Bargaining with Optimism: A Structural Analysis of Medical Malpractice Litigation. Abstract
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- Arabella Carpenter
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1 Bargaining wih Opimism: A Srucural Analysis of Medical Malpracice Liigaion Anonio Merlo and Xun Tang 1 Deparmen of Economics Rice Universiy Augus 3, 2015 Absrac We sudy ideni caion and esimaion of a srucural model of bargaining wih opimism where players have heerogeneous beliefs abou he nal resoluion of a dispue if hey fail o reach an agreemen. We show he disribuion of he players beliefs and he sochasic bargaining surplus are nonparamerically ideni ed from he probabiliy of reaching an agreemen and he disribuion of ransfers in he nal resoluion of he dispue. We use a Maximum Simulaed Likelihood approach o esimae he beliefs of docors and paiens in medical malpracice dispues in Florida during he 1980s and 1990s. We nd srong evidence ha beliefs for boh paries vary wih he severiy of he injury and he quali caion of he docor named in he lawsui, even hough hese characerisics are saisically insigni can in explaining wheher he cour rules in favor of he plaini or of he defendan. We also quanify he reducion in selemen amouns ha would resul from he inroducion of a (counerfacual) policy ha imposes caps on he oal compensaion for plaini s. Key words: Bargaining, opimism, nonparameric ideni caion, medical malpracice liigaion 1 We hank seminar paricipans a Brown, HKUST, Ohio Sae, SUFE (Shanghai), Tsinghua and aendans of Economeric Sociey Norh America Summer Meeing 2014 and Cowles Foundaion Conference 2015 for useful feedbacks. This research is funded by NSF Gran # SES We hank Devin Reily and Michelle Tyler for capable research assisance. 1
2 1 Inroducion Excessive opimism is ofen invoked as a possible explanaion for why paries involved in a negoiaion someimes fail o reach an agreemen even hough a compromise could be muually bene cial. For example, consider a medical malpracice dispue where a paien (he plaini ) su ered a damage allegedly caused by a docor s (he defendan) negligence or wrongdoing. If he plaini and he defendan are boh overly opimisic abou heir chances of geing a favorable jury verdic, here may no be any selemen ha can saisfy boh paries exaggeraed expecaions. The general argumen daes back o Hicks (1932), and was laer developed by Shavell (1982), among ohers, in he conex of legal dispues. A recen heoreical lieraure originaed by he work of Yildiz (2003, 2004), exends his insigh and sudies a general class of bargaining models wih opimism (see Yildiz (2011) for a survey). These models have also been used in a variey of empirical applicaions ha range from prerial negoiaions in medical malpracice lawsuis (Waanabe (2006)), o negoiaions abou marke condiions (Thanassoulis (2010)), and cross-license agreemens (Galasso (2012)). Despie he recen surge of ineres in he heory and applicaion of bargaining wih opimism, none of he exising conribuions formally addresses he issue of ideni caion in his class of models. Tha is, under wha condiions can he srucural elemens of he model be unambiguously recovered from he hisory of bargaining oucomes repored in he daa? This is he main quesion we address in his paper. We inroduce an empirical framework for srucural esimaion of bargaining models wih opimism, and show ha he model elemens are nonparamerically ideni ed. We consider a bilaeral bargaining environmen where players are opimisic abou he probabiliy ha a sochasic oucome favors hem if hey fail o reach an agreemen. The players have a one-ime opporuniy for reaching an agreemen a an exogenously scheduled dae during he bargaining process, and make decisions abou wheher or no o sele and, if so, he amoun of he selemen based on heir beliefs and ime discoun facors. We show ha all srucural elemens in he model are ideni ed nonparamerically from he probabiliy of reaching an agreemen and he disribuion of ransfers in he nal resoluion of he dispue. The ideni caion sraegy does no rely on any paramerizaion of he srucural primiives such as players beliefs or he bargaining surplus disribuion. We hen propose a Maximum Simulaed Likelihood (MSL) esimaor based on exible paramerizaion of he join beliefs of he players, and esimae he model using daa on medical malpracice dispues in he Sae of Florida during he period Sieg (2000) and Waanabe (2006) also use he same source of daa for heir empirical analyses of medical malpracice liigaion. Sieg (2000) esimaes a bargaining model wih one-sided incomplee informaion; Waanabe (2006) a bargaining model wih opimism and learning. Neiher sudy addresses he issue of ideni caion, which is he main focus of our paper. 2
3 The bargaining environmen we consider is simpler han he one sudied by Yildiz (2004). Raher han allowing for muliple rounds of o ers and counero ers, in our model here is a single selemen opporuniy for he players o reach an agreemen. Hence, in our bargaining environmen here are no dynamic learning consideraions in he players decisions, and he daes of he nal resoluion of he bargaining episodes are solely deermined by he players opimism, heir paience, and heir percepion of he surplus available for sharing. 3 There are boh heoreical and empirical reasons ha moivaed our choice of he bargaining environmen. Daa limiaions would preven us from deriving robus (paramerizaionfree) argumens for he ideni caion of srucural elemens in general models of bargaining wih opimism ha admi muliple rounds of o ers and counero ers. For insance, none of he daa ses which are used in empirical applicaions of bargaining models wih opimism conains informaion on he sequence of proposers in a negoiaion or he iming and size of rejeced o ers. By absracing from he dynamic learning aspecs ha would be inroduced ino he heoreical analysis if we were o consider a more general bargaining environmen wih muliple rounds of (unobserved) o ers and counero ers, we ake a pragmaic approach and specify a model ha is ideni able under realisic daa requiremens and mild economeric assumpions. A he same ime, despie his simpli caion, our model capures he key insigh of bargaining wih opimism in ha he iming of agreemen is deermined by he players opimism and heir paience. Thus, our work represens a rs imporan sep oward addressing he issue of nonparameric ideni caion in more general models of bargaining wih opimism. 4 Our modeling choice is also moivaed by he speci c empirical applicaion we consider here, which analyzes medical malpracice dispues in Florida. The law of he Sae of Florida (Florida Saues, Tile XLV, Chaper 766, Secion 108), requires ha a one-ime, mandaory selemen conference beween he plaini and he defendan be held a leas hree weeks before he dae se for rial. The selemen conference is scheduled by he couny cour, is held before he cour, and is mediaed by cour-designaed legal professionals. Our ideni caion sraegy builds on he following insighs. Firs, if he lengh of ime beween he scheduled selemen conference and he rial (henceforh, he wai-ime ) 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 wai-ime. Second, he 3 Sieg (2000) also considers a bargaining environmen where here is a one-ime opporuniy for he players o sele ou of cour. Raher han sudying a model of bargaining wih opimism, Sieg (2000) assumes ha he defendan has an informaional advanage over he plaini. In oher words, his analyasis of medical malpracice dispues is based on a bargaining model wih one-sided incomplee informaion, where he defendan knows he acual probabiliy of a vefrdic in his/her favor, while he plaini does no. 4 Waanabe (2006) sudies medical malpracice dispues in he conex of a dynamic model of bargaining wih opimism and learning. As we already menioned, however, his analysis is fully parameric and does no address he issue of ideni caion. 3
4 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 of he plaini and he defendan and he cour decision. Third, because he acceped selemen o er re ecs a plaini s imediscouned expecaion of his or her share of he oal surplus, we can idenify he condiional disribuion of he plaini s belief given ha here is a selemen using he disribuion of acceped selemen o ers given he lengh of he wai-ime. This is done hrough a deconvoluion argumen using he disribuion of surplus recovered above. Finally, 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. A key challenge for implemening his ideni caion sraegy in our empirical applicaion is ha he wai-ime is no repored in he daa. In order o solve he issue of unobserved wai-ime, we ap ino a recen lieraure ha uses eigenvalue decomposiion o idenify nie mixure models or srucural models wih unobserved heerogeneiy (see, for example, Hall and Zhou (2003), Hu (2008), Hu and Schennach (2008), Kasahara and Shimosu (2009), An, Hu and Shum (2010) and Hu, McAdams and Shum (2013)). In paricular, we rs exploi he insiuional deails of our environmen o group lawsuis ino smaller clusers de ned by he couny and he monh in which a lawsui is led. We argue ha he lawsuis wihin each cluser can be plausibly assumed o share he same, albei unobserved wai-ime beween he mandaory selemen conference and he rial. We hen use he cases in he same cluser as insrumens for each oher and apply 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 wai-ime. Then, he argumens in he previous paragraph apply o idenify he join disribuion of beliefs. 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 plaini s and he defendans vary wih observed characerisics of he lawsuis such as he severiy of he injury arising ou of medical malpracice and he professional quali caion of he docor named in he lawsui. On he oher hand, we nd ha he observed case characerisics are saisically insigni can in explaining he cour and jury decisions. We use our esimaed model o assess he e ecs of a counerfacual or reform which limis he liabiliy of defendans by imposing caps on he oal compensaion received by plaini s. For each level of severiy of he injury arising ou of malpracice, we se a cap equal o he 75h percenile of he oal compensaion paid by he defendan following a jury verdic observed in he daa. Our calculaions show hese caps can induce sizeable reducions in he average selemen amouns. Speci cally, he reducions are 33%, 45% 4
5 and 24% for low, medium and high severiy cases, respecively. There is also clear evidence ha he impac of hese caps varies wih he quali caion of he defendan: docors who are board ceri ed would bene from a signi canly more sizable reducion in he average selemen amouns hey would have o pay o he plaini s han heir non-board-ceri ed counerpars in medium- and high-severiy cases. The res of he paper is organized as follows. Secion 2 inroduces he model of bilaeral bargaining wih opimism. Secion 3 esablishes ideni caion of he srucural elemens in he model. Secion 4 presens he Maximum Simulaed Likelihood (MSL) esimaor. Secion 5 describes he daa and he insiuional deails of he empirical applicaion which focuses on medical malpracice lawsuis in Florida. Secions 6 and 7 presen and discuss our esimaion resuls and policy analysis, respecively. We conclude wih Secion 8. All he proofs and a mone carlo sudy are conained in he appendices. 2 The Model Consider a lawsui following an alleged insance of medical malpracice involving a plaini (he paien) and a defendan (he docor). The oal amoun of poenial compensaion relaed o he injury arising ou of he malpracice, C, is assumed o be common knowledge among he plaini and he defendan. This amoun can be inerpreed as a sunk cos for he defendan, which he defendan may or may no be able o recover, in par or in oal, depending on he nal oucome of he legal dispue. Afer he ling of he lawsui, he plaini and he defendan are noi ed of a dae for a one-ime selemen conference, which is mandaory by sae law, according o Tile XLV, Chaper 766, Secion 108 of he Florida Saues. The conference, which is held before he cour, requires aendance by boh paries (and heir aorneys), as well as legal professionals designaed by he couny cour where he lawsui is led. This conference mus ake place a leas hree weeks before he dae se for rial. 5 During he selemen conference, he defendan has he opporuniy o make a selemen o er of S o he plaini. If he plaini acceps i, hen he case is seled ou of cour wih he plaini receiving S and he defendan reclaiming C S. Oherwise, he case is aken o cour and decided by a jury. Boh he defendan and he plaini are aware ha he rial mus ake place a leas hree weeks afer he selemen conference and are informed of he exac dae of he rial, which is deermined by he cour schedule and he backlogs of he couny cour judges. 6 Le T denoe he lengh of ime beween he selemen conference 5 The curren Florida Saues peraining o medical malpracice and relaed maers are available online a hp:// www. senae. gov/ Laws/ Saues/ 2014/ Chaper766 6 Cases are assigned randomly among all he couny cour judges in he couny cour where he suis are led depending on heir availabiliy. 5
6 and he rial (henceforh, he wai-ime ). Le A 1 denoe he even ha a selemen is reached a he conference, and A 0 ha he case goes o rial. In he laer case, a he end of he rial, he jury makes a binary decision, D, and rules eiher in favor of he plaini, D = 1, in which case he plaini is awarded he compensaion C, or in favor of he defendan, D = 0, in which case he defendan is no awarded any compensaion. 7 The plaini and he defendan have heerogeneous beliefs abou he probabiliy ha he jury would rule in heir favor in he even ha he case goes o rial. We le p ; d 2 [0; 1], denoe he subjecive probabiliy of winning he rial believed by he plaini and he defendan, respecively. Excessive opimism arises from he assumpion ha he join suppor of ( p ; d ) is f(; 0 ) 2 (0; 1] 2 : 1 < + 0 2g. The realized value of ( p ; d ) is common knowledge beween boh paries paricipaing in he selemen conference. We also 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 densiy over. (ii) Condiional on A = 0, he jury decision D is orhogonal o C and T. Assumpion 1 allows he beliefs of he plaini and he defendan o be correlaed and be asymmeric wih di eren marginal disribuions. This is empirically relevan because he marginal disribuion of beliefs may very well di er beween paiens and docors due o facors such as informaional asymmeries (e.g., docors may be beer informed abou he cause and severiy of he damage arising ou of medical malpracice) or unobserved individual heerogeneiies. The beliefs of he plaini and he defendan are also likely o be correlaed hrough unobserved heerogeneiy a he case level. For example, he docor and he paien may boh know aspecs relaed o he cause and severiy of he damage arising ou of medical malpracice ha are no recorded in daa. Such aspecs lead o correlaions beween paiens and plaini s beliefs from he perspecive of he economerician. Assumpion 1 also accommodaes correlaion beween ( p ; d ) and C. The independence beween he wai ime T and he beliefs of he paries in he lawsui is plausible because he wai-ime T is mosly deermined by he availabiliy of judges and juries in he couny cour where he sui is led. This, in urn, depends on he cour schedule and he backlogs of he couny cour judges, which are idiosyncraic and orhogonal o he paries beliefs ( p ; d ). The orhogonaliy of C from D given T and A = 0 in condiion (ii) is also plausible. On he one hand, C is a moneary measure of he magniude of he damage su ered by he plaini regardless of is cause; on he oher hand, D capures he jury s judgemen abou 7 We absrac here from he legal coss associaed wih he lawsui. 6
7 he cause of he damage based on he evidence presened a rial. I is likely ha he jury decision is correlaed wih speci c feaures of he lawsui ha are repored in he daa and ha may also a ec he beliefs of boh paries. Neverheless, once we condiion on such observable feaures, jury decisions are likely o be orhogonal o he 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 he daa. We now characerize he Nash equilibrium a he selemen conference. The plaini acceps an o er if and only if S T p C, where is a consan ime discoun facor xed hroughou he daa-generaing process. The defendan o ers he plaini S = T p C if and only if C S T d C. Hence, in equilibrium a selemen occurs during he conference if and only if: C T p C T d C which is equivalen o d + p T. I hen follows ha he disribuion of selemen amouns, condiional on he wai-ime beween he selemen conference and he rial being T =, is: Pr (S s j A = 1; T = ) = Pr p C s j d + p, (1) where he lower case leers denoe realized values for random variables, and he equaliy follows from par (i) in Assumpion 1. Besides he disribuion of he poenial compensaion, condiional on here no being a selemen in a conference T = periods ahead of he rial 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) where he equaliy follows from boh condiions in Assumpion 1. The daa we use in his paper repors characerisics of plaini s and defendans, such as he professional quali caion (board ceri caion) of he defendan and he age of he plaini. I also repors he severiy of he injury arising ou of medical malpracice. These variables, denoed by a vecor X, are correlaed wih he poenial compensaion C and he beliefs ( p ; d ). The model described above can incorporae such observed case-heerogeneiy by leing he primiives (i.e., he disribuion of ( p ; d ), compensaions C, jury decisions D, and he wai-ime T ) depend on X. If boh resricions in Assumpion 1 hold condiional on X, hen he srucural links beween he daa and he model elemens are characerized in he same way as above, excep ha all disribuions need o be condiioned on X. Since hese characerisics ha vary across lawsuis are repored in he daa, our ideni caion argumen in Secion 3 should be inerpreed as condiional on X. We suppress dependence of he srucural elemens on X only for he sake of noaional simpliciy. We only make he dependence explici when needed. 7
8 3 Ideni caion This secion shows how o recover he disribuion of boh paries beliefs from he probabiliy of 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 each case seled a he conference, he daa repors he amoun paid by he defendan o he plaini (S). For each of he oher cases ha were aken o he cour, he daa repors he jury decision (D) and, if he cour rules in favor of he plaini, he amoun of oal compensaion paid by he defendan (C). However, he exac daes of selemen conferences and he scheduled cour hearings (if necessary) are never repored in he daa. 8 Thus, he wai-ime T beween selemen conferences and scheduled cour hearings, which is known o boh paries a he ime of he conference, is no available in daa. To address his issue wih unrepored wai-ime, we propose sequenial argumens ha exploi an implici panel srucure of he daa. In paricular, we noe ha lawsuis led wih he same couny cour during he same period (week) pracically share he same wai-ime 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 couny cour, and are assigned on a rs come, rs served basis. Thus, he selemen conferences for he cases led wih he same couny cour a he same ime are pracically scheduled for he same period. Besides, he daes for poenial cour hearing are deermined by he schedule and backlog of judges a he couny cour. Hence, he 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 wai-ime, despie unobservabiliy of T in daa. We formalize his implici panel srucure as follows. Assumpion 2 The daa is pariioned ino known clusers, each of which consiss of muliple (poenially more han hree) cases ha share he same wai-ime T. Across he cases wihin a cluser, ( p ; d ; C) and 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 8 For example, he daa we use in Secion 5 repors he dae of nal disposiion for each case. However, for a case seled ouside he cour, his dae is de ned no as he exac dae of he selemen conference, bu as he day when o cial adminisraive paperwork is nished and he claim is declared closed by he insurer. 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 were aken o he cour in ha he repored daes of nal disposiion do no correspond o he acual daes of cour hearings. 8
9 in he lawsuis wihin he same cluser as insrumens for each oher, and apply eigendecomposiion-based argumens in Hu (2008) and Hu and Schennach (2008) o recover he join selemen probabiliy and he disribuion of acceped selemen o ers condiional on he unobserved T. We hen use hese quaniies o back ou he join disribuion of beliefs using 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. We mainain ha here is posiive probabiliy ha a cluser conains a leas hree cases. 3.1 Condiional disribuion of selemen o ers We rs recover he selemen probabiliy and he disribuion of selemen o ers given he wai-ime T before cour hearings. Le S; T denoe he uncondiional suppor 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 [0; c]. Tha he suppor T is known is empirically relevan in our seing. Wihou loss of generaliy, denoe he 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 one-period discoun facor is so low ha he compounded discoun facor is less han one half. Condiion (i), ogeher wih he non-increasingness of E[A i j T = ] over 2 T under Assumpion 1, pin down he index for eigenvalues and eigenvecors in he aforemenioned decomposiion. Condiion (ii) is a mild resricion on he suppor of poenial compensaion. I is implied if C is orhogonal from ( p ; d ) wih a bounded suppor. 9 The role of (ii) will become clear as we discuss he ideni caion resul below. 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 2 T and s 2 S. This inermediae resul uses argumens similar o ha in Hu, McAdams and Shum (2013) for idenifying rs-price sealed-bid aucions wih non-separable aucion heerogeneiies. I explois he condiional independence of beliefs across lawsuis wihin a cluser in Assumpion 2. These condiions allow us o break down he join disribuion of he incidence of 9 I is worh noing ha our ideni caion argumen remains valid even wih c being unbounded, as long as he full-rank condiion in Lemma B1 in Appendix B holds for some pariion of S. 9
10 selemen and he size of acceped 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 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 wai-ime 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 equaliy follows from Assumpion 1, he fac ha S = T p C whenever A = 1 and A = 1 if and only if p + d T and ha ( p ; d ; C) are independen draws across he lawsuis i; j; k under 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 M nondegenerae inervals, each of which is denoed by d m. 10 For a given pariion D M, le L Si ;S k be a M-by-M marix whose (m; m 0 )-h enry is he probabiliy ha S i 2 d m and S k 2 d m 0 condiional on A i;k = 1 (selemens are reached in he wo cases i, k); and le Si ;S k be a M- by-m 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 M-by-jT j marix wih (m; )-h enry being Pr(S i 2 d m j A i = 1; T = ); j be a jt j-by-jt j diagonal marix wih diagonal enries being [E(A j j T = )] =1;:;jT j ; and L T;Sk be a jt j-by-m 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 non-singular (see Lemma B1 in Appendix B for deails). Then (4) and (5) imply Si ;S k (L Si ;S k ) 1 = L Si jt j L Si jt 1 (6) 10 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 where he lef-hand side consiss of direcly ideni able quaniies. The righ-hand side of (6) akes he form of an eigen-decomposiion 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 nd ou he speci c value of 2 T ha corresponds o each diagonal enry in j ). The scale in he eigen-decomposiion 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 S (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. Assumpion 4 The join disribuion of ( p ; d ) is independen from 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 = ; and le ' (s) denoe he probabiliy ha a selemen is reached when he wai-ime beween he selemen conference and he rial is and ha he acceped selemen o er is no greaer han s. 11 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 non-negaiviy of C and ( p ; d ) and an applicaion of he law of oal probabiliy on he righ-hand side of (8) implies: Z c Z c 1 ' (s) = Pr c p s 1 ; f d + C (c)dc = h (c=s) f C (c)dc (9) p 0 11 In general, we could also allow S, T and S o depend on observed heerogeneiy of lawsuis. Throughou his secion, we refrain from such generalizaion in order o simplify exposiion. 11 0
12 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). 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 well-de ned condiional densiy wih suppor [0; c=s]. 12 Le F V ja=1;t = denoe he disribuion of V given T = and A = 1, 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 =. 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 full-rank condiion on if he condiional suppors of S and V were nie. 13 Proposiion 1 Under Assumpions 1-5, 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 12 This is because (v; s) > 0 for any v 0, s > 0, and ha R 1 (v; s)dv = R c=s sf 0 0 C (vs)dv = 1 for any s. 13 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
13 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, in our model 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 eigen-decomposiion ideni ed in he rs sep. Thus he issue wih indexing eigenvalues is also solved. The remaining sep of idenifying he join disribuion of 1= p and 1=( p + d ) from f S (: j A = 1; T = ) and E[A j T = ] follow from he same argumen as in he discree case. An addiional sep based on Jacobian ransformaion leads o ideni caion of he join disribuion of ( p ; d ) when T is coninuously disribued. 4 Maximum Simulaed Likelihood Esimaion A nonparameric esimaor based on he ideni caion resul in Secion 3 would require a large daa se, and he curse of dimensionaliy aggravaes if he daa also repor caselevel variables ha may a ec boh paries beliefs (such as he severiy of he injury arising ou of medical malpracice and he quali caion of he defendan) and herefore should be condiioned on in esimaion. To deal wih case-heerogeneiy in moderae-sized daa, we propose in his secion a Maximum Simulaed Likelihood (MSL) esimaor based on a exible paramerizaion of he join belief disribuion. Consider a daa conaining N clusers. A cluser is indexed by n and consiss of m n 1 cases, each of which is indexed by i = 1; :::; m n. For each case i in cluser n, le A n;i = 1 when here is an agreemen for selemen ouside he cour and A n;i = 0 oherwise. De ne Z n;i S n;i if A n;i = 1; Z n;i C n;i if A n;i = 0 and D n;i = 1; and Z n;i 0 oherwise. Le T n denoe he wai-ime beween he selemen conference and he scheduled dae for cour decisions, which is shared by all cases in cluser n. We propose an MSL esimaor for he join disribuion of ( p ; d ) ha also use variaion in he heerogeneiy of lawsuis repored in he daa. Throughou his secion, we assume he idenifying condiions hold once condiional on such observed heerogeneiy of he lawsuis. Le x n;i denoe he vecor of case-level variables repored in he daa ha a ecs he disribuion of C. (We allow x n;i o conain a consan in he esimaion below.) The oal poenial compensaion C in a lawsui wih observed feaures x n;i is drawn from an exponenial disribuion wih he rae parameer given by: (x n;i ; ) expfx n;i g for some unknown consan vecor of parameers. In he rs sep, we pool all observaions 13
14 where he jury is observed o rule in favor of he plaini o esimae : P ^ arg max n;i d n;i(1 a n;i ) [x n;i expfx n;i gc n;i ]. Nex, le w n;i denoe he vecor of case-level variables in he daa ha a ecs he join belief disribuion. (The wo vecors x n;i and w n;i are allowed o have overlapping elemens.) We suppress he subscrips n; i for simpliciy when here is no ambiguiy. In he second sep, we esimae he belief disribuion condiional on such a vecor of case-level variables W using ^ above as an inpu in he likelihood. To do so, we adop a exible paramerizaion of he join disribuion of ( p ; d ) condiional on W as follows. For each realized w, le (Y 1 ; Y 2 ; 1 Y 1 Y 2 ) be drawn from a Dirichle disribuion wih concenraion parameers j expfw j g for j = 1; 2; 3 for some consan vecor ( 1 ; 2 ; 3 ). In wha follows, we suppress he dependence of j on w o simplify he noaion. Le p = 1 Y 1 and d = Y 1 + Y 2. The suppor of ( p ; d ) is f(; 0 ) 2 [0; 1] 2 : g, which is consisen wih our model wih opimism. (Table C1 in Appendix C shows how exible such a speci caion of he join disribuion of ( p ; d ) is in erms of he range of momens and he locaion of he model i allows.) Also noe Y 2 = p + d 1 by consrucion, so i is a measure of opimism. Under his speci caion, he marginal disribuion of Y 1 condiional on W = w is Bea( 1 ; ), where of course j s are funcions of w. The condiional disribuion Y 2 j Y 1 = ; W = w is he same as he disribuion of (1 )Bea( 2 ; 3 ). For any y and 2 (0; 1), we can wrie: Y2 PrfY 2 y j Y 1 = ; W = wg = Pr 1 y 1 Y 1 = ; W = w where he righ-hand side is he c.d.f. of a Bea( 2 ; 3 ) evaluaed a y=(1 Le q n;i Pr(D n;i = 1 j A n;i = 0; W n;i = w n;i ). Recall ha we mainain D is orhogonal o (T; C) condiional on A = 0 and W. Hence q n;i does no depend on c n;i. This condiional probabiliy is direcly ideni able from he daa. Le h n (:; ) denoe densiy of he wai-ime T n in cluser n. This densiy in general depends on cluser-level variables repored in he daa, and is speci ed up o an unknown vecor of parameers. The log-likelihood of our model is: L N (; ; ) P N n=1 ln P 2T h n(; ) Q m n i=1 f n;i(; ; ) where f n;i (; ; ) is shorhand for he condiional densiy of Z n;i ; A n;i ; D n;i given T n =, W n;i = w n;i and wih parameer, evaluaed a (z n;i ; a n;i ; d n;i ). Speci cally, f n;i (; ; ) [g 1;n;i (; ; )] a n;i fg 0;n;i (; ) [1 p n;i (; )] q n;i g (1 a n;i)d n;i f[1 p n;i (; )] (1 q n;i )g (1 a n;i)(1 d n;i ) ). 14
15 where p n;i (; ) Pr(A n;i = 1 j T n = ; W n;i = w n;i ; ) = Pr( p;n;i + d;n;i j w n;i ; ) = Pr(Y 2 1 = j w n;i ; ); g 0;n;i (; ) g 0 (z n;i ; x n;i ; ; Pr(C n;izja n;i =0;T n=;x n;i =x n;i = f C (z n;i j x n;i ; ); (11) Z=zn;i wih f C (: j x n;i ; ) being he condiional densiy of he poenial compensaion given X n;i = x n;i ; and g 1;n;i (; ; ) g 1 (z n;i ; w n;i ; x n;i ; ; ; Pr(S n;iz;a n;i =1jT n=;w n;i ;x n;i Z 0 Pr Y 1 1 Z=(c ); Y 2 1 wn;i ; Z=zn;i f C (c j x n;i ; )dc (12). Z=zn;i In he derivaion above, we have used he condiional independence beween C n;i and D n;i, T n, ( p;n;i ; d;n;i ) condiional on W n;i ; X n;i. Under regulariy condiions ha allow for he change of he order of inegraion and di ereniaion in (12), g 1;n;i (; ; ) equals: Z 1 z n;i n Pr Y 2 1 Y1 = 1 z n;i =c; w n;i ; o f Y1 (1 z n;i =c j w n;i ; ) f C(c j x n;i ; ) c dc where he lower limi is z n;i because he inegrand is nonzero only when 1 z n;i =c 2 (0; 1), c 2 ( z n;i ; +1).Changing variables beween c and 1 z n;i =c for any i; n and xed, we can wrie g 1;n;i (; ; ) as: Z 1 0 Y2 Pr 1 1 (1 ) Y 1 = ; w n;i ; fy1 ( j w n;i ; )f C zn;i (1 (1 ) j x ) n;i; d 1 where he rs condiional probabiliy in he inegrand is a Bea c.d.f. evaluaed a (1 ) and parameers ( 2 (w n;i ; 2 ); 3 (w n;i ; 3 )) and he second erm f Y1 ( j w n;i ; ) is he Bea p.d.f. wih parameers ( 1 (w n;i ; 1 ); 2 (w n;i ; 2 ) + 3 (w n;i ; 3 )). For each n, i, and a xed vecor of parameers (; ), le ^g 1;n;i (; ; ) be an esimaor for g 1;n;i (; ; ) using S > N simulaed draws of. (We experimen wih various forms of densiy for simulaed draws.) I follows from he Law of Large Numbers ha ^g 1;n;i (; ; ) is an unbiased esimaor for each n; i and (; ). sep is Our Maximum Simulaed Likelihood esimaor for he belief parameer in he second (^; ^) arg max ; ^L N (; ; ^). (13) 15
16 where ^L N (; ; ) is an esimaor for L N (; ; ) by replacing g 1;n;i (; ; ) wih ^g 1;n;i (; ; ) and replacing q n;i wih a parameric (logi or probi) esimae ^q n;i ; and ^ is he esimaes for he parameers in he disribuion of poenial compensaion in he rs sep. Under appropriae regulariy condiions, (^; ^) converge a a p N-rae o a zero-mean mulivariae normal disribuion wih some nie covariance as long as N! 1, S! 1 and p N=S! 0. The covariance marix can be consisenly esimaed using he analog principle, which involves he use of simulaed observaions. (See equaion (12.21) in Cameron and Trivedi (2005) for a deailed formula.) 5 Daa Descripion Since 1975 he Sae of Florida has required all medical malpracice insurers o le repors on heir resolved claims o he Florida Deparmen of Financial Services. Using his source, we consruc a sample ha consiss of 13,351 medical malpracice lawsuis led in Florida beween 1984 and Our sample includes hose cases ha are eiher resolved hrough he mandaory selemen conference or by a jury decision following a rial. For each lawsui, he daa repors he dae when he sui is led (Sui_Dae) and he couny cour wih which i is led (Couny_Code), he dae of he nal disposiion (Year_of_Disp) corresponding o he dae when he claim is closed wih he insurer, and wheher he case is resolved by a selemen a he selemen conference or by a jury decision in cour (A = 1 or A = 0). The daa also repors he size of he ransfer from he defendan o he plaini upon he resoluion of he lawsui. This ransfer is equal o he selemen amoun acceped by he plaini (S), if he case is seled ou of cour, or he oal compensaion awarded o he plaini according o he cour decision (C), oherwise. In addiion, we also observe case-level variables ha may be relevan o he join disribuion of beliefs and/or o he disribuion of poenial compensaions. These variables include he severiy of he injury arising ou of medical malpracice (Severiy), he age (Age) of he paien who su ered he injury and wheher he docor named in he lawsui is board-ceri ed (Board_Code), where Board_Code = 1 denoes ha he docor is repored o be ceri ed by a leas one professional board and 0 oherwise. The daes of he selemen conference and of he scheduled jury rial for each lawsui are no repored in he daa, regardless of wheher he case is seled ou of cour or decided by a jury. In fac, he recorded dae of he nal disposiion of a case only repors when he claim is closed wih he insurer, which ypically occurs laer han he acual dae when an agreemen is reached in he selemen conference or when a decision is made by he jury in cour, and includes adminisraive delays which may vary across cases and are no 14 Sieg (2000) and Waanabe (2009) also use he same source of daa for heir empirical analyses of medical malpracice lawsuis. 16
17 direcly measurable. Therefore, he wai-ime beween he selemen conferences and he rials canno be recovered from daa. Despie hese daa limiaions, we de ne clusers wihin which he cases could be reasonably assumed o share he same lengh of wai-ime. I is, in fac, plausible ha he lawsuis led wih he same couny cour in he same monh would be scheduled for cour proceedings in he same monh, since he schedule for hearings in a couny cour is mosly deermined by he backlog of unresolved cases led wih ha cour, and by he availabiliy of judges and oher legal professionals from ha cour. By he same oken, he schedule for selemen conferences, which require he presence of cour o cials who have he auhoriy o faciliae a selemen, are also mosly deermined by he backlog of cases as well as he availabiliy of aorneys represening boh paries. Based on hese consideraions, we mainain ha he lenghs of wai-ime beween selemen conferences and cour hearings for all lawsuis led wih he same couny cour in he same monh are he same. As explained in Secion 3, he disribuion of selemen decisions and acceped o ers in lawsuis from hese clusers are su cien for recovering he join beliefs of plaini s and defendans. The daa consiss of 3,545 clusers de ned by monh-couny pairs. In oal here are 1,344 clusers which repor a leas hree medical malpracice lawsuis. Abou half of hese clusers (661 clusers) conain a leas six cases. Moreover, among hese 1,344 clusers, 1,294 have a leas wo lawsuis ha were seled ou of cour during he mandaory selemen conference. These feaures of he daa con rm ha we can apply our ideni caion sraegy from Secion 3 o recover he join disribuion of paiens and docors beliefs. I is worh menioning ha in our MSL esimaion, he likelihood includes all 3,545 clusers o improve he e ciency of he esimaor, even hough in heory ideni caion only requires he join disribuion of selemen decisions and acceped o ers from he subse of clusers ha have a leas wo selemens ou of hree or more cases. Table 1(a): Selemen probabiliy and acceped o ers Board Cer n Severiy # obs ^p sele s:e:(^p sele ) ^ SjA=1 ($1k) s:e:(^ SjA=1 ) ($1k) ceri ed low 1, medium 2, high 2, unceri ed low 1, medium 2, high 2, Nex, we repor some evidence from he daa ha he beliefs of he plaini s and he defendans are a eced by cerain observed characerisics ha vary across lawsuis. Table 17
18 1(a) summarizes he selemen probabiliy and he average size of acceped selemen o ers in he sample afer conrolling for he docors quali caion and he level of severiy of he injury arising ou of medical malpracice. There is evidence ha boh he selemen probabiliy and he size of acceped selemen o ers di er sysemaically across he subgroups. Table 1(b) repors he p-values of wo-sided -ess (using he unequal variance formula) for he equaliy of selemen probabiliies in sub-groups. We le (u,c) and (l,m,h) be shorhand for he realized values of (unceri ed, ceri ed) in Board_code and (low, medium, high) in Severiy respecively. Wih he excepion of hree pair-wise ess, he nulls in he oher ess are all rejeced a he 2% signi cance level. Among he hree excepions, he null for equal selemen probabiliy beween (u,m) and (u,h) is also rejeced a he 10% level. The only wo cases where he null can no be rejeced even a he 10% signi cance level are (u,l) versus (c,m) and (u,l) versus (c,h). This is consisen wih he inuiion ha a plaini may be relaively more opimisic ha he jury would rule in his or her favor when he injury su ered is relaively more severe, or when he docor s quali caion is no suppored by board ceri caion. Our esimaes in he nex secion are also consisen wih his inuiion. The failure o rejec he null of equal selemen probabiliy beween he wo subgroups (u,l) and (c,h), for example, may be due o he fac ha he impac of severiy and of board ceri caion on he plaini s belief o se each oher. Pairwise -ess for he equaliy of average acceped selemen o ers beween he sub-groups de ned by severiy and docor quali caion also demonsrae similar paerns. Speci cally, he null of equal average selemen o ers is almos always rejeced a he 1% signi cance level for all pair-wise -ess using unequal variances, wih he only excepion being he es comparing (u,l) versus (c,l). Table 1(b): p-values for -ess: selemen probabiliy u,l u,m u,h c,l c,m c,h u,l < < < u,m < < u,h < < < c,l < < c,m c,h The daa also conains some evidence ha he disribuion of oal compensaion may be parly deermined by he age of he plaini and he severiy of he injury. Ou of he oal 2,298 lawsuis which were no resolved hrough selemen, 359 were ruled in favor of he plaini by he cour. The observaions of he realized oal compensaion in hese cases 18
19 are useful for inference on he disribuion of C. Figure 1(a) and 1(b) in Appendix A repor hisograms of he acceped o ers (S) from he cases seled ouside he cour and he oal compensaion (C ) from he cases where he cour ruled in favor of he plaini, condiioning on he informaion abou he plaini s. The variable Age is discreized ino hree caegories: young (Age < 33), older (Age > 54) and middle, wih he wo cuo s being he 33rd and he 66h perceniles in he daa. Figure 1(a) suggess he younger plaini s end o receive higher ransfers eiher hrough acceped o ers in selemen or hrough he oal compensaion paid by he defendan when he cour rules in favor of he plaini. Figure 1(b) shows he cases wih more severe injuries in general are associaed wih higher ransfers. Boh paerns are inuiive, and consisen wih our esimaes presened in he nex secion below. To furher compare he disribuion of acceped selemen o ers wih ha of oal compensaions awarded by he cour, we compare he perceniles of boh variables condiional on Age and Severiy. We nd ha he 10h, 25h, 50h, 75h and 90h condiional perceniles of he acceped selemen o ers are consisenly lower han hose of he oal compensaions awarded by he cour. This is consisen wih he noion ha an acceped selemen o er is equal o he discouned expecaion of he oal compensaion ha could be awarded by he cour. The quali caion of he docors does no seem o have any noiceable e ec on he disribuion of he oal compensaion. Figure 1(c) repors he hisogram of he oal compensaion for he cases where he cour ruled in favor of he plaini, condiioning on he board ceri- caion saus of he docors. A -es for he equaliy of he average compensaion for he wo subgroups wih and wihou board ceri caion repors an asympoic p-value of (assuming unequal populaion variance). Furhermore, a one-sided Komolgorov-Simirnov es agains he alernaive ha he disribuion of C is sochasically lower when he defendan is board-ceri ed yields a es saisic of and an asympoic p-value of Thus, in eiher es he null can no be rejeced even a he 15% signi cance level. On he oher hand, i is reasonable o posulae ha he oal poenial compensaion in a malpracice lawsui is posiively correlaed wih he conemporary income level in he couny where he lawsui is led. In order o conrol for such an income e ec, we collec daa on household income in all counies in Florida beween 1981 and We collec daa on he median household income in each Florida couny in 1989, 93, 95, 97, 98 and 99 from he Small Area Income and Povery Esimaes (SAIPE) produced by he U.S. Census Bureau. 15 We also collec a ime series of sae-wide median household income in Florida each year beween 1984 and 1999 from he U.S. Census Bureau s Curren Populaion Survey. We combine his laer sae-wide informaion wih he couny-level informaion from SAIPE o exrapolae he median household income in each Florida couny in he years , 15 See hp:// 19
20 92, 94 and We hen incorporae his yearly daa on household income in each couny while esimaing he disribuion of oal compensaion nex year. 6 Esimaion Resuls As he rs sep in esimaion, we use a logi regression o he cour decisions in hose lawsuis ha are resolved hrough cour hearings. The goal is o provide some evidence abou wheher he jury decisions were a eced by case characerisics repored in he daa. Moreover, he prediced probabiliy for D = 1 (he jury ruled in favor of he plaini ) from he logi regression will be used in he MSL esimaion of he join beliefs of docors and paiens. Table 2. Logi Esimaes for Cour Decisions (Response Variable: D. Toal # of observaions: 2,289 cases.) (1) (2) (3) Board_Code (0.120) (0.282) (0.288) Severiy ** (0.023) (0.033) (0.056) Age (0.003) (0.003) 0.021* (0.012) SeveriyBoard_Code (0.046) (0.046) Age (0.011) SeveriyAge (0.011) Consan *** (0.189) *** (0.228) *** (0.391) Log likelihood Pseudo-R p-value for L.R.T Noes: Sandard errors are repored in parenheses. *** signi can a 1%; ** signi can a 5%; * signi can a 10%. Age 2 is repored in unis of 100 yr 2. Table 2 repors he logi regression esimaes under di eren speci caions, using 2,289 lawsuis from he daa ha were no seled ouside he cour and hus had o be resolved 16 The exrapolaion is done based on a mild assumpion ha a couny s growh rae relaive o he saewide growh rae remains seady in adjacen years. For example, if he raio beween he growh rae in Couny A beween 1993 and 1995 and he conemporary sae-wide growh rae is, hen we mainain he yearly growh raes in Couny A in (and ) are boh equal o p imes he sae-wide growh raes in (and respecively). Wih he yearly growh rae in Couny A beeen calculaed, we hen exrapolae he median household income in Couny A in 1994 using he daa from he SAIPE source. 20
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