Bargaining with Heterogeneous Beliefs: A Structural Analysis of Florida Medical Malpractice Lawsuits. Abstract


 Georgiana Ross
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1 Bargaining wih Heerogeneous Beliefs: A Srucural Analysis of Florida Medical Malpracice Lawsuis Anonio Merlo and Xun Tang 1 Deparmen of Economics Rice Universiy May 16, 2015 Absrac We propose a srucural bargaining model where players hold heerogeneous beliefs abou he nal resoluion if no selemen is reached ouside he cour. We show he disribuion of heir beliefs and he sochasic surplus are nonparamerically ideni ed from he probabiliy for reaching an selemen and he disribuion of nal ransfers beween players. We hen use a Simulaed Maximum Likelihood (SML) approach o esimae he beliefs of docors and paiens in medical malpracice lawsuis in Florida in he 1980s and 1990s. We nd srong evidence ha he beliefs for boh paries vary wih he severiy of he injury and he quali caion of he docors in he lawsuis, even hough hese characerisics are saisically insigni can in explaining wheher he cour rules in favor of he plaini or he defendan. Key words: Bargaining wih heerogeneous beliefs, nonparameric ideni caion, medical malpracice lawsuis 1 We hank paricipans a Brown, OSU and Economeric Sociey Norh America Summer Meeing 2014 for useful feedbacks. This research is funded by NSF Gran # xxxxxx. We hank Devin Reily and Michelle Tyler for capable research assisance. Errors (if any) are our own. 1
2 1 Inroducion A major heme in recen developmen of bargaining heory is o raionalize he delay in reaching an agreemen ha are prevalen in realworld bargaining episodes. (See, for example, Cho (1990), Merlo and Wilson (1995) and Yildiz (2004).) One explanaion for he delay is ha he paries involved in bargaining are oo opimisic abou heir respecive bargaining power in he absence of a common prior belief. Speci cally, consider a bilaeral bargaining episode where players could learn abou each oher s bargaining power hrough hisory of negoiaion. In he presence of opimism, a player i will decide o wai in hopes ha he oher player j will learn abou his (i s) selfperceived srong posiion and agree o his (i s) erms. As ime passes, he learning slows down, and i becomes no longer worhwhile o wai for he oher paries learning. Tha is when hey reach an agreemen. Yildiz (2004) inroduced his model and showed ha here is a deerminisic selemen dae, which is predeermined by he prior and he discoun facor of he players, such ha players will wai unil his dae o reach an agreemen. Since is inroducion, he model of bargaining wih opimism has been applied in a wide range of empirical conex, such as prerial negoiaions in medical malpracice lawsuis (Waanabe (2006)), negoiaion abou marke condiions (Thanassoulis 2010), and crosslicense agreemens (Galasso (2006)). Despie he recen surge in he heory and he applicaion of bargaining wih opimism, we are no aware of any exising work which addresses he ideni caion quesion in such a model formally. Tha is, under wha condiions can he srucural elemens of he model be unambiguously recovered from he hisory of bargaining repored in he daa. One of he objecives of our paper is o ll in his gap beween heory and empirical work by inroducing a framework for he srucural esimaion of bargaining wihou a common prior. In paricular, we propose a model for bilaeral bargaining where players have opimism abou he sochasic nal oucome in case no agreemen is reached. The players have a oneime opporuniy for reaching an agreemen a an exogenously scheduled dae during he bargaining process, and make decisions abou he selemen based on heir beliefs and ime discoun facors. We show ha all srucural elemens in his model are ideni ed nonparamerically from he probabiliy for reaching an agreemen and he disribuion of ransfer in he nal resoluion. The ideni caion sraegy does no rely on any paramerizaion of he srucural primiives such as players beliefs or he surplus disribuion. We hen propose a Simulaed Maximum Likelihood (SML) esimaor based on exible paramerizaion of he join beliefs. The model we inroduce is a simpli caion of he model of bargaining wihou a common prior in Yildiz (2004). We model a oneime selemen opporuniy for he players o reach an agreemen. Consequenly here is no dynamic learning consideraion in players decisions, and he daes of he nal resoluion of he bargaining episodes are deermined by players opimism, heir paience and heir percepion of he surplus o be shared. 2
3 There are boh heoreical and empirical moivaion for such simpli caion and modi  caion. Firs o, in realworld bargaining episodes, daa limiaion prevens researchers from deriving robus (paramerizaionfree) argumens for he ideni caion of srucural elemens in a full edged model of bargaining wih uncommon prior. For insance, he ideniy of proposers and he iming or he size of rejeced o ers in negoiaions are seldom repored in he daa available o researchers. By absracing away from he dynamic learning aspecs in Yildiz (2004), we adop a pragmaic approach o build a model ha is ideni able under less sringen daa requiremens and mild economeric assumpions. Despie his simpli caion, our model capures a key aspec of models wihou a common prior in ha he iming of he agreemen is deermined by he players opimism and heir paience. Thus our work provides a benchmark for undersanding wha addiional daa or economeric assumpions are needed o recover he primiives in more elaborae models of bargaining wihou common priors. Our modeling choices are also moivaed by he empirical quesion addressed in his paper: In medical malpracice lawsuis, wha do he selemen decisions by paiens and docors ell us abou heir respecive percepion of how likely he cour would rule in heir favor in case a cour hearing is necessary? The law of he Sae of Florida requires ha here should be a oneime mandaory selemen conference beween he plaini and he defendan, which is scheduled by he couny cour someime prior o he hearing and is mediaed by courdesignaed legal professionals. Besides, unlike Yildiz (2004) where a player s bargaining power is modeled as his chance for being he proposer, we model bargaining power as a player s percepion of he probabiliy ha he will receive he favored oucome in he nal resoluion in case no agreemen is reached. This modi caion is mean o be a beer approximaion of he acual decision environmen in he legal conex. Our sraegy for idenifying his model builds on a couple of insighs: Firs o, if he lengh of ime beween he scheduled selemen conference and he cour hearing (a.k.a. he waiime ) were repored in he daa, we would be able o recover he disribuion of opimism by observing how he condiional selemen probabiliy varies wih he lengh of waiime. Second, he disribuion of he poenial surplus o be divided beween players can be recovered from he disribuion of oal compensaion awarded o he plaini by he cour decision, provided he surplus disribuion is orhogonal o he beliefs. Third, because he acceped selemen o ers re ecs a plaini s imediscouned expecaion of his share of he oal surplus, we can idenify he disribuion of he plaini s belief condiional on selemen using he disribuion of acceped selemen o ers given he lengh of he waiime. This is done hrough a deconvoluion argumen using he disribuion of surplus recovered above. Las, since opimism is de ned as he sum of boh paries beliefs minus one, he objecs ideni ed from he preceding seps can be used o back ou he join disribuion of he beliefs hrough a sandard Jacobian ransformaion. 3
4 A key challenge for implemening he ideni caion sraegy in he environmen of malpracice lawsuis is ha he lengh of waiime is no direcly repored in he daa. In order o solve his issue of unobserved waiime, we ap ino a branch of recen lieraure ha uses an approach based on eigenvalue decomposiion o idenify nie mixure models or srucural models wih unobserved heerogeneiy. (See for example Hall and Zhou (2003), Hu and Schennach (2008), Kasahara and Shimosu (2009), An, Hu and Shum (2010) and Hu, McAdams and Shum (2013).) To do so, we rs exploi he insiuional deails in our environmen o group lawsuis ino smaller clusers (de ned by he couny and he monh in which a lawsui is o cially led) ha can be plausibly assumed o share he same (albei unobserved) lengh of waiime beween selemen conference and cour hearings. We hen use he cases in he same cluser as insrumens for each oher and apply an eigenvalue decomposiion o he join disribuion of selemen decisions and acceped o ers wihin he cluser. This allows us o recover he selemen probabiliy and he disribuion of acceped o ers condiional on he unobserved waiime. Then he argumens from he preceding paragraph applies o idenify he join disribuion of beliefs. The inference of docors and paiens beliefs in medical malpracice lawsuis is an ineresing empirical quesion in is own righ. In paricular, a cenral issue in he reform of U.S. healh care sysem is how o minimize he liigaion coss in medical malpracice lawsuis, which are known o consiue a large porion of he soaring insurance expenses. Knowing how selemen ouside he cour depends on paiens and docors opimism in he bargaining process could shed lighs on policy design. Using daa from medical malpracice lawsuis in Florida in he 1980s and 1990s, we nd clear evidence in our esimaes ha he beliefs of he docors and paiens vary wih observed characerisics of he lawsuis such as he severiy of he injury and he quali caion of he docors. This conrass wih he realiy ha he cour and jury decisions depend mosly on he naure and he cause of he malpracice and no so much on hese observed case characerisics (which is anoher fac revealed in our esimaes in he applicaion). Our esimaes can be used for answering fuure policy design quesions such as how he selemen probabiliy would change if he disribuion of he waiime is changed or some caps on pu on he poenial compensaion possible. The res of he paper is organized as follows: Secion 2 inroduces he model of bilaeral bargaining wih uncommon beliefs. Secion 3 esablishes he ideni caion of srucural elemens in he model. Secion 4 de nes he Simulaed Maximum Likelihood (SML) esimaor. Secion 5 describes he daa and he insiuional deails in he applicaion of medical malpracice lawsuis in Florida. Secion 6 presens and discusses he esimaion resuls. Proofs and a mone carlo sudy are presened in he appendices. 4
5 2 The Model Consider a lawsui following an incidence of medical malpracice involving a plaini (or paien) and a defendan (or docor). The oal amoun of poenial compensaion C is common knowledge among he plaini and he defendan. (I should be inerpreed as a sunk cos for he defendan, analogous o he money paid by he defendan for bailou.) Afer he ling of a lawsui, he plaini and he defendan are noi ed of a dae for a oneime selemen conference, which is mandaory by he Sae Saues in Florida. The conference requires aendance by boh paries (and heir aorneys), as well as legal professionals designaed by he couny cour where he lawsui is led. Such selemen conferences ake place wihin 120 days afer he ling of he lawsui. 2 During he conference, he defendan has he opporuniy o make an selemen o er of S C o he plaini. If he plaini acceps i, hen he legal process ends wih plaini receiving S and he defendan reclaiming C S. Oherwise he case needs o go hrough a cour hearing process ha culminaes in jury decisions. Boh he defendan and he plaini are aware ha he cour hearing needs o be a leas hree weeks laer han he selemen conference; and he exac dae is deermined by he schedule and he backlogs of all judges available a he coun cour. Le T denoe he lengh of ime beween he selemen conference and he scheduled cour hearing dae. Le A 1 if a selemen is reached a he conference; and A 0 oherwise. In he laer case, a he end of he cour hearing process, he jury makes a binary decision D as o wheher he plaini ges compensaed wih he full amoun C (i.e. D = 1) or he defendan is acquied wih no compensaions o he plaini required (i.e. D = 0). The plaini and he defendan believe heir chances of winning are p and d 2 [0; 1] respecively. These beliefs are common knowledge beween he paries, bu are no repored in daa. The join suppor of beliefs is f( p ; d ) 2 (0; 1] 2 : 1 < p + d 2g. This means excessive opimism always occurs (i.e. p + d > 1 wih probabiliy 1). We mainain he following assumpion hroughou he paper. Assumpion 1 (i) ( p ; d ) and C are independen from he wai ime T ; and he disribuions of ( p ; d ) is coninuous wih posiive densiies over. (ii) Condiional on A = 0, he jury decision D is orhogonal o C and T. Assumpion 1 allows plaini s and defendans beliefs o be correlaed wih each oher and asymmeric wih di eren marginal disribuions. This is empirically relevan because he marginal disribuion of beliefs may well di er beween paiens and docors due o 2 See Secion 108 in Chaper 776 of Florida Saues Web link: hp:// www. senae. gov/ Laws/ Saues/ 2012/
6 facors such as informaional asymmeries (e.g. docors are beer informed abou he cause and severiy of he malpracice) or unobserved individual heerogeneiies. Beliefs of plaini s and defendans are also likely o be correlaed hrough unobserved heerogeneiy of he case of malpracice. For example, hey may boh observe aspecs relaed o severiy or cause of he malpracice ha are no recorded in daa. Such aspecs lead o correlaions beween paiens and plaini s beliefs from an ousider s perspecive. Assumpion 1 also accommodaes correlaion beween ( p ; d ) and C. The independence beween he wai ime T and he beliefs is a plausible condiion, because he wai ime T is mosly deermined by availabiliy of judges and juries in he couny cour during he lawsuis. This depends on he schedule and backlogs of judges, which are idiosyncraic and orhogonal o paries beliefs ( p ; d ). The orhogonaliy of C from D given T and A = 0 in condiion (ii) is also jusi ed. On he one hand, C is a moneary measure of he magniude of he damage in iced on he plaini regardless of is cause; on he oher hand, D capures he jury s judgemen abou he cause of damage based on cour hearings. I is likely ha he jury decision is correlaed wih speci c feaures of he lawsui ha are repored in daa and ha may also a ec he beliefs of boh paries. Neverheless, once condiional on such observable feaures, jury decisions are mos likely o be orhogonal o measure of damage capured by C. A he end of his secion, we discuss how o exend our model o accoun for heerogeneiies across lawsuis repored in daa. We now summarize how he disribuions ha are direcly ideni able from daa are linked o model primiives under he assumpion ha boh paries follow raional sraegies. A he selemen conference, he plaini acceps an o er if and only if S T p C, where is a consan ime discoun facor xed hroughou he daageneraing process and available in daa. The defendan o ers he plaini S = T p C if he remainder of he poenial compensaion C S exceeds T d C. Hence a selemen occurs during he conference if and only if: C T p C T d C, d + p T. The resuled disribuion of selemens, condiional on he wai ime beween he selemen conference and scheduled cour hearing being T =, is: Pr (S s j A = 1; T = ) = Pr p C s j d + p. (1) where lower cases denoe realized values for random variables; and he equaliy follows from par (i) in Assumpion 1. Besides, he disribuion of poenial compensaion, condiional on he absence of selemen in he conference T = periods ahead of he cour hearing and condiional on he jury ruling in favor of he plaini, is: Pr(C c j A = 0; D = 1; T = ) = Pr(C c j d + p > ) (2) 6
7 where he equaliy follows from boh condiions in Assumpion 1. In pracice, he daa repors di erences in he characerisics of plaini s and defendans, such as he professional quali caion of he defendan or he demographics of he plaini. Besides, he daa also repors feaures relaed o he cause and he severiy of malpracice in quesion. Such informaion available in daa (denoed by a vecor X) are correlaed wih oal compensaion C and beliefs ( p ; d ). The simplisic model above can incorporae such observed case heerogeneiies by leing he primiives (i.e. disribuions of beliefs ( p ; d ), compensaions C, jury decisions D and he waiime T ) depend on X. If boh resricions in Assumpion 1 hold condiional on X, hen raional sraegies are characerized in he same way as (1) and (2) excep ha all disribuions needs o be condiioned on X. More imporanly, he ideni caion sraegy proposed in Secion 3 below are applicable when daa repors heerogeneiies across lawsuis. Formally, he resuls in Secion 3 (Lemma 1 and Proposiion 1) hold afer condiioning on X, provided he idenifying condiions (Assumpions 2, 3, 4 and 5) are formulaed as condiional on X. Noneheless, in order o simplify exposiion of he main idea for ideni caion, we choose o suppress dependence on observable case characerisics in Secion 3, and only incorporae hem explicily laer in he esimaion secion. 3 Ideni caion This secion shows how o recover he disribuion of boh paries beliefs from he probabiliy for reaching selemens and he disribuion of acceped selemen o ers. We consider an empirical environmen where for each lawsui he daa repors wheher a selemen occurs during he mandaory conference (A). For cases seled a he conference, he daa repors he amoun paid by he defendan o he plaini (S). For he oher cases ha underwen cour hearings, he daa repors jury decisions (D) and, if he cour rules in favor of he plaini, he amoun of oal compensaions paid by he defendan (C). However, exac daes of selemen conferences and scheduled daes for cour hearings (if necessary) are never repored in daa. 3 Thus he waiime T beween selemen conference and scheduled cour hearings, which is known o boh paries a he conference, is no available in daa. 3 For example, he daa we use in Secion 5 repors Daes of Final Disposiion for each case. However, for cases seled ouside he cour, hese daes are de ned no as he exac dae of he selemen conference, bu as he day when all o cial adminisraive paperwork are concluded. There is a subsanial lengh of ime beween he wo. For insance, for a large proporion of cases ha are caegorized as Seled wihin 90 days of he ling of lawsuis, he repored daes of nal disposiion are acually more han 150 days afer he iniial ling. Similar issues also exis for cases ha underwen cour hearings in ha he repored daes of nal disposiion are no idenical o he acual dae of cour hearings. 7
8 To address his issue wih unrepored waiime, we propose sequenial argumens ha exploi an implici panel srucure of he daa in he curren conex. In paricular, we noe ha lawsuis led wih he same couny cour during he same period (week) pracically share he same waiime T. The reason for such a paern is as follows: Firs, he daes for selemen conferences are mosly deermined by availabiliy of auhorized legal professionals a liaed wih he coun cour, and are assigned on a rscome, rs served basis. Thus selemen conferences for cases led wih he same couny cour a he same ime are pracically scheduled for he period. Besides, he daes for poenial cour hearing are deermined by he schedule and backlog of judges a he couny cour. Hence cases led wih he same couny cour simulaneously can be expeced o be handled in cour in he same period in he fuure. This allows us o e ecively group lawsuis ino clusers wih he same T, despie unobservabiliy of T in daa. We formalize his implici panel srucure as follows. Assumpion 2 Researchers have su cien informaion o divide he daa ino clusers, each of which consiss of a leas hree lawsuis sharing he same waiime T. Across he cases wihin he same cluser, he beliefs ( p ; d ), he oal compensaion C and he poenial jury decision D (if necessary) are independen draws from he same disribuion. This implici panel srucure in our daa allows us o use acceped selemen o ers in he lawsuis wihin he same cluser as insrumens for each oher, and apply eigendecomposiionbased argumens in Hu and Schennach (2008) o recover he join selemen probabiliy and disribuions of acceped selemen o ers condiional on he unobserved T. We hen use hese quaniies o back ou he join disribuion of beliefs using exogenous variaions in T. For he res of his secion, we rs presen argumens for he case where T is discree (i.e. jt j < 1). A he end of his secion, we explain how o generalize hem for ideni caion when T is coninuously disribued. 3.1 Condiional disribuion of selemen o ers An inermediae sep for idenifying he join disribuion of beliefs is o recover he condiional selemen probabiliy and he disribuion of selemen o ers given he waiime before cour hearings T. Le S; T denoe he uncondiional suppors of S; T respecively. Assumpion 3 (i) The suppor of T is nie (jt j < 1) wih a known cardinaliy and inff : 2 T g 1=2. (ii) Given any ( p ; d ), he poenial compensaion C is coninuously disribued wih posiive densiy over a conneced suppor [0; c]. 8
9 We focus on he model wih nie T wih known cardinaliy because of is empirical relevance. Wihou loss of generaliy, denoe elemens in T by f1; 2; :; jt jg. Condiion (i) also rules ou unlikely cases where a cour hearing is scheduled so far in he fuure or he oneperiod discoun facor is so low ha he compounded discoun facor is less han one half. Condiion (i), ogeher wih he nonincreasingness of E[A i j T = ] over 2 T due o Assumpion 1, pin down he index for eigenvalues and eigenvecors in he aforemenioned decomposiion. Par (ii) in Assumpion 3 is a mild condiion on he condiional suppor of poenial compensaion. A su cien condiion for his is ha C is orhogonal from ( p ; d ) wih a bounded suppor. 4 resul below. The role of par (ii) will become clear as we discuss he ideni caion Lemma 1 Under Assumpions 1, 2 and 3, E (A j T = ) and f S (s j A = 1; T = ) are joinly ideni ed for all and s. This inermediae resul uses argumens similar o ha in Hu, McAdams and Shum (2013) for idenifying rsprice sealedbid aucions wih nonseparable aucion heerogeneiies. I explois he panel srucure of he daa and he condiional independence of beliefs across lawsuis in Assumpion 2. These condiions allow us o break down he join disribuion of he incidence of selemen and he size of acceped selemen o ers across muliple lawsuis wihin one cluser ino he composiion of hree linear operaors. More speci cally, le f R1 (r 1, R 2 = r 2 j :) be a PrfR 1 ~r, R 2 = r 2 j :gj ~r=r1 for any discree random vecor R 2 and coninuous random vecor R 1. For any hree lawsuis i; j; k sharing he same waiime T, le A i;k = 1 be a shorhand for A i = A k = 1. By consrucion, f Si ;S k (s; s 0 ; A j = 1 j A i;k = 1) = P f Si (s j S k = s 0 ; A j = 1; T = ; A i;k = 1)E[A j j S k = s 0 ; T = ; A i;k = 1]f T;Sk (; s 0 j A i;k = 1) 2T = P 2T f Si (s j A i = 1; T = )E[A j j T = ]f T;Sk (; s 0 j A i;k = 1). (3) The second equaliies follow from Assumpion 1; from S = T p C whenever A = 1 and A = 1 if and only if p + d T ; and from he fac ha beliefs ( p ; d ) and poenial compensaion C are independen draws across he lawsuis i; j; k according o Assumpion 2. To illusrae he ideni caion argumen, i is useful o adop marix noaions. Le D M denoe a pariion of he uncondiional suppor of acceped selemen o ers S ino 4 I is worh noing ha our ideni caion argumen remains valid even wih c being unbounded, as long as he fullrank condiion in Lemma A1 holds for some pariions of S. 9
10 M inervals. Each of he inervals has a nondegenerae inerior and is denoed by d m. 5 For a given pariion D M, le L Si ;S k probabiliy ha S i 2 d m and S k 2 d m 0 be a MbyM marix whose (m; m 0 )h enry is he condiional on A i;k = 1 (selemens are reached in cases i and k); and le Si ;S k be a MbyM marix wih is (m; m 0 )h enry being f(s i 2 d m ; A j = 1; S k 2 d m 0 j A i;k = 1). Noe ha boh Si ;S k and L Si ;S k are direcly ideni able from daa. Thus a discreized version of (3) is : Si ;S k = L Si jt j L T;Sk (4) where L Si jt be a MbyjT j marix wih (m; )h enry being Pr(S i 2 d m j A i = 1; T = ); j be a jt jbyjt j diagonal marix wih diagonal enries being [E(A j j T = )] =1;:;jT j ; and L T;Sk be a jt jbym marices wih is (; m)h enry being Pr (T = ; S k 2 d m j A i;k = 1). Besides, due o condiional independence in Assumpion 2. L Si ;S k = L Si jt L T;Sk (5) Par (ii) in Assumpion 3 implies he supreme of he condiional suppor of acceped o ers given T = is c and hence decreases in. This, in urn, guaranees here exiss a pariion D jt j such ha L Si jt as well as L Si ;S k are nonsingular (as proved in Lemma A1 in Appendix B). Then (4) and (5) imply Si ;S k (L Si ;S k ) 1 = L Si jt j L Si jt 1 (6) where he L.H.S. consiss of direcly ideni able quaniies. The R.H.S. of (6) akes he form of an eigendecomposiion of a square marix, which is unique up o a scale normalizaion and unknown indexing of he columns in L Si jt and diagonal enries in j (i.e. i remains o pin down a speci c value of 2 T for each diagonal enry in j ). The scale in he eigendecomposiion is implicily xed because he eigenvecors in L Si jt are condiional disribuions and needs o sum up o one. The quesion of unknown indices is solved because in our model E[A j j T = ] is monoonically decreasing in over T provided he paries follow raional sraegies described in Secion 2. This is again due o he independence beween iming and he beliefs in Assumpion 1 and he moderae compounded discouning in Assumpion 3. This esablishes he ideni caion of j and L Si jt, which are used for recovering L T;Sk and hen he condiional densiy of acceped selemen o ers over is full suppor (see proof of Lemma 1 in Appendix B). 3.2 The join belief disribuion We now explain how o idenify he join disribuion of beliefs ( p ; d ) from he quaniies recovered from Lemma 1 under he following orhogonaliy condiion. 5 Tha is, d m [s m ; s m+1 ] for 1 m M, wih (s m : 2 m M) being a vecor of ordered endpoins on S such ha s 1 < s 2 < :: < s M < s M+1 and s 1 inf S, s M+1 sup S. 10
11 Assumpion 4 The join disribuion of beliefs ( p ; d ) is independen from poenial compensaions C. This condiion requires he magniude of poenial compensaion o be independen from plaini and defendans beliefs. This condiion is plausible because C is mean o capure an objecive moneary measure of he severiy of damage in iced upon he paien. On he oher hand, he beliefs ( p ; d ) should depend on he evidence available as o wheher he defendan s neglec is he main cause of such damage. I hen follows from (2) ha he disribuion of C is direcly ideni ed as: Pr(C c) = Pr(C c j A = 0; D = 1). (7) Le S [0; c ] denoe he condiional suppor of acceped selemen o ers S = T p C given A = 1 and T = ; 6 and le ' (s) denoe he probabiliy ha a selemen is reached when he lengh of waiime beween he selemen conference and he dae for cour hearing is and ha he acceped selemen o er is no greaer han s. Tha is, for all (s; ), ' (s) Pr (S s; A = 1 j T = ) = Pr p C s= ; d + p 1= (8) where he equaliy is due o Assumpion 1. The nonnegaiviy of C and ( p ; d ) and an applicaion of he law of oal probabiliy on he righhand side of (8) implies: ' (s) = Z c 0 1 Pr c p s 1 ; f(c)dc = d + p Z c 0 h (c=s) f C (c)dc (9) where f C (c) is he densiy of C and h (v) Prfp 1 v ; ( d + p ) 1 g; and he rs equaliy is due o orhogonaliy beween C and ( p ; d ). Changing variables beween C and V C=S for any xed and s, we can wrie (9) as: ' (s) = Z 1 0 h (v)(v; s)dv (10) where (v; s) sf C (vs)1fv c=sg. Wih he disribuion (and hence densiy) of C recovered from (7), he kernel funcion (v; s) is considered known for all (v; s) hereinafer for ideni caion purposes. Also noe for any s > 0, (:; s) is a wellde ned condiional densiy wih suppor [0; c=s]. 7 Le F V ja=1;t = denoe he disribuion of V given T = and A = 1 (or equivalenly d + p T ), whose suppor is denoed as V. Assumpion 5 For any and g(:) such ha E[g(V ) j A = 1; T = ] < 1, he saemen R 1 0 g(v)(v; s) = 0 for all s 2 S implies he saemen g(v) = 0 a.e. F V ja=1;t =. 6 In general, we could also allow suppors S;T and S o depend on observed heerogeneiies of lawsuis as well. Noneheless, hroughou his secion, we refrain from such generalizaion in order o simplify exposiion. 7 This is because (v; s) > 0 for any v 0, s > 0. Besides R 1 (v; s)dv = R c=s sf 0 0 C (vs)dv = 1 for any s. 11
12 This condiion, known as he compleeness of kernels in inegral operaors, was inroduced in Lehmann (1986) and used in Newey and Powell (2003) for ideni caion of nonparameric regressions wih insrumenal variables. Andrews (2011) and Hu and Shiu (2012) derived su cien condiions for various versions of such compleeness condiions when g(:) is resriced o belong o di erence classes. This condiion is analogous o a fullrank condiion on if he condiional suppors of S and V were nie. 8 Proposiion 1 Under Assumpions 15, Pr( p ; p + d ) is ideni ed for all 2 (0; 1] and 2 T. For he res of his secion, we discuss how o generalize resuls above when T is in nie (T is coninuously disribued over a known inerval). Firs o, he key idea of using eigendecomposiions in Secion 3.1 remains applicable, excep ha L Si jt and L T;Sk become linear inegral operaors, and heir inveribiliy needs o be saed as an assumpion as opposed o being derived from resricions on model primiives and implicaions of raional sraegies (as is he case when T is discree). Under he suppor condiion ha inff : 2 T g 1=2, he eigenvalues in he decomposiion E[A j j T = ] remains sricly monoonic over he inerval suppor T when T is coninuously disribued. On he oher hand, he argumen ha uses monooniciy of he eigenvalues over a nie suppor T o index hem is no longer applicable when T is coninuously disribued. However, raional sraegies in our model imply he supremum of he suppor of acceped selemen o ers given T = mus be c. Wih he supremum of he suppor of compensaions c ideni ed and known, his means can be expressed hrough a known funcional of he eigenvecors f Si (: j A i = 1; T = ) in he eigendecomposiion ideni ed in he rs sep. Thus he issue wih indexing eigenvalues is also solved. The remaining sep of idenifying he join disribuion of ( p ; p + d ) from f S (: j A = 1; T = ) and E[A j T = ] follow from he same argumen above. I is worhy of noe ha an addiional sep based on Jacobian ransformaion leads o ideni caion of he join disribuion of ( p ; d ) when T is coninuously disribued. 8 If he suppor of poenial compensaion is unbounded, here are pleny of examples of parameric families of densiies ha saisfy he compleeness condiions. For example, suppose poenial compensaions follow a Gamma disribuion wih parameers ; > 0. Tha is, f C () = () 1 expf g. Then, wih s > 0, he kernel (v; s) sf C (vs) = [s] () v 1 expf v (s)g is a densiy of a Gamma disribuion wih a shape parameer > 0 and a scale parameer s > 0. Tha is, (v; s) remains a condiional densiy wihin he exponenial family, and sais es he su cien condiions for he compleeness condiion in Theorem 2.2 in Newey and Powell (2003). 12
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