)con aining all he columns o he ansi ion ma ix. The ma ix comp ises elemen s kτ(x|x) o each x∈Ajτ and x∈ Aτ+1. The Aτ+1sys em o equa ions exhibi s ρ-pe iod dependence, ha is κτ+1(xτ+1|x  1)=κτ+1(xτ+1|x 2)wi h τ= +ρ, i and only i he e exis ec o s Ωkτ(Ajτ1)and Ωkτ(Ajτk) o each k∈{2J}sol ing: Kτ+1≡F1τ(A1τ)K1τ(A1τ)−F1τ(A2τ)K2τ(A2τ)=HτDτ(4.6) whe e he (Aτ+1−1)×(J −1)[A1τ+A2τ]ma ix Hτ,and he(J −1)[A1τ+A2τ] ec o Dτ, a e espec i ely de ined by25 Hτ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ F2τ(A2τ)−F1τ(A2τ)    FJτ(A2τ)−F1τ(A2τ) F1τ(A1τ)−F2τ(A1τ)    F1τ(A1τ)−FJτ(A1τ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  Dτ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ω2τ(A2τ2)◦K2τ(A2τ)    ΩJτ(A2τ2)◦K2τ(A2τ) Ω2τ(A1τ1)◦K1τ(A1τ)    ΩJτ(A1τ1)◦K1τ(A1τ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.7) Appealing o Hadley (1961, pp. 168–169) yields necessa y and su icien condi ions o he exis ence o a solu ion o his linea sys em, which we s a e as a heo em. Theo em 2. De ine he (Aτ+1−1)×{(J −1)[A1τ+A2τ]+1}ma ix H∗ τ≡[Hτ   K +1], ob ained by adding an ex a column K +1 o Hτ.Fini e dependence om x wi h espec o choices iand jis achie ed in ρ=τ− pe iods i and only i he e exis weigh s om +1 o τ−1such ha he ank o Hτequals he ank o H∗ τ. Theo em 2shows ha es ablishing one-pe iod dependence when he e a e mo e han wo choices is a s aigh o wa d ex ension o he case in which J=2. Howe e , non- linea i y in he weigh s en e (4.6)whenρ>1because Kjτ(Ajτ)depends on Ωks(A2sj), he weigh on ac ion k∈{2J} o e e y pe iod s<τgi en ini ial choice j∈{12}. Deno e  Kjτ(Aτ)as he Aτ ec o con aining he weigh s o ansi ioning o each o he Aτs a es, ha is, he a ainable s a es om ei he pa h-gi en he ini ial choice o j. Kjτ(Ajτ)is hen he nonze o en ies o  Kjτ(Aτ). The ollowing ecu si e s uc u e is hen e iden :  Kjτ(Aτ)=⎡ ⎢ ⎣ F2τ−1(Ajτ−1)     FJτ−1(Ajτ−1) ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ Ω2τ−1(Ajτ−1j)◦Kjτ−1(Ajτ−1)    ΩJτ−1(Ajτ−1j)◦Kjτ−1(Ajτ−1) ⎤ ⎥ ⎦(4.8) 25One o he equa ions is edundan because i all o he s a es ha e he same weigh assigned o hem ac oss he wo pa hs hen he las one mus be lined up as well, implying ha i he ank o Hτis Aτ+1−1 hen ini e dependence holds in ρpe iods. 870 A cidiacono and Mille Quan i a i e Economics 10 (2019) Taking he nonze o elemen s ou o  Kjτ(Aτ) o o m Kjτ(Ajτ)and subs i u ing in o Kjτ(Ajτ)using (4.8)in(4.7)and(4.6) demons a es ha c oss p oduc s o elemen s in Ω2τ−1(Ajτ−1j) and Ω2τ(A2τ2)en e (4.8). Fo mally, he sys em is bilinea , no lin- ea . To see ha he sys em is bilinea , suppose J=2and w i e ωτ(xτj) ≡ω2τ(xτj): expanding (4.6) e m by e m p o es ha wo-pe iod dependence exis s o some gi en x i and only i : X  x +2=1 X  x +1=1 1 +2(x +3|x +2) 1 +1(x +2|x +1) 1 (x +1|x )− 2 (x +1|x ) = X  x +2=1 X  x +1=1 2 +2(x +3|x +2)− 1 +2(x +3|x +2) × 2 +1(x +2|x +1)− 1 +1(x +2|x +1) ×ω +2(x +22)ω +1(x +12) 2 (x +1|x ) −ω +2(x +21)ω +1(x +11) 1 (x +1|x ) + X  x +2=1 X  x +1=1 2 +2(x +3|x +2)− 1 +2(x +3|x +2) × 1 +1(x +2|x +1)ω +2(x +11) 1 (x +1|x ) ×ω +2(x +12) 2 (x +1|x )−ω +2(x +11) 1 (x +1|x ) + X  x +2=1 X  x +1=1 1 +2(x +3|x +2) 2 +1(x +2|x +1)− 1 +1(x +2|x +1) ×ω +1(x +12) 2 (x +1|x )−ω +1(x +11) 1 (x +1|x )(4.9) o all x +3∈X. Since p oduc s o weigh s appea in (4.9), bilinea solu ion echniques a e equi ed o sol e his p oblem. Mo e gene ally, c oss p oduc s o he powe o ρen e in o he equa ion sys em de ining ρ-pe iod dependence. We exploi he special s uc u e o his nonlinea p oblem by di iding i in o wo pa s, each ha ing a ini e numbe o ope a ions. The second pa is he linea in e sion p oblem o which Theo em 2applies. The i s pa delinea es he subse s o nodes in X ha can be eached by pe iod +ρwi h nonze o weigh by a pa h om each o he wo ini ial choices being conside ed. Ha ing es ablished exis ence, we can ob ain weigh s sa is ying (3.1) as a by-p oduc . The e a e an in ini e numbe o weigh ing schemes, each o which migh concei ably es ablish ini e dependence, a ac ha migh explain why esea che s ha e op ed o guess and e i y me hods when designing models exhibi ing his compu a ionally con- enien p ope y. Ou nex heo em, howe e , p o ed by cons uc ion in he Appendix, Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 871 shows ha an exhaus i e sea ch o a se o weigh s ha es ablish ini e dependence can be achie ed in a ini e numbe o s eps. The key o he p oo is ha al hough he de ini ion o Hτdoes indeed depend on he weigh s, many se s o weigh s p oduce he same A1τand A2τ(and hence he same Aτ+1). Since he in e sion o Hτhinges on he a ainable s a es, and he se s o all possible a ainable s a es is ini e, a ini e numbe o ope a ions is needed o es ablish whe he a ini e dependence pa h exis s. Theo em 3. Fo each τ∈{ +1 +ρ}, he ank o Hτand H∗ τcan be de e mined in a ini e numbe o ope a ions. Theo em 3applies o any dynamic disc e e choice p oblem desc ibed in Sec ion 2. Howe e , he numbe o calcula ions equi ed o de e mine ρ-pe iod dependence is speci ic o he numbe o choices, J, in pe iods be ween +1and +ρ, henumbe o s a es in each o hose pe iods, and he ansi ion ma ices. As ρinc eases, so oo will he se s o possible a ainable s a es, inc easing compu a ional complexi y in inding he ini e dependence pa h. Inc easing he numbe o choices, J, also will inc ease he se s o possible a ainable s a es. A he same ime, inc easing Jgi es mo e con ol o line up he s a es. When examining ini e dependence o a pai o ini ial choices, he mini- mum ρmus be weakly dec easing as mo e choices a e a ailable as one could always se he weigh on hese addi ional choices o ze o. Finally, he complexi y o he s a e space does no necessa ily equi e mo e calcula ions o de e mine ini e dependence o wo easons. Fi s , i is only he s a es ha can be eached in ρpe iods om he cu en s a e ha a e ele an o de e mining ini e dependence. Second, as he se s o a ainable s a es inc ease, he esea che also has mo e op ions o inding pa hs ha exhibi ini e dependence. 5. Fini e dependence in games Applica ions o ini e dependence in he empi ical li e a u e on games a e sca ce. One excep ion a e models wi h exi decisions, which ha e he e minal s a e p ope y. Al- hough ini e dependence is usually no exploi ed in hese models (bu see Beauchamp (2015)andMazu (2017), Colla d-Wexle (2013), Dunne, Klimek, Robe s, and Xu (2013), and Ryan (2012)) all exhibi he ini e dependence p ope y ha could be used o sim- pli y es ima ion. In p inciple, he me hods de eloped abo e a e di ec ly applicable o dynamic games o sho panels, ha is, a e de ining j (x +1|x )wi h (2.12). Le F(n) kτ (Ajτ)deno e he i s Aτ−1columns o he ansi ion ma ix om Ajτ o Aτ+1gi en choice kby playe na ime τwhen e e yone else plays hei equilib ium s a egy and le  F(n) kτ (Ajτ)de- no e he ansi ion ma ix con aining all he columns. These a e de ined analogously o Fkτ(Ajτ)and Fkτ(Ajτ)in he indi idual op imiza ion case. Also le Ω(n) kτ(A2τj)deno e a ec o o weigh s on choice k o each o he Ajτ s a es in Ajτ. Finally, le K(n) jτ (Ajτ)de- no e he τ-pe iod ansi ion p obabili ies o Ajτ when nini ially chooses jand ollows he weigh s when e e ybody else plays hei equilib ium s a egies. Analogous o (4.6), 872 A cidiacono and Mille Quan i a i e Economics 10 (2019) ρpe iod dependence holds o he i s wo ac ions i ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ F(n) 2τ(A2τ)−F(n) 1τ(A2τ)    F(n) Jτ (A2τ)−F(n) 1τ(A2τ) F(n) 1τ(A1τ)−F(n) 2τ(A1τ)    F(n) 1τ(A1τ)−F(n) Jτ (A1τ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ω(n) 2τ(A2τ2)◦K(n) 2τ(A2τ)    Ω(n) Jτ (A2τ2)◦K(n) 2τ(A2τ) Ω(n) 2τ(A1τ1)◦K(n) 1τ(A1τ)    Ω(n) Jτ (A1τ1)◦K(n) 1τ(A1τ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ =F(n) 1τ(A1τ) −F(n) 1τ(A2τ)K(n) 1τ(A1τ) K(n) 2τ(A2τ)(5.1) In p ac ice, es ablishing ini e dependence is gene ally mo e one ous in games han in indi idual op imiza ion p oblems. Fini e dependence in a game is playe speci ic; in p inciple ini e dependence migh hold o some playe s bu no o o he s. Fu he - mo e, he ansi ion o he s a e a iables o any one playe aking a pa icula ac- ion depends on he equilib ium decisions o all he o he playe s. Thus, ini e depen- dence in games is ul ima ely a p ope y ha de i es no jus om he game p imi i es, bu also equilib ium play. Consequen ly, games do no ypically exhibi one-pe iod i- ni e dependence: i wo di e en choices o na ime a ec he o he playe s’ equi- lib ium choices in +1(o la e ), i is gene ally no easible o line up all he s a es x +2≡(x(0) +2x(1) +2x(N) +2)ac oss bo h pa hs emana ing om he espec i e ini ial choices o nwi hin wo pe iods. A key ea u e o he incomple e in o ma ion games se ings we conside is ha a , when he playe s o he han ncollec i ely choose d(∼n) , hey condi ion on he lagged choice o n(i.e., how d(n) −1a ec s x(n) ), bu no on d(n) , he cu en choice o n.Ou ap- p oach o de e mining ini e dependence in games exploi s his ea u e in he ollowing way. Fi s , we ob ain necessa y and su icien condi ions o playe n o ake a sequence o weigh ed ac ions inducing, say a e ρ−1pe iods, he o he playe s o ake ac ions a +ρ ha ma ch up he weigh dis ibu ions o x(∼n) +ρ+1, condi ional on x(∼n) , mean- ing:26 κ +ρ+1x(∼n) +ρ+1|x i=κ +ρ+1x(∼n) +ρ+1|x j(5.2) Clea ly, (5.2) is a necessa y condi ion o (3.3) o hold. Second, wi h one las choice o weigh pai s a +ρ,playe nlines up he join dis ibu ion o he s a es o all he play- e s, se ing ω(n) k +ρ(x +ρi) and ω(n) k +ρ(x +ρj), and inco po a ing he es ic ions ha gi e (5.2), so ha (3.3) simul aneously holds. 26This inducemen is based on he o he playe s ollowing hei equilib ium s a egies. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 873 5.1 Fini e dependence o s a e componen s con olled by o he playe s F om (3.1), ini e dependence a τ equi es X  xτ=1 J  k=1 (∼n) τx(∼n) τ+1|xτ (n) kτ x(n) τ+1|x(n) τω(n) kτ (xτj)κ(n) τ(xτ|x j) = X  xτ=1 J  k=1 (∼n) τx(∼n) τ+1|xτ (n) kτ x(n) τ+1|x(n) τω(n) kτ (xτi)κ(n) τ(xτ|x i) (5.3) Necessa y and su icien condi ions o (5.3) o hold a e ound in he same way as ini e dependence is de e mined o indi idual op imiza ion p oblems. They a e based on he in ui ion ha om pe iods h ough τ−1playe n akes pai s o weigh ed ac ions s a - ing wi h iand j ha induce he o he o he playe s o align he p obabili y dis ibu ions o x(∼n) τ+1 h ough hei equilib ium choices. A necessa y condi ion o τdependence comes om summing (5.3)o e hex(n) τ+1 ou comes. No ing ha X(n)  x(n) τ+1=1 X  xτ=1 (∼n) τx(∼n) τ+1|xτJ  k=1 ω(n) kτ (xτj) (n) kτ x(n) τ+1|x(n) τκ(n) τ(xτ|x j) = X  xτ=1 (∼n) τx(∼n) τ+1|xτJ  k=1 ω(n) kτ (xτj)X(n)  x(n)=1 (n) kτ x(n) τ+1|x(n) τκ(n) τ(xτ|x j) = X  xτ=1 (∼n) τx(∼n) τ+1|xτκ(n) τ(xτ|x j) (5.4) we simpli y he sum (5.3)o e x(n) τ+1using (5.4) oob ain X  xτ=1 (∼n) τx(∼n) τ+1|xτκ(n) τ(xτ|x j)−κ(n) τ(xτ|x i)=0(5.5) This p o es ha whe he (5.5) holds o no depends on he weigh s assigned o nin pe iods +1 hough τ−1,bu no on hepe iodτweigh s. To de i e a ank condi ion unde which (5.5) holds, i is no a ionally con enien o ocus on he i s wo choices as be o e. Suppose (5.5)holdsa τ+1. Then he e mus be decision weigh s a τ−1wi h he ollowing p ope y: he s a es ha esul in τlead he o he playe s o make (equilib ium) decisions a τso ha each o hei own s a es ha e he same weigh ac oss he wo pa hs a τ+1. Fo mally, le Ajτ−1⊆Xdeno e he se o a ainable s a es a τ−1 o he weigh sequence beginning wi h nchoosing j∈{12}.Le Aτ⊆Xdeno e he se o a ainable s a es a τ o he weigh sequence beginning wi h n ei he choosing 1o 2.Le A(∼n) τ+1⊆X(∼n) deno e he a ainable s a es o he o he playe s a τ+1gi en he wo weigh sequences. Le A(∼n) τ+1deno e he numbe o elemen s in 874 A cidiacono and Mille Quan i a i e Economics 10 (2019) A(∼n) τ+1.Le P(∼n) τ(Aτ)deno e he anspose o he i s A(∼n) τ+1−1columns o he ansi ion ma ix om Aτ o he se o compe i o s a es A(∼n) τ+1. Finally, de ine H(∼n) τand K(∼n) τ+1as H(∼n) τ≡P(∼n) τ(Aτ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣  F(n) 2τ−1(A2τ−1)− F(n) 1τ−1(A2τ−1)     F(n) Jτ−1(A2τ−1)− F(n) 1τ−1(A2τ−1)  F(n) 1τ−1(A1τ−1)− F(n) 2τ−1(A1τ−1)     F(n) 1τ−1(A1τ−1)− F(n) Jτ−1(A1τ−1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  (5.6) K(∼n) τ+1≡P(∼n) τ(Aτ) F(∼n) 1τ−1(A1τ−1)K(n) 1τ−1(A1τ−1)− F(∼n) 1τ−1(A2τ−1)K(n) 2τ−1(A2τ−1)(5.7) Fini e dependence equi es weigh ing ules om +1 o τ−1so ha when he o he playe s ake equilib ium ac ions a τon he wo pa hs he s a es o he o he playe s a e lined up a τ+1. The e ec s o hese equilib ium ac ions on he s a e ope a e h ough P(∼n) τ(Aτ)in (5.6). Thus he simila i y o H(∼n) τand Hτis e iden om compa ing (5.6) wi h (4.7); likewise he simila i ies be ween K(∼n) τ+1and Kτ+1a e ob ious om (4.6)and (5.7). Following he same logic as Theo em 2, we ob ain he ollowing esul . Theo em 4. Gi en an ini ial pe iod and s a e (  x ),and ini ial choices 1and 2,(5.5) holds o all x(∼n) ∈X(∼n) i and only i he e exis s a pai o weigh sequences de ining H(∼n) τand K(∼n) τ+1such ha H(∼n) τand [H(∼n) τ   K(∼n) τ+1]ha e he same ank. 5.2 Aligning he join dis ibu ions Fo one specializa ion, checking he condi ions o Theo em 4su ices o de e mine whe he (3.3) holds o no . Suppose ha o each x∈Aτ, he e is an ac ion d(n)(x) yield- ing some ixed x(n) ∈X(n) o su e.27 Then sa is ying he condi ions o Theo em 4imply he condi ions o Theo em 2a e me oo. In his specializa ion, he join dis ibu ion ac oss he wo pa hs is aligned in τ+1because (i) κ(∼n) τ(x(∼n) τ+1|x j), he ma ginal weigh dis ibu ion o he o he playe s’ s a es is aligned, and (ii) he s a e o playe ndoes no a y ac oss he s a es o he o he playe s. Thus e i ying ini e dependence educes o inding condi ions ha sa is y (5.2) in his case. Renewal and e minal ac ions p o ide examples because he enewal o e minal s a e, x(n) ∈X(n), can be eached om any x(n) ∈X(n) in one pe iod wi h ce ain y. Sec ion 6.2 illus a es ou s ep-by-s ep p oce- du e o es ablishing ini e dependence in a coo dina ion game. A second special case occu s when he ank o H(∼n) τis A(∼n) τ+1−1and he se o weigh s is unique. We i s de i e he unique se o weigh s ω(n) kτ (xτ1)and ω(n) kτ (x τ2) o τ∈in his linea subp oblem; hen ollowing he app oach in he p eceding subsec ion, we show below, as a special case o a mo e gene al esul , ha whe he a se o weigh s 27Mo e gene ally, he e exis s one ac ion d(n)(x), o some weigh ed mix u e o ac ions, ha when applied o ei he sequence, yields he same weigh dis ibu ion o e x(n) ∈X(n) o all x∈A(n) τ. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 875 exis s es ablishing τ- pe iod ini e dependence o no , educes o sol ing a second linea p oblem in ω(n) kτ (xτ1)and ω(n) kτ (x τ2) o k∈{2J}and xτ∈A1τ+1and x τ∈A2τ+1, simila o hose analyzed in he single agen p oblems. I he se o weigh s equalizing he ma ginal dis ibu ions o x(∼n) +ρ+1∈X(∼n) is no unique, hen an uncoun able numbe do, since any con ex combina ion o say wo se s o weigh s also equalize he ma ginal dis ibu ions. Since he weigh s de e mining he solu ion o he s a es o he o he playe s also help de e mine he condi ional dis ibu- ion o x(n) +ρ+1, he selec ion o a solu ion o he o he playe s may impac whe he he condi ional dis ibu ions o x(n) +ρ+1can be aligned o no . To ea bo h cases o mally, le Ω(n) kτ−1(A2τ−1j)deno e an Ajτ−1dimensional ow ec o o unknown weigh s assigning a eal numbe o choice k∈{2J} o each s a e in Ajτ−1a τ−1, gi en s ic ly posi i e weigh s K(n) 2τ−1(A2τ−1).Deno e1J−1as a (J −1) column ec o o ones. De ine Ω(n) τ−1and K(n) τ−1as Ω(n) τ−1≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ω(n) 2τ−1(A2τ−1j)    Ω(n) Jτ−1(A2τ−1j) Ω(n) 2τ−1(A1τ−1j)    Ω(n) Jτ−1(A1τ−1j) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  K(n) τ−1≡1J−1⊗K(n) 2τ−1(A2τ−1) 1J−1⊗K(n) 1τ−1(A1τ−1)(5.8) whe e ⊗is he K onecke p oduc . I he e exis s weigh s Ω(n) τ−1sol ing K(∼n) τ+1=H(∼n) τΩ(n) τ−1◦K(n) τ−1(5.9) hen a necessa y condi ion o ini e dependence embodied in (5.5), ela ing he weigh s o all he playe s aside om n, is sa is ied. In he special case whe e (A1τ−1+A2τ−1)(J − 1)=A(∼n) τ+1−1and H(∼n) τ=H(∼n) τin e s, om (5.9), Ω(n) τ−1=H(∼n) τ−1K(∼n) τ+1◦!K(n) τ−1(5.10) Mo e gene ally, le D(n) τ−1deno e an A(∼n) τ+1−1dimensional ec o gi en by Ω(n) τ−1◦K(n) τ−1and D(n) τ−1a ec o o dimension (A1τ−1+A2τ−1)(J −1)−(A(∼n) τ+1−1)gi en by Ω(n) τ−1◦K(n) τ−1. Also pa i ion H(∼n) τa he (A(∼n) τ+1−1) h column, w i ing H(∼n) τ=[H(∼n) τH(∼n) τ],whe eH(∼n) τ con o ms o D(n) τ−1,andH(∼n) τ o D(n) τ−1.Then(5.9) can be exp essed as K(∼n) τ+1=H(∼n) τD(n) τ−1+H(∼n) τD(n) τ−1(5.11) F om (5.11), i is e iden ha whe he a solu ion o D(n) τ−1exis s hinges on H(∼n) τ,bu no on he alues o D(n) τ−1. Fo example, a su icien condi ion o a solu ion o he i s s ep 876 A cidiacono and Mille Quan i a i e Economics 10 (2019) is ha he ank o H(∼n) τequals A(∼n) τ.28 Mo eo e , om (5.11) when he ank condi ion o H(∼n) τis sa is ied, D(n) τ−1 a ies wi h D(n) τ−1; speci ically D(n) τ−1=H(∼n) τ−1K(∼n) τ+1−H(∼n) τD(n) τ−1(5.12) To es ablish ini e dependence a pe iod τ+1 o he join sys em, i su ices o show ha he e exis s some D(n) τ−1and a se o weigh s on he choices made by nin pe iod τsol ing he join sys em when we inco po a e he e ec s o D(n) τ−1ope a ing h ough K(n) jτ−1.Modi y(4.8)bysupe sc ip ingwi hn he  F(n) jτ−1(Ajτ−1) ansi ions, as well as he weigh ec o s  Kjτ(Aτ)and Kjτ−1(Ajτ−1), o indica e he playe o whom ini e dependence is being checked. Also eplace he ec o o med om he elemen s Ωkτ−1(Ajτ−1j)◦Kjτ−1(Ajτ−1)wi h D(n) τ−1.Then(4.8) becomes  K(n) jτ (Ajτ)=⎡ ⎢ ⎢ ⎣ F(n) 2τ−1(Ajτ−1)     F(n) Jτ−1(Ajτ−1) ⎤ ⎥ ⎥ ⎦  D(n) τ−1(5.13) Fo m he ec o  D(n) τ−1by eplacing he elemen s in D(n) τ−1wi h he linea mappings de ined in (5.12), and subs i u e o K(n) jτ (Ajτ)using he nonze o elemen s o (5.13) in o (5.1). These ope a ions yield a bilinea sys em o equa ions o be sol ed in Ω(n) τand D(n) τ−1.We hen check o a solu ion by minimizing a quad a ic no m o he equa ion sys em. Fini e dependence is achie ed when he quad a ic no m a ains a alue o ze o o some Ω(n) τ and D(n) τ−1. The p oduc ion quali y game conside ed in Sec ion 6.3 illus a es his s epwise p ocedu e. 6. Applica ions This sec ion p o ides h ee illus a ions, new o he li e a u e, ha apply ou ini e de- pendence ep esen a ion. The i s is a job sea ch model. Es ablishing ini e dependence in a sea ch model would seem di icul gi en ha he e is no gua an ee one will ecei e ano he job o e in he u u e i an o e is u ned down oday, and hence lining up, o example, u u e expe ience le els would seem di icul . We show ha ou ep esen a ion applies di ec ly o his case, and in he p ocess highligh he p ac ical impo ance o us- ing nega i e weigh s. The second is a coo dina ion game whe e we apply he esul s o Theo em 4 o show ha we can achie e wo-pe iod ini e dependence in a s a egic se - ing when he condi ions o he specializa ion discussed in he p e ious sec ion hold. Thi d, we analyze a p oduc quali y game ha does no sa is y he condi ions o he specializa ion and de ies a guess and e i y app oach. 6.1 A sea ch model The ollowing simple sea ch model shows why nega i e weigh s a e use ul in es ablish- ing ini e dependence, and uses he algo i hm o exhibi an e en less in ui i e pa h o 28Necessa y and su icien condi ions a e ound in he same way as he single agen op imiza ion case. Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 883 Table 1. Weigh s ha gene a e ini e dependence. P obabili y o d(n) 2τ=1 Time S a e d(n) 1 =1d(n) 2 =1 +1(45)47859 (46)20178 (55)13928 (56)10139 +2(45)1 (46)1 (47)1 (55)047475 (56)0110759 (57)0457515 (65)0 (66)0 (67)0 Ω(n) +2: F(n) 2 +2(A2 +2)−F(n) 1 +2(A2 +2)Ω(n) 2 +2(A2 +22)◦K(n) 2 +2(A2 +2) +F(n) 1 +2(A1 +2)−F(n) 2 +2(A1 +2)Ω(n) 2 +2(A1 +21)◦K(n) 1 +2(A1 +2) +F(n) 1 +2(A2 +2)K(n) 2 +2(A2 +2)−F(n) 1 +2(A1 +2)K(n) 1 +2(A1 +2)2(6.8) In his example, we exploi he bilinea p ope y o (6.8)bysol ing o Ω(n) +2as a linea sys em in he scala D(∼n) +1, hen subs i u ing he solu ion o Ω(n) +2back in o (6.8), and inally esol ing he esul ing sys em in he scala D(∼n) +1. Weigh s gi ing a ze o alue o (6.8) a e displayed in Table 1. Thus wo-pe iod dependence is es ablished by cons uc- ion. 7. Conclusion CCP me hods p o ide a compu a ionally cheap way o es ima ing dynamic disc e e choice models in bo h single-agen and mul iagen se ings. This pape p ecisely de- linea es and expands he class o models ha exhibi he ini e dependence p ope y used in CCP es ima o s, whe eby only a- ew-pe iod-ahead condi ional choice p obabil- i ies a e used in es ima ion. Ou app oach applies o a wide class o p oblems lacking s a iona i y, and is ee o assump ions abou he s uc u e o he model and he belie s o playe s ega ding e en s ha occu a e he (sho ) panel has ended. Thus ou me h- ods p o ide an app oach o es ima ing nons a iona y in ini e ho izon games e en when he e a e no e minal ac ions. 884 A cidiacono and Mille Quan i a i e Economics 10 (2019) Appendix:P oo s P oo o Theo em 1. Wi h (bounded) nega i e weigh s he ini e ho izon esul s o Theo em 1 o A cidiacono and Mille (2011) is easily adap ed, since he posi i i y o neg- a i i y o he weigh s is no used in ha p oo . P oo o Theo em 3.Deno eby A≡{x(1) Ax(A) A}whe e x(a) A∈X o all a∈ {1A}.ThusA∈S, he se con aining 2Xelemen s o all subse s o X. Also de ine he se Aa ains a τby B≡x(b) B∈Xsuch ha jτx(b) B|x=0 o some x∈Aand some j=1J Thus B={x(1) Bx(B) B} o some B≤X. Fo each a∈{1A}, de ine he (J −1)×1 weigh ec o : ωτx(a) A=ω1τx(a) AωJ−1τx(a) A whe e |ωjτ(x(a) A)|<∞and ωJτ(x(a) A)≡1−J−1 j=1ωjτ(x(a) A).Le KA≡(K(1) AK(A) A)de- no e an A×1weigh ec o o e he s a es in A, ha is sa is ying A x=1K(a) A=1wi h |K(a) A|<∞and K(x) A=0. We also de ine K(b) B≡ A  a=1 J  j=1 jτx(b) B|x(a) Aωjτx(a) AK(a) A and no e ha B  b=1 K(b) B= B  b=1 A  a=1 J  j=1 jτx(b) B|x(a) Aωjτx(a) AK(a) A = A  a=1 J  j=1 ωjτx(a) AK(a) A= A  a=1 K(a) A=1(A.1) Depending on KA, and also he choice o ωτA≡(ωτ(x(1) A)ωτ(x(A) A)),someele- men s o KB≡(K(1) BK(B) B)may be ze o. We say ha A eaches A∗⊆Aa τ o he ec o weigh ing KAi , o some choice o ωτA, e e y elemen in A∗is a ained (has nonze o weigh ), and e e y elemen in he complemen o A∗is no a ained (has ze o weigh ). Theo em 2, and i s p oo in he ex , shows ha only a ini e numbe o ope a ions a e equi ed o de e mine whe he o no ini e dependence can be achie ed in one pe iod om wo gi en se s A1 +ρand A2 +ρ. In pa icula , i is e iden om he cons uc ion o Hτ, ha he ope a ions do no depend on he ωτA1 +ρand ωτA2 +ρ, he espec i e weigh s on elemen s in A1 +ρand A2 +ρ.Gi enj∈{12}, and a sequence o weigh s de ined om +1 o +ρ, a unique sequence o se s is de e mined: say {Ajτ}ρ τ= +2.Al- hough he e a e an uncoun able numbe o pa hs, since Ajτ ∈Sand Scon ains (only) 2Xelemen s, he e a e a mos 2(ρ−1)X se s ha any weigh sequence can successi ely each, om Aj +1≡{x∈X: j (x|x )>0}up o and including Aj +ρ. The e o e, he Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 885 p oo is comple ed by showing ha a ini e numbe o ope a ions su ice o de e mine whe he o no a gi en A⊆A jτ+1can be eached om any Ajτ ∈S, o all possible (nonze o) weigh s KA. To de e mine whe he A eaches A∗a τ, we ex end simila a gumen s gi en in he ex o checking whe he ρ=2in he special case whe e J=2. Wi hou loss o gene ali y, we ocus on he case whe e A∗is migh be eached because he i s A∗elemen s o KA∗a e nonze o and he emaining B∗−A∗a e ze o. (The o he cases a e co e ed by a eo de ing o he s a es.) Thus KB≡(K(1) BK(B) B)is a weigh ing o A∗i and only i K(b) B= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1− A∗  b=2 K(b) B o b=1 any nonze o alue o b∈2A∗ subjec o he cons ain A∗  b=2 K(b) B=1 0 o b∈A∗+1B (A.2) The exis ence o a solu ion o an uncons ained linea sys em, comp ising B−1 equa ions in (J −1)A unknowns, de e mines whe he A eaches A∗a τo no . The unknown a iables in he linea sys em a e he Achoice weigh ec o s ωτ(x(a) A), each o dimension J−1.TheB−1equa ions co espond o he nonze o weigh s placed on he s a es {x(2) Bx(A∗) B}and he ze o weigh ing placed on he las B−A∗s a es, which belong o Bbu no A∗. All choice weigh s sa is ying he equa ions co esponding o {x(2) Bx(A∗) B}also sa is y he i s s a e in Bby (A.1)and(A.2). Gi en K(b) Bsa is ying (A.2), a solu ion o his linea sys em exis s i he e exis s A choice weigh ec o s ωτ(x(a) A) o each b∈{2B}sol ing K(b) B= A  a=1 Jτx(b) B|x(a) AK(a) A+ A  a=1 J−1  j=1 jτx(b) B|x(a) A− Jτx(b) B|x(a) Aωjτx(a) AK(a) A(A.3) Le Fjτ(A)deno e he A×(B −1) ansi ion ma ix o Ain o all bu he i s s a es in B o choice j∈{12J−1}. De ine [KA◦ωτ(A)]as he A(J −1)×1 ec o o med om he elemen -by-elemen p oduc K(a) Aωjτ(x(a) A).Deno e he(B −1)×A(J −1)con- ca ena ed ma ix o ansi ions by Fτ(A)≡F1τ(A)··· FJ−1τ(A) =⎡ ⎢ ⎣ 1τx(2) B|x(1) A··· 1τx(2) B|x(A) A··· J−1τx(2) B|x(1) A··· J−1τx(2) B|x(A) A     ···      1τx(B) B|x(1) A··· 1τx(B) B|x(A) A··· J−1τx(B) B|x(1) A··· J−1τx(B) B|x(A) A⎤ ⎥ ⎦ De ining K∗ Bas a (B −1)×1 ec o o med om all bu he i s elemen o KBsa is ying (A.2) hen(A.3) may be exp essed in ma ix no a ion as K∗ B=FJτ(A)KA+Fτ(A)−FJτ(A)KA◦ωτ(A)(A.4) 886 A cidiacono and Mille Quan i a i e Economics 10 (2019) Appealing o Hadley (1961, pp. 168–169), o a gi en K∗ B, a solu ion o (A.4)in [KA◦ω∗ τ(A)]exis s i and only i he ank o [Fτ(A)−FJτ(A)]equals he ank o he aug- men ed ma ix o med by adding he column [K∗ B−FJτ(A)KA] o [Fτ(A)−FJτ(A)].By cons uc ion, he augmen ed ma ix ei he has he same ank as, o one plus he ank o [Fτ(A)−FJτ(A)]. Since de e mining he ank o a ini e dimensional ma ix equi es only a ini e numbe o ope a ions, and he e a e only a ini e numbe o s eps, he he- o em is p o ed. P oo o Theo em 5. The p oo is by cons uc ion. In his game, each playe n∈{12} con ols wo s a es, namely he choices o he p e ious pe iod “in” o “ou ,” so om (5.9) a su icien condi ion o wo-pe iod dependence is he exis ence o a solu ion o H(∼n) +2Ω(n) 2 +1(A2 +12)◦K2 +1(A2 +12) Ω(n) 2 +1(A1 +11)◦K1 +1(A1 +11) =P(∼n) +2A(n) +2F(n) 1 +1(A1 +1) −F(n) 1 +1(A2 +1)K1 +1(A1 +1) K2 +1(A2 +1)(A.5) whe e he de ini ions o H(∼2) τ, gi en in (5.6), Kj +1(A2 +1)and Ω(n) 2 +1(A2 +1j),gi en abo e (5.1)andP(∼n) +2(A +2), gi en abo e (5.6) specialize o30 H(∼n) +2≡P(∼n) +2(A +2)F(n) 2 +1(A2 +1)−F(n) 1 +1(A2 +1) F(n) 1 +1(A1 +1)−F(n) 2 +1(A1 +1)  Ω(n) 2 +1(A2 +1j)=ω(n) +1(j2)ω(n) +1(j1) K2 +1(A2 +1)=K1 +1(A2 +1)=p(∼n) 2 (x )p (∼n) 1 (x ) P(∼n) +2(A +2)=p(∼n) 2 +2(22)p (∼n) 2 +2(21)p (∼n) 2 +2(12)p (∼n) 2 +2(11) and in his example: F(n) 1 +1(A1 +1) −F(n) 1 +1(A2 +1) =⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 00 0 0 00 0 0 p(∼n) 2 +1(12)p (∼n) 2 +1(11)−p(∼n) 2 +1(22)−p(∼n) 2 +1(12) p(∼n) 1 +1(12)p (∼n) 1 +1(11)−p(∼n) 1 +1(22)−p(∼n) 1 +1(21) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦  30Since ma ching he weigh on one s a e au oma ically ma ches he weigh on he o he , we can elimi- na e he las ow o P(∼n) +2(A(n) +2). Quan i a i e Economics 10 (2019) Nons a iona y dynamic models 887 F(n) 2 +1(A2 +1)−F(n) 1 +1(A2 +1) F(n) 1 +1(A1 +1)−F(n) 2 +1(A1 +1) =⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p(∼n) 2 +1(22)p (∼n) 2 +1(21)−p(∼n) 2 +1(12)−p(∼n) 2 +1(11) p(∼n) 1 +1(22)p (∼n) 1 +1(21)−p(∼n) 1 +1(12)−p(∼n) 1 +1(11) −p(∼n) 2 +1(22)−p(∼n) 2 +1(21)p (∼n) 2 +1(12)p (∼n) 2 +1(11) −p(∼n) 1 +1(22)−p(∼n) 1 +1(21)p (∼n) 1 +1(12)p (∼n) 1 +1(11) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A.6) No ing ω(n) +1(x +1j)≡ω(n) +1((jd(∼n) 2 )j) we now de ine ω(n) +1(x +1)≡ω(n) +1(x +1j) o elimina e he no a ional edundancy, and subs i u e he exp essions abo e in o he le - hand side o (A.5) oob ain ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p(∼n) 2 +2(22) p(∼n) 2 +2(21) p(∼n) 2 +2(12) p(∼n) 2 +2(11) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p(∼n) 2 +1(22)p (∼n) 2 +1(21)−p(∼n) 2 +1(12)−p(∼n) 2 +1(11) p(∼n) 1 +1(22)p (∼n) 1 +1(21)−p(∼n) 1 +1(12)−p(∼n) 1 +1(11) −p(∼n) 2 +1(22)−p(∼n) 2 +1(21)p (∼n) 2 +1(12)p (∼n) 2 +1(11) −p(∼n) 1 +1(22)−p(∼n) 1 +1(21)p (∼n) 1 +1(12)p (∼n) 1 +1(11) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ×⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω(n) +1(22)p(∼n) 2 (x ) ω(n) +1(21)p(∼n) 2 (x ) ω(n) +1(12)p(∼n) 2 (x ) ω(n) +1(11)p(∼n) 2 (x ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A.7) Since p(∼n) 2 (x )>0, we can es ablish wo-pe iod dependence by equa ing (A.7)wi h he igh -hand side o (A.5) and sol ing o he unknowns. By inspec ion, (A.7)is1×1,and (A.5) educes o a single equa ion, wi h ou unknowns ha con o m o he 1×4 ow ec o H(∼n) +2. To comple e he p oo , i is use ul o de ine o i∈{12} he exp ession: Ci≡p(∼n) 2 +2(21)−p(∼n) 2 +2(11) +p(∼n) 2 +1(2i)p(∼n) 2 +2(22)+p(∼n) 2 +2(11)−p(∼n) 2 +2(21)−p(∼n) 2 +2(12)(A.8) We now p o e C2=0i C1=0. No e ha C2−C1=p(∼n) 2 +1(22)−p(∼n) 2 +1(21) ×p(∼n) 2 +2(22)+p(∼n) 2 +2(11)−p(∼n) 2 +2(21)−p(∼n) 2 +2(12)(A.9) I he second b acke ed e m is ze o, hen C1=C2 om (A.9), and hence om (A.8) C1=0because by assump ion p(∼n) 2 +2(21)=p(∼n) 2 +2(11). The e o e, i C1=0 he b ack- e ed e m is nonze o. In ha case, C2=C1by (A.9) because p(∼n) 2 +1(21)=p(∼n) 2 +1(22)by assump ion. We conside wo possibili ies, in which ω(n) +2(x(n) +2j)=1 o j∈{12}and ω(n) +1(1i)= 0 o i∈{12} o bo h possibili ies. Also se ω(n) +1(22)=0i C1=0,andse ω(n) +1(21)= 888 A cidiacono and Mille Quan i a i e Economics 10 (2019) 0i C1= 0. Using (A.8) and no ing p(∼n) 1 +1(22)=1−p(∼n) 2 +1(22), simpli y (A.7) o Cip(∼n) 2 (x )ω(n) +1(2i). Sol ing o he only nonze o weigh , ake he quo ien o he scala (A.7)andCip(∼n) 2 (x ) o ob ain ω(n) +1(2i)=P(∼n) +2A(n) +2F(n) 1 +1(A1 +1) −F(n) 1 +1(A2 +1)K(n) 1 +1(A1 +1) K(n) 2 +1(A2 +1)◦ p(∼n) 2 (x )Ci(A.10) whe e he ma ices in (A.10) a e gi en abo e. Thus ω(n) +1(21)is de e mined by se ing i=1in (A.10)whenC1= 0and ω(n) +1(22)is de e mined by se ing i=2in (A.10)when C1=0. Two-pe iod dependence can now be es ablished by di ec e i ica ion. Re e ences Agui egabi ia, V. and A. Magesan (2013), “Eule equa ions o he es ima ion o dynamic disc e e choice s uc u al models.” Ad ances in Econome ics, 31, 3–44. [854,861] Agui egabi ia, V. and A. Magesan (2017), “Solu ion and es ima ion o dynamic disc e e choice s uc u al models using Eule equa ions.” Wo king pape . [854,861] Agui egabi ia, V. and P. 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