Option Value and Dynamic Programming Model Estimates of Social Security Disability Insurance Application Timing

Transcription

1 DISCUSSION PAPER SERIES IZA DP No. 941 Opion Value and Dynamic Programming Model Eimae of Social Securiy Diabiliy Inurance Applicaion Timing Richard V. Burkhauer J. S. Buler Gulcin Gumu November 2003 Forchunginiu zur Zukunf der Arbei Iniue for he Sudy of Labor

2 Opion Value and Dynamic Programming Model Eimae of Social Securiy Diabiliy Inurance Applicaion Timing Richard V. Burkhauer Cornell Univeriy J. S. Buler Univeriy of Kenucky Gulcin Gumu IZA Bonn Dicuion Paper No. 941 November 2003 IZA P.O. Box 7240 D Bonn Germany Tel.: Fax: Thi Dicuion Paper i iued wihin he framework of IZA reearch area Welfare Sae and Labor Marke. Any opinion expreed here are hoe of he auhor and no hoe of he iniue. Reearch dieminaed by IZA may include view on policy, bu he iniue ielf ake no iniuional policy poiion. The Iniue for he Sudy of Labor IZA in Bonn i a local and virual inernaional reearch cener and a place of communicaion beween cience, poliic and buine. IZA i an independen, nonprofi limied liabiliy company Geellchaf mi bechränker Hafung uppored by Deuche Po World Ne. The cener i aociaed wih he Univeriy of Bonn and offer a imulaing reearch environmen hrough i reearch nework, reearch uppor, and viior and docoral program. IZA engage in i original and inernaionally compeiive reearch in all field of labor economic, ii developmen of policy concep, and iii dieminaion of reearch reul and concep o he inereed public. The curren reearch program deal wih 1 mobiliy and flexibiliy of labor, 2 inernaionalizaion of labor marke, 3 welfare ae and labor marke, 4 labor marke in raniion counrie, 5 he fuure of labor, 6 evaluaion of labor marke policie and projec and 7 general labor economic. IZA Dicuion Paper ofen repreen preliminary work and are circulaed o encourage dicuion. Ciaion of uch a paper hould accoun for i proviional characer. A revied verion may be available on he IZA webie or direcly from he auhor.

3 IZA Dicuion Paper No. 941 November 2003 ABSTRACT Opion Value and Dynamic Programming Model Eimae of Social Securiy Diabiliy Inurance Applicaion Timing Thi paper develop dynamic rucural model - an opion value model and a dynamic programming model - of he Social Securiy Diabiliy Inurance SSDI applicaion iming deciion. We eimae he ime o applicaion from he poin a which a healh condiion fir begin o affec he kind or amoun of work ha a currenly employed peron can do. We ue Healh and Reiremen Sudy HRS and rericed acce Social Securiy earning daa for eimaion. Baed on e of boh in-ample and ou-of-ample predicive accuracy, our opion value model perform beer han boh our dynamic programming model and our reduced form hazard model. JEL Claificaion: H31, H55 Keyword: Social Securiy Diabiliy Inurance, Healh and Reiremen Survey, opion value, dynamic programming Correponding auhor: Gulcin Gumu IZA Bonn P.O. Box Bonn Germany Tel.: gumu@iza.org We hank Ben Jeper Chrienen and oher paricipan a he Conference on Social Inurance and Penion Reearch, in Aarhu, Denmark for heir commen and uggeion. Thi reearch i funded in par by he Unied Sae Deparmen of Educaion, Naional Iniue on Diabiliy and Rehabiliaion Reearch, cooperaive agreemen No

4 Inroducion Rapid growh in he number of Social Securiy Diabiliy Inurance SSDI beneficiarie in he early 1990 ogeher wih a parallel decline in male labor force paricipaion rae produced exenive reearch on he behavioral effec of policy variable on SSDI applicaion 1. Thi empirically baed reearch ha primarily ued reduced form model o e he imporance of he effec of ize and availabiliy of SSDI benefi on worker deciion o leave he labor force and apply for benefi. While uch model are ueful approximaion of he relaionhip beween pa SSDI policie and pa applicaion behavior, fuure policy change may no yield he ame reduced form repone. A beer heoreical approach o pecify how change in SSDI policy will change fuure behavior i o incorporae explicily SSDI incenive wihin a rucural model. In hi paper, we develop and e dynamic rucural model of he iming o SSDI applicaion, once a healh condiion begin o affec he kind or amoun of paid work a currenly employed worker can do. Worker deciion o apply for SSDI can be made a he one of a work limiaion or can be poponed. Hence, SSDI applicaion deciion are inrinically dynamic and ochaic. Following he eminal work by Sock and Wie 1990 and Lumdaine, Sock, and Wie 1992 [LSW, hereafer], we inveigae hi dynamic deciion uing boh opion value and dynamic programming model in addiion o a convenional reduced form hazard model. The opion value approach wa fir formalized by LSW o model reiremen deciion. The opion value model i imilar in piri and rucure o he dynamic programming model. However, i ha ome analyical difference and i compuaionally le inenive. Opion value and dynamic programming model are heoreically more powerful approximaion of individual behavior han are reduced form pecificaion. Reduced form model are mo appropriae for udying he implicaion of policy change when he underlying behavioral rucure remain 1

5 unchanged Luca, In conra, rucural model are be when he propoed policy change i large enough o ignificanly aler incenive and individual behavior. By rucural model, we mean ha he eimaion of underlying behavioral elaiciie i baed on a formal economic model. A priori, we do no differeniae beween he wo rucural model we ue in hi paper, raher we eimae boh and compare hem uing e of predicive performance. Previou Reearch Previou udie have focued on he reponivene of labor upply o SSDI benefi level or o replacemen rae uing reduced form model, e.g. Paron 1980, Haveman and Wolfe 1984, Slade 1984, and Bound Thee udie how ha labor force paricipaion i negaively relaed o SSDI benefi level, bu he magniude of hi relaionhip remain unreolved. Oher reearcher Paron,1991; Bound and Waidmann, 1992; Gruber and Kubik, 1997, have udied he effec of SSDI accepance rae on he decline in male labor force paricipaion rae. They find ha labor force paricipaion rae are negaively relaed o SSDI accepance rae. Halpern and Hauman 1986 employ a rucural eimaion approach o analyze boh hee policy variable. Uing a wo-period model hey find ha SSDI applicaion are more reponive o change in he benefi level han o change in accepance rae. Kreider 1998 analyze he effec of wage and eligibiliy uncerainy on SSDI applicaion deciion. He argue ha uncerainy abou fuure earning increae applicaion probabiliie a individual may apply for benefi in order o avoid labor marke rik. Kreider 1999 ue a rucural model of SSDI applicaion, award, and income o analyze he effec of hi program on male labor force paricipaion wihin a lifeime framework. He find ha increae in he level of SSDI benefi modely reduce male labor force paricipaion rae. Kreider and Riphahn 2000 udy 2

6 he deerminan of SSDI applicaion uing a emi-parameric dicree facor procedure. They ue hi mehod o approximae a dynamic opimizaion model and find ha facor uch a benefi level, pa labor earning, and benefi eligibiliy affec applicaion behavior. They alo find ha men and women have ignifican difference in heir reponivene o policy change. Boh Kreider 1999 and Kreider and Riphahn 2000 udie recognize he imporance of modeling he iming of applicaion bu did no do o. Thee udie meaured applicaion elaiciy over an eigh-year period for a group of healh limied worker a rik. We argue ha hi i a ueful approximaion of one par of he impac of policy change on caeload, bu one mu model he iming of applicaion over he enire lifeime. Ru, Buchinky, and Beniez- Silva 2003 propoed o develop and eimae a dynamic programming model of SSDI program ogeher wih he Old Age and Survivor Inurance OASI, and Supplemenal Securiy Income SSI. In hi paper, we develop dynamic rucural model of he SSDI applicaion deciion ha are adapaion of dynamic rucural model ued o udy reiremen. Ru 1989, Berkovec and Sern 1991, Ru and Phelan 1997, and Heyma forhcoming have all ued dynamic programming model o analyze reiremen deciion. Sock and Wie 1990b, baed on Sock and Wie 1990a, developed an opion value reiremen model ha hey argue i cloe in piri o he dynamic programming rule bu more convenien o eimae. The difference beween he wo approache i ha he opion value model evaluae he fuure a he maximum of he expeced value of uiliy, wherea he dynamic programming model ue, heoreically preferred, he expeced value of he maximum, which i necearily larger. Their opion value model predic he age of reiremen bu undereimae reiremen a age 65. They and oher hen provide evidence of he advanage of opion value reiremen model in a erie of paper. 3

7 LSW find ha opion value and dynamic programming model work equally well in predicing he effec of a window plan, which i a emporary reiremen incenive offered by firm o i employee. Bu hey noe ha he opion value model i eaier o eimae. A probi pecificaion i alo eimaed for comparion purpoe, and hey find ha boh rucural model ouperform probi model. Daula and Moffi 1995 develop a dynamic programming model of army reenlimen wih wo period. They add a vecor of obervable variable ino heir model in order o allow uch variable o reflec valuaion of non-moneary characeriic of applicaion and work ae. They alo ue wo impler-o-compue rucural model, an opion value model and an annualized co of leaving model, and compare heir predicion wih heir dynamic programming model. They conclude ha all model produce imilar predicion in-ample, bu dynamic programming produce more plauible predicion ou-of-ample. Our paper which develop a rucural model of he deciion o apply for SSDI conribue o he lieraure in everal way. The preen work focue on explici modeling of ime o applicaion. Once people ge on he SSDI roll, hey end o ay, o he iming of applicaion i an imporan facor in deermining he SSDI caeload and program co. Uing rucural modeling, we how ha he policy variable are imporan for he raniion ono he SSDI roll following he one of a work-limiing condiion. Mo of he lieraure uing opion value and dynamic programming model focue on reiremen deciion. We eimae and compare opion value and dynamic programming model of SSDI applicaion. A a echnical improvemen, we add o he lieraure by eimaing rucural model of SSDI for up o 16 period LSW include a mo a hree-period analyi. Finally, following Daula and Moffi 1995, we alo include fixed uiliy difference a dicued below. 4

8 Dynamic programming and i alernaive I can be argued ha opion value model more cloely reflec how individual acually behave. If people inuiively make deciion by comparing he conequence of an applicaion hi year wih an applicaion made in one, wo, or five year, hen opion value model mimic ha proce. Furher, opion value model are cloe o dynamic programming model when he relevan choice e a oppoed o he complee choice e i circumcribed. I i, a he very lea, an open queion a o how well opion value model perform in uch cae. Thi i one of he iue we explore in hi paper. Bu he main reaon o inveigae boh opion value and dynamic programming model i ha lile i known abou he ucce of he wo ype of model in deciion oher han reiremen. The fac ha opion value model are more convenien o eimae, and hu migh implify more complex model of governmen program and he imulaion of policy change, i an addiional reaon o conider ha queion. However, one hould noe ha he reul of he comparion we carry ou criically depend on he modeling aumpion we make. In hi paper, our focu i o model he deciion wheher and when o apply for SSDI by worker who experience a work limiing healh condiion. Therefore, we choe o abrac from modeling deciion abou reurning o work, applying for reiremen or oher program uch a SSI, SSDI appeal, aving and labor upply choice. Thu, a more formal model which incorporae all hee apec could provide differen reul of comparion beween he wo model. Of coure imilar argumen regarding convenience can be made wih repec o he value of reduced form model, uch a hazard model, which are even le difficul o eimae, primarily becaue hey are available in andard compuer package. Thee reduced form model can be criicized for being unable o capure he conequence of a changing rucure, bu hey 5

9 offer uch convenience in pracice ha hey hould alo be conidered. The only way o know how reduced form model compee wih rucural model in heir abiliy o predic he behavioral conequence of policy change i o eimae boh reduced form and rucural model of real and complex problem. Thi i anoher iue we explore in hi paper, by comparing he predicive power of reduced form hazard model o our rucural model. How SSDI Work SSDI i a ocial inurance program ha provide benefi baed on previou Social Securiy covered employmen. The program i financed by he Social Securiy payroll axe. In December 2000, SSDI paid 5,042,334 diabled worker an average monhly benefi of $786 U.S. Social Securiy Adminiraion, Here we provide a brief overview of SSDI program rule. A much fuller decripion can be found in he Annual Saiical Supplemen o he Social Securiy Bullein. To be eligible for SSDI benefi worker mu be judged o have a medically deerminable phyical or menal condiion ha ha laed or i expeced o la a lea 12 monh or reul in deah, and ha preven hem from performing any ubanial gainful aciviy SGA. In 2001, earning of more han $740 a monh ordinarily demonraed ha an individual i engaged in SGA The SGA level i auomaically adjued annually baed on increae in he naional average wage index.. They mu alo be in inured au. Fully inured au depend on age a he ime of one and ime in Social Securiy covered employmen. They mu alo mee a ubanial recen work aciviy e. In general, hi e require being in Social Securiy covered employmen for one-half of he quarer over he previou 10 year. 6

10 Succeful applican ar o receive heir monhly benefi Primary Inurance Amoun, PIA following a five monh non-work period. Thi i he auory waiing period before benefi can be received following he one of diabiliy and during hi period applican need o be almo compleely wihdrawn from he labor marke. Beniez-Silva, Buchinky, Chan, Ru, and Sheidvaer 1999 repor ha he rejecion or award proce on average ake abou he ame amoun of ime. PIA are baed on worker covered earning hiory Average Indexed Monhly Earning, AIME. Worker can apply for SSDI a any age prior o age 65. Worker wih a ufficien work hiory will become eligible for acuarially reduced reiremen Old-Age and Survivor Inurance, OASI benefi a age 62 and full OASI benefi a age 65. We conider a ample of individual wih work limiing healh condiion and aume ha everyone in hi ample who chooe no o apply for SSDI prior o age 62 applie for OASI benefi a age SSDI benefi may be erminaed for everal reaon. In ome cae, beneficiarie condiion improve and hey reurn o work. In oher cae, hey are found capable of SGA. However, beneficiarie rarely reurn o work, and when hey do, heir wage are uually lower han hey were before ee Bound, 1989 and Bound, Burkhauer, and Nichol, 2003 for evidence. We aume ha once worker ge SSDI, hey ay on he roll unil hey are auomaically moved o he OASI program a age 65. The deciion o apply for SSDI i far more difficul han i he deciion o apply for OASI. Eligibiliy o boh depend on pa payroll conribuion, however, OASI eligibiliy i baed on age and hu i fairly eay o deermine wherea eligibiliy for SSDI i harder o verify. Applicaion o SSDI i merely he beginning of a muliple ep eligibiliy proce and a proraced appeal proce which can be long and whoe final oucome i uncerain. Thu while a probabiliy of accepance i no required in reiremen model, model of SSDI applicaion mu 7

11 include i becaue applican may be eiher rejeced or acceped. The probabiliy ha an applicaion for SSDI i approved ha varied dramaically over ime and ae ee Burkhauer, Buler, and Weaher, In our model, we ue he rae of approval by ae and year, which varied from 25 percen o 75 percen beween 1974 and Applican who are iniially rejeced for benefi can file appeal a variou level. In heory, our eup i adapable for appeal proce a well. However, appeal are no a random ample and herefore we chooe no o model hem in hi paper a he eimaion would become much more complicaed. The main focu of hi paper i modeling he iming of fir applicaion for SSDI. For he ame reaon, we alo abrac from reurn o work eiher afer being acceped or rejeced. Bound and Burkhauer 1999 repor ha acceped individual rarely reurn o permanen work. Bound 1989 how ha reurn o work i no likely for he rejeced group ince he relaive reward for reurning o work are mall. The Opimal Timing of SSDI Applicaion Our opion value and dynamic programming model follow LSW. We fir pecify he choice and i poenial conequence and hen decribe he uiliy funcion and he diribuion of he ochaic elemen. The Opion Value Model Time i dicree and he horizon i finie. The choice in each period i o coninue o work or o apply for SSDI a long a one i eligible for hee benefi. Thu, an eligible individual can eiher chooe o apply for SSDI, or never apply. The conequence of an applicaion i eiher rejecion or accepance and receip of benefi unil reiremen or deah. If rejeced, one may 8

12 eiher appeal o he deciion, or reurn o work, or no work and live excluively on non-labor income which are no explicily modeled here. A dicued above, we aume everyone who doe no apply for SSDI by age 62, reire a age 62. We will pecifically model he conequence of 1 coninued work no applicaion, 2 applicaion and accepance, and 3 applicaion and rejecion. Le he curren period be. One can apply for SSDI in period r, r. The end period i called d, 62 year of age in our cae. The probabiliy of urviving o period given urvival o period,, i π. If one applie for SSDI, he probabiliy of being approved i α, and π and α are no eimaed in he model. Earning while ill working i W, and expeced income if one applie for SSDI i D. Noe ha, in our model, income in he SSDI accepance ae become OASI income a age 62. Income i Y if one i urned down for SSDI, and income i B if one i acceped. Hereafer, we will no wrie ou all of hee erm bu ue D o and for he weighed average of repecively. Le B and Y where he weigh are α and 1 α, U denoe he uiliy funcion and β denoe he dicoun facor. We follow LSW in pecifying ha uiliy i a funcion of labor earning and income in he SSDI accepance ae. Thi aumpion of forced conumpion i clearly rericive. However, here we focu on he SSDI applicaion deciion and herefore we abrac from borrowing, aving and conumpion moohing iue by auming incomplee marke a in Ru and Phelan In general, include a yemaic predeermined porion and a ochaic, random porion 3. A in LSW, we aume ha income may produce more or le uiliy afer applicaion for SSDI. Applicaion iming deciion depend on everal incenive which coni of facor affecing individual U 9

13 preference for conumpion and leiure, labor earning, SSDI benefi, healh condiion, ociodemographic characeriic, job characeriic and work condiion uch a employer accommodaion following one, and iniuional deail of he SSDI program. Furher dicuion on he relevance of hee variable for he applicaion iming deciion can be found below. In hi conex, a any given period, poponing applicaion may provide higher curren conumpion and higher fuure poenial benefi due o coninued labor marke aciviy, bu may alo lead o lower curren leiure conumpion and higher dicomfor from work. Uiliy in he pre-applicaion ae i: U W W = W γ + ω, 1 and uiliy in he po-applicaion ae i: U D γ γ D = κ D + ξ, 2 where κ i he uiliy funcion parameer which repreen he relaive value of income in he applicaion ae o income in he work ae. Income in differen ae may be valued differenly a no-work ae may alo imply higher leiure or igma from applicaion. We aume ha hi parameer i he ame, regardle of which oucome approval or rejecion occur. A a implificaion, we choe o approximae he preference by enering D a he argumen of he po-applicaion ae uiliy funcion wherea a more complee formal model would define preference over loerie. Therefore, he uiliy funcion parameer γ only repreen rik averion wih repec o income variabiliy and no he rik averion wih repec o applicaion for he SSDI program. Our implificaion can be juified by argumen made by Kreider 1998 who find ha he rik averion wih repec o rejecion by he program i weak compared o he rik averion wih repec o labor earning variabiliy in he conex of SSDI applicaion. Noe ha, our implificaion doe no necearily imply rik neuraliy wih repec o rejecion in our 10

14 11 model, a κ can alo be hough a repreening he relaive value of he loery i.e. he applicaion ae o he work ae. So, if individual are rik avere wih repec o rejecion, hi would imply a relaively lower κ which would indicae ha he non-work income i valued le relaive o work income due o i uncerainy. A priori, we do no impoe any rericion on κ and allow i eimae o ake any value. The diurbance are aumed o be independen over people and ime. One can calculae he uiliy of applying for SSDI paymen a variou period. The uiliy value a ime of applying for SSDI a ime r i denoed a r V. = = + = d r D r W r D U W U r V 1 β π β π 3 The problem i o maximize ] [ r V E over r. The value of applying for SSDI now period i = = d D D U V E β π. To define he problem more convenienly, define he expeced value of applying for SSDI in year r minu he expeced value of doing o now a r G. V E r V E r G = 4 Thi i he gain, evaluaed a period, from poponing SSDI applicaion unil period r. Subiuing in he above formulae lead o: = = = + = d d r r D E r D E W E r G 1 γ γ γ γ γ κ β π κ β π β π = + 1 r E ν β π 5 where. ξ ω ν = Thu, r G coni of he following wo par:

15 12 = = = + = d r d r r D E r D E W E r g 1 γ γ γ γ γ κ β π κ β π β π 6 = = 1 r E r ν β π φ. 7 Then r g i he yemaic erm, he exogenou porion of uiliy aociaed wih applying for SSDI in period r, and r φ i he ochaic porion of uiliy. If we define ] [ arg max * r V E r r =, hen he peron popone SSDI applicaion if 0 * * > = V E r V E r G, i.e. if he opion value * r G i poiive. The applicaion rule can be explained a follow: one applie for SSDI in period r > if 0 < + g ν for d r +1 and for r < ', uch ha 0 ' ' > + g ν. Thu, one mu compue j g i for d j i <. Thi ep i recurive and enail Taylor expanion of ] [ γ X E unle he analyical expecaion exi. LSW ue he following Taylor erie expanion: γ γ γ γ ] [ 2 X E E X E X X E X E + 8 Therefore, we have γ γ γ γ } { ] [ X E X Var X E + 9 LSW doe no expand γ γ κ D in a Taylor erie on he aumpion ha he variance of income i mall in he non-work ae. We aler he formulae of LSW by expanding γ ] [ X E 2 2 ] [ln ln 1 ] [ X E X E X E EX X E X E + + = γ γ γ 10

16 γ γ and imilarly for κ D. Thi expanion increae he accuracy of he approximaion and improve numerical performance. The nex ep i o find he maximum g i j over j, hi i g j * i, where j * i he period in which applicaion for SSDI occur. To calculae he likelihood funcion, we define he following probabiliie, which all add o uniy: Probabiliy of applying for SSDI a period : Pr SSDI * = = Pr g j < ν 11a Probabiliy of applying for SSDI a period j > i : Pr SSDI * * = j = Pr g i > ν, g j < ν 11b Probabiliy ha he individual doe no apply for SSDI before he end period d : Pr SSDI * * > d = Pr g j > ν, g j > ν 11c d d The Dynamic Programming Model The dynamic programming verion of hi model ue mo of he above equaion, bu omewha differen probabiliy compuaion. The maximizaion in he opion value model i he maximum of he expeced value of uiliy, wherea he maximizaion in he dynamic programming approach i he expeced value of he maximum uiliy. The laer i necearily larger and he opion value model underae he expeced value of waiing. Therefore, dynamic programming i heoreically preferred o opion value a i provide a more formal oluion o he ineremporal uiliy maximizaion problem. Thee wo approache are he ame only if he maximum uiliy i guaraneed o be in one cerain year, whoe expeced value i hen he maximum. Alernaively, imagine a e of nonrivial zero-mean random variable which are no perfecly correlaed. The maximum expeced value of fuure diurbance indeed, every 13

17 expeced value i zero, bu he expeced maximum exceed zero, he more o, he more period here are o come. A period, he pre-applicaion and po-applicaion uiliie are given by U W W + ω and U + D D ξ, where value funcion a ime i given by: ω and ξ are aumed o be independen over people and ime. The V d r = max E [ UW W + ω + βπ + 1 V r ], E β τ = + ξ τ + 1 π τ [ U D Dτ ] 12 Again we define he probabiliy of urvival a π τ and we obain V { V r + ω, V + ξ } = r max d τ where V1 r = UW W + βπ + 1 EV + 1 r, and V2 = E β π τ U D Dτ τ = The applicaion rule here i: if V 1 r + ω < V2 + ξ, hen he individual will apply for SSDI in period, oherwie he or he will coninue working. Therefore, he probabiliy of SSDI applicaion i Pr V1 r + ω < V2 + ξ. The calculaion in he dynamic programming model involve he expeced maxima of uiliy over all poible SSDI applicaion ime. Daa In hi ecion, we will briefly decribe he daa e we ue and define he variable in our analyi 4. Our daa come from he fir hree wave of he Healh and Reiremen Sudy HRS 5. The HRS i a longiudinal udy of he healh, wealh, income, and employmen of primary reponden aged in 1992 and econdary reponden poue or parner of hee primary reponden who were inerviewed regardle of heir age. Reponden born beween he year of 1931 and 1941 are conidered age eligible. Individual were inerviewed 14

18 biennially, and five wave of daa are currenly available, hree in final form. HRS daa can be linked o rericed acce SSA adminiraive daa 6. Three rericed acce file are ued in hi udy: The HRS Covered Earning File, The Summary of Earning and Projeced Benefi SEPB File, and The Wage and Self-Employmen Income in Covered and Non-Covered Job File. The HRS i an excellen ource of daa for analyzing policy iue relaed o SSDI. I include a module on diabiliy wih deailed reropecive queion abou SSDI applicaion and award. Daa on individual' demographic characeriic, labor force paricipaion, employmen, and healh au are alo available in eparaely deigned ecion. The income ecion provide daa on benefi, income, and wealh holding. We alo ue addiional ource of daa. The Lewin Group creaed a Public Ue File which include ae level daa on SSDI and SSI program a well a ae level decripive variable for he year 1974 hrough The daa conain iniial SSDI allowance rae for each ae compued a he number of people awarded SSDI benefi a he iniial ae level creening proce divided by he oal number of iniial SSDI applicaion in ha ae. Thee daa are ued o form he probabiliie of accepance for SSDI applicaion 7. The rericed HRS daa e Wave 1 Geographic Indicaor Verion 1.0 file provide ae geographic idenifier variable from HRS Wave 1, including informaion on Wave 1 ae of reidence and ae or counry of birh. Thee variable are maked in he public HRS file. We obained pecial permiion from he HRS aff a he ISR a he Univeriy of Michigan o be able o merge he geographic ae idenifier variable wih he Lewin Group Public Ue File on allowance rae 8. In our udy we need daa on he probabiliie of deah for individual wih work limiing healh condiion, and for hi purpoe we ue life able daa provided in Zayaz

19 We draw our ample from boh age eligible and age ineligible peron who repored a work limiing healh problem in Wave or Wave of he HRS a defined by a poiive repone o he queion Do you have an impairmen or healh problem ha limi he kind or amoun of paid work you can do? Beniez-Silva, Buchinky, Chan, Cheidvaer, and Ru forhcoming how ha he elf-repored diabiliy meaure in HRS i an unbiaed indicaor of he rue diabiliy au. In he fir wave, 2,717 peron 1,324 men and 1,393 women repored ha hey had uch an impairmen or healh problem 9. To hi populaion we added 340 peron 140 men and 200 women who were no in he ample drawn from Wave 1 bu who repored having a work limiing healh condiion in Wave 2. Of hee 3,057, we kep hoe wih permanen condiion impairmen expeced o la for more han hree monh who were working for omeone ele no elf-employed a he one of heir work limiing healh condiion. Thi iniial creening yielded a ample of 1,653 individual 924 men and 729 women. Individual were aked when heir condiion fir began o boher hem, and hi dae i ued a he one of he healh problem. They were alo aked if hey applied for SSDI benefi. For hoe who applied, heir pell end a he year of applicaion. SSDI benefi award au can be obained uing he income ecion of he urvey. We excluded individual wih a miing one or SSDI applicaion dae, or wih miing SSDI applicaion or award au informaion. We kep hoe wih an one dae afer 1950 and before age 61. Finally, only hoe who were eligible for SSDI benefi in erm of being fully inured in a lea one period following one were kep in he final ample. Thi way, we guaranee ha he individual in our ample whoe applicaion i rejeced are denied SSDI benefi olely due o medical creening reul. Individual who became eligible afer 1993 were alo dropped from he analyi ince i wa no 16

20 poible o oberve hem applying for SSDI. Applying hee crieria, our final ample conied of 1,085 individual 592 men and 493 women. Table 1 provide decripive aiic for he key variable ued in our analyi. The ime uni in our analyi i a biennial period ince he dae on SSDI award au and income during he urvey period are known over biennial period. We calculae uiliy from he ream of labor earning in differen ae and he poenial SSDI benefi which would reul from applicaion for each period of poenial applicaion. We conruc oher inpu which are aumed o be exogenou o he model. We include all of hee meauremen and predicion in our opion value and dynamic programming model in order o analyze he deciion o apply for SSDI. Table 2 decribe he diribuion of pell lengh from one o applicaion by gender for our ample. The fir column how he number of period ince he fir period of eligibiliy afer he one of a work limiing healh condiion. The nex five column how he number of men who apply wihin he period; who are cenored wihin he period; heir hazard rae; heir probabiliy of no applying before he beginning of he period; and he eimaed probabiliy ma funcion. The nex five column how hee ame value for women. The hazard rae i greae in he fir period boh for men and for women. Nonehele, only abou a quarer of he ample apply for SSDI in he fir period fir wo year following one. The va majoriy of worker who experience he one of a work limiing healh condiion do no apply for SSDI in he fir wo year bu chooe o wai o apply. The hazard rae decline afer he fir period, and he longe oberved pell i 16 period 32 year for men and 15 period 30 year for women. Thee life able demonrae he ubanial variaion in pell lengh from one o applicaion bu no he reaon for uch grea variaion in oucome. 17

21 Explanaory Variable All of he following variable are aumed o be exogenou inpu o he model: labor earning, SSDI benefi, ocio-demographic variable, he ype of healh condiion, accommodaion by employer following one, iniuional deail of he SSDI and OASI program including he probabiliy of accepance in he SSDI program, and he probabiliy of deah. Economic Variable. In order o eimae a dynamic programming or opion value model, i i neceary o eimae fuure labor earning and poenial benefi level. Expeced real labor earning are inended o capure he opporuniy co of applying for benefi. We need o predic labor earning profile for each individual for each of he hree ae we will inveigae: 1 no applicaion, 2 applied and rejeced, and 3 applied and acceped. Our aim i o ge good labor earning predicion raher han o eimae a rucural model of lifeime earning. Toal labor earning are defined a he um of covered earning covered by OASDI axe and non-covered wage, and hey are adjued for inflaion. Non-covered wage had o be impued in ome cae ee Gumu, Covered earning are cenored a he OASDI axable earning maximum. To obain expeced value of labor earning in uch cae, we fied a eparae log-normal diribuion for each year. Once hee iue were addreed, an auoregreion wa ued o conruc expeced earning profile for he hree ae decribed above. Following Burkhauer, Buler, Kim, and Weaher 1999, we prediced labor earning uing an auoregreion which include: a conan and four lagged value of labor earning alone and ineraced wih a dummy variable ha conrolled for having a limiaion a o he kind or amoun of work ha one can do; pell lengh defined a he ime lag beween each period and he fir period of eligibiliy afer he one year; dummie for being in he applied and rejeced 18

22 group or he applied and acceped group ; age, age quare, and unemploymen rae a each period. Once we obained hee hree ime-varying earning predicion, we hen compued heir correponding andard deviaion, which we inerpre a he uncerainy of he earning ream hemelve. We, hen conider he probabiliy of having zero labor earning. A probi equaion i eimaed o obain he prediced probabiliie of having zero earning. A final predicion of labor earning i done by incorporaing he prediced value of covered wage from he auoregreion, he probi, and he prediced value of he non-covered wage. Labor earning eimaion for individual wih no adminiraive record i done in he ame way, excep ha he elf-repored labor earning value from he fir hree wave of he HRS are ued inead of he rericed acce earning hiorie. Uing acual earning hiorie and predicion of hem when hiorie are no available, we hen compue poenial PIA following SSDI program rule. Deail of he PIA compuaion rule can be found in he Annual Saiical Supplemen o he Social Securiy Bullein. The benefi compuaion i decribed in deail in an unpublihed daa appendix ee Gumu, We need o projec SSDI benefi rule for year afer 2000, and for hi purpoe we aumed ha he iniuional deail of he SSDI program do no change excep a decribed by aue in We aume poenial SSDI recipien know and ac on hi informaion. Annual SSDI benefi are hen convered ino real 1967 dollar. Demographic Variable. We include everal demographic variable uch a race, educaion, and marial au. Thee variable reflec labor marke aachmen and dicriminaion, and hu are relevan o he uiliy funcion pecificaion. Healh Variable. To be included in our ample, a worker mu experience he one of a 19

23 work limiing impairmen or healh problem. Bu he condiion vary in ype and everiy. The ype of condiion i included in he HRS daa, a well a comorbidiy, he preence of oher menal and phyical condiion. Thee are facor affecing wage and hu he deciion o apply for SSDI benefi. They will alo be conidered a facor affecing he uiliy difference beween he work and applicaion ae. Policy Variable. We are inereed in everal policy variable. Employer accommodaion i baed on a queion o he individual aking if he employer did anyhing pecial for he individual a one o ha he or he could remain a work. Accommodaion by employer can increae he lengh of ime during which an employee ay on a job and doe no apply for SSDI Burkhauer e al., 1999; Burkhauer e al., We include he iniial SSDI allowance rae in our compuaion of expeced income in he applicaion ae. In our economeric model, he labor earning in he no applicaion ae, and W i D i he weighed average of he income for he applied and acceped ae B and he labor earning for he applied and rejeced ae Y. The weigh are given by he iniial allowance rae. Thee rae are available a he ae level. Higher allowance rae are expeced o increae he peed a which individual apply for SSDI benefi. Opion Value and Dynamic Programming Eimaion Reul To eimae our opion value model we eimae he yemaic uiliy g uing Taylor expanion of he nonlinear funcion, and hen eimae he univariae or mulivariae normal likelihood funcion for he opimal iming of SSDI applicaion. The laer ep produce parameer of he uiliy funcion and he variance. To eimae our dynamic programming model, we fir calculae he expeced uiliy of permanen labor marke exi a he erminal 20

24 period d age 62. We hen calculae he expeced maximum uiliy of applying for SSDI a period d 1 which enail a maximum over wo independen diurbance. The probabiliy of applying a d 1 i he probabiliy ha he uiliy of doing o exceed he uiliy of waiing. The uiliy of exiing a d i V = U D. Tha i no a choice a ha ime ince i i he end period. d D d The uiliy of working a d 1 i he value funcion U W + βu D W d 1 D d, where β i a dicoun facor whoe eimaion i ofen no ucceful in hee model. We did no eimae β, o hi parameer will be e ouide he model a a value baed on LSW The uiliy of applying a d 1 i U D + βu D D d 1 D d 10. All of he uiliie involve diurbance and we obain he probabiliie in he following way. By aumpion, Pr exi a d no applicaion before ha = Then, we have Pr exi a d 1 no applicaion before ha = Pr UW Wd + β U D Dd < U D Dd 1 + βu D 1 D d. Define V d 1 a he value funcion a d 1, including boh poible pah. We coninue hi backward recurion one more period. The uiliy of working a d 1 i U W Wd 2 β E d 2V 1. d 2 + D d 1 D d + d 2 The uiliy of applying for SSDI a d 2 i U D β U D + β U D D. Finally, Pr exi a d 2 no applicaion before ha = Pr UW Wd + β Vd 1 < U D Dd 2 βu D Dd β U D D d. 2 + Thi recurive proce coninue wih ever-growing formulae back o ime, hereby defining he probabiliie for maximum likelihood or oher mehod of eimaion. Each ime he value funcion V i defined. Evenually one reurn o, he preen, and he fir probabiliy in ime, bu he la in calculaion. The probabiliy ha working now in ha lower uiliy i he + + probabiliy ha chooing nex period UW W β EV 1 ha lower uiliy han applying for SSDI 21

25 d now U D + β U D +... β U D. The normal diribuion, or indeed mo D D D d diribuion, of ochaic uiliy make hi equenial approach difficul, becaue he maximum of a e of random variable ha a diribuion ha i no known in cloed form. Therefore, in all cae we direcly inegrae he expeced maximum uiliy numerically. Following Daula-Moffi 1995, we add an obervable variable vecor x o he original uiliy funcion decribed in equaion 1 and we denoe he effec of hi vecor by δ. Thu, he uiliy funcion in he work ae ake he form U W γ = Y + x' δ + ε. Here he idea i o include ome fixed uiliy difference beween he pre-applicaion ae and po-applicaion ae. Thee are differen from he marginal uiliy difference we impoed originally hrough κ, bu hey alo reflec he relaive valuaion of he pre-applicaion ae and po-applicaion ae. The x vecor include year of educaion, dummie for race, marial au, employer accommodaion, healh condiion and for being a whie collar worker. We call he parameer correponding o hee variable Daula-Moffi parameer. Noe ha poiive Daula-Moffi parameer dicourage applicaion and negaive one encourage applicaion. The parameer of he opion value and dynamic programming model o be eimaed are: κ he relaive value of income in he non-work ae o income in he work ae, γ uiliy funcion parameer repreening rik averion wih repec o income variabiliy where he coefficien of relaive rik averion= γ 1, β dicoun facor, and δ Daula-Moffi parameer. The eimae of he dynamic programming model are obained by auming a normal diribuion for he diurbance. We eimaed wo verion of he opion value model, one auming a normal and he oher a Weibull diribuion. Table 3 and Table 4 preen eimaion reul for men and women, repecively. 22

26 In Table 3, boh our opion value and dynamic programming model produce aiically ignifican eimae of κ and γ, when β i e equal o Eimaed γ i le han one in each of he hree e of reul and ugge ha people are rik avere wih repec o income variaion. The Arrow-Pra coefficien of relaive rik averion wih repec o income variabiliy i abou 0.6 in he dynamic programming and abou 0.55 and 0.4 in he opion value model. The eimae of κ i greaer han 1 in he dynamic programming model and he normal verion of he opion value model, which ugge ha every dollar earned wihou work i worh more han every dollar earned from work. The eimae i around 1.4 in he dynamic programming model. Thi mean ha he average man in our ample would be indifferen beween geing a dollar from work and geing 71 cen under SSDI. Uing he Weibull verion of he opion value model, however, we find ha he average man value non-work income le relaive o work income ince eimaed κ i maller han 1 in hi model -- i i around Thi apparenly large difference doe no however, have much of an effec on he likelihood or oher parameer in he model. The eimae of he Daula-Moffi parameer are almo alway ignifican and hey are imilar acro model. Addiional educaion or accommodaion by employer increae he uiliy of earning relaive o SSDI, dicouraging applicaion. Being African-American or being married encourage applicaion wih one excepion: he effec of marial au i poiive bu mall in he normal verion of he opion value model. Being a whie collar worker a he one of a work limiaion ha no aiically ignifican effec. There i evidence ome healh condiion lead o more rapid applicaion han oher. Muculokeleal condiion, uch a back, neck and pine problem, or arhrii lead o a relaively longer duraion unil SSDI applicaion ince hee condiion end o be chronic. Noe, however, ha eimaed effec are no alway aiically 23

27 ignifican. On he oher hand, cardiovacular condiion, uch a roke or hear aack, lead o horer duraion o SSDI applicaion ince hey end o be acue. Thi effec i alway aiically ignifican. The omied caegory i all oher healh condiion. All hee reul are conien wih reduced form model finding ee Burkhauer e al., 1999; and Burkhauer e al., Table 4 preen he eimaion reul for women. The eimaed γ ake a wide range of value. I i around 0.5 in he dynamic programming model. I i eimaed o be around 1.7 in he normal verion of he opion value model and abou 1 in he Weibull verion of he opion value model. Thi wide range of γ again doe no affec he general naure of he reul when one look a he cerainy equivalen 12. The eimae of κ i greaer han 1 in each model. I i abou 2.1 in he dynamic programming model and abou 2 and 1.8 in he opion value model. The eimae of he Daula-Moffi parameer are imilar o hoe of men wih a few excepion. Educaion, marial au, and accommodaion are no alway ignifican. Being a whie collar worker a he one dicourage applicaion ignificanly. The effec of arhrii and cardiovacular condiion are no ignifican bu he effec of muculokeleal condiion i ignifican, and i lead o a relaively longer duraion unil SSDI applicaion. In-Sample and Ou-of-Sample Predicion Table 5 and Table 6 how model predicion of SSDI applicaion rae for men and women, repecively. The fir column of each model give he prediced applicaion rae, and he nex column give he difference beween he life able and he prediced rae. The opion value and dynamic programming model produce imilar predicion. Separae analye of men and women change he value bu no he general paern. 24

28 We e he fied diribuion reuling from he opion value and dynamic programming model veru he diribuion eimaed uing life able mehod in he original daa. Fir, a in Table 2, we eimae he hazard of applicaion and he implied diribuion of ime o applicaion in he original daa. Then we compare he fied diribuion from opion value and dynamic programming in Table 5 and 6 uing a mulinomial likelihood raio e baed on he dicree period. The Pearon chi-quare goodne of fi i a Taylor Serie approximaion o hi, bu we ue he acual likelihood raio. We prefer hi meric for comparing predicive power acro our model o comparing log-likelihood value baed on non-need hypohee. To avoid mall cell, we conider he fir 15 period 30 year for men and he fir year for women. The criical value of he chi-quare e aiic are percen and percen for men and percen and percen for women. The reul are a follow: for men, dynamic programming normal, 63.3, opion value normal, 41.7, opion value Weibull, 39.5; for women, dynamic programming normal, 41.5, opion value normal, 29.5, and opion value Weibull, The concluion are ha none of he fied diribuion are precie, bu he opion value model fi beer han he dynamic programming model, and here i lile difference beween he wo opion value model. In all he model, we undereimae he applicaion rae in he fir period. In he nex everal period, we overeimae he applicaion rae. Thi ugge ha individual who apply in he earlier period are expeced o do o, bu hey are no expeced o apply for SSDI immediaely in he fir period. Thi overapplicaion problem in he fir period reemble he problem ha LSW encounered wih he age 65 reiremen effec. A hey alo poin ou, hi kind of a paern i clearly no relaed o he earning or benefi ream LSW, In our cae, i may be due o our inabiliy o idenify cae where he one condiion i o evere ha no 25

29 real choice of applicaion i feaible. For ome ignifican minoriy of worker, he one condiion compel immediae applicaion. Thi overapplicaion problem in he fir period could alo be he reaon for our failure o eimae he dicoun facor β i eimae goe o zero o fi early applicaion beer. Alo afer he fir period, he predicion are very imilar among he hree e of reul. Boh he opion value and he dynamic programming model overeimae he number of people who chooe no o apply for SSDI, bu he opion value model are more accurae. We alo e ou-of-ample predicive performance of our model baed on revied daa. To do o, we randomly divided our daa e ino wo eparae ample: wo-hird of he daa were ued for parameer eimaion only, and he re wa ued o ae he predicive performance 13. Thi i done eparaely for men and women. The diribuion of pell lengh by gender for our wo-hird ample are given in Appendix Table 3A. Thee diribuion are very imilar o hoe for he original ample. The fied diribuion reuling from he opion value and dynamic programming model uing he wo-hird ample are compared o he diribuion eimaed uing life able mehod in he one-hird ample. Thee comparion are preened in Table 7 and 8. We conider he fir 12 period 24 year for men and he fir 11 period 22 year for women. The criical value of he chi-quare e aiic are percen and percen, and percen and percen, repecively. The e aiic are: for men, dynamic programming normal, 24.1, opion value normal, 17.5, opion value Weibull, 16.7; for women, dynamic programming normal, 20.9, opion value normal, 14.7, and opion value Weibull, The Weibull verion of he opion value model provide he be ou-ofample foreca for boh men and women even hough here i lile difference beween he Weibull and he normal verion of he opion value model in erm of predicive performance. 26

30 Noe ha, for boh model, he e aiic i le han he criical value a he five percen ignificance level for boh ample. Therefore, we canno rejec he null hypohei ha he opion value model i empirically correc. The prediced applicaion rae from he opion value and dynamic programming model are imilar overall. However, he comparion of he predicive performance of he wo model baed on he mulinomial likelihood raio e ugge ha he opion value model i uperior o he dynamic programming model. Noe ha he opion value model provide a beer fi ou-ofample han in-ample. Thi poin i very imporan in he conex of policy analyi, ince policy imulaion require model ha are able o predic acual repone o fuure policy change. Given he eimaion co of dynamic programming model, he opion value model clearly ha an addiional advanage ince i i able o produce good predicion even hough i i compuaionally le inenive. Thi concluion i imilar o ha of LSW for reiremen who find ha he more complex dynamic programming model doe no produce beer predicion han he opion value model. Comparion wih Hazard Model In hi ecion, we dicu hazard model reul. The hazard model i baed on Burkhauer e al We define new variable in order o make he hazard pecificaion comparable o he opion value and dynamic programming approache. Hazard model are explicily reduced form model of he ime unil an even occur, here applicaion for SSDI. In he ame way ha one migh compare rucural model of wage or macroeconomic oucome wih model deigned excluively o make predicion, one can compare hazard model wih opion value and dynamic programming model. A priori, hazard model would be expeced o 27

31 have higher likelihood funcion in ample becaue hey impoe lile in he way of aumpion, bu hey would be upec a he bai for policy imulaion of ubanial change in program rule. Becaue hazard model are available in andard compuer package and hey have been ued in empirical reearch before, a lea a brief comparion i ueful for pracical policy analyi. We neverhele prefer rucural modeling a priori becaue i i baed on he uiliy heoreic bai of individual behavior. Here, he hazard rae i defined a he probabiliy of applying for SSDI benefi once a work limiing condiion fir begin o boher he worker. Noe ha hi probabiliy i imedependen, i i condiional on no having applied earlier, and he pell ar when he worker become eligible. An inerval hazard ha conrol for unoberved heerogeneiy i ued o eimae he raniion o SSDI applicaion. The unoberved heerogeneiy, due o omied variable or difference in he diribuion funcion acro individual, i inegraed ou of he likelihood funcion by auming a log normal form. We define ln W a he logarihm of he curren earning in he no applicaion ae. LnD i he naural logarihm of a weighed average of curren income in he applied and rejeced and applied and acceped ae, where he weigh are baed on he probabiliy of accepance, i.e. ln D = ln[ α * B + 1 α * Y ]. The variable we ue in he hazard model i he difference {lnw ln D }. The la column in Table 5 and 6 how he prediced applicaion rae uing he hazard model 14. The hazard model predic he fir period applicaion relaively well for men. However, i overeimae he applicaion rae for almo all oher period a he economic and policy incenive are capured only indirecly. Moreover, uncerainy abou he fuure i no fully modeled. The mulinomial likelihood raio e aiic i 74.2 for men and 62.9 for women for he in-ample predicion and he opion value model fi beer han he hazard model. 28

32 Concluion In hi paper, we model he iming of SSDI applicaion. We do o uing wo dynamic rucural model, opion value and dynamic programming. Thee model enable u no only o decribe bu alo o explain how iming o SSDI applicaion i affeced by boh healh condiion and policy variable. We hen inveigae wheher he reul from he eimaed model differ qualiaively or quaniaively. The aim i o ee which of hee model perform beer in erm of predicive performance. The meaure we ue o evaluae he predicive performance i a mulinomial likelihood raio e baed on he difference beween he opimal iming predicion and he life able applicaion rae. Our predicive power comparion uing boh in-ample and ou-ofample daa how ha he opion value model perform beer han he dynamic programming model in erm of predicive power even hough i may be le conien wih heoreical individual behavior. In addiion, i compuaional impliciy i an advanage over he dynamic programming approach. Neverhele, dynamic programming i clearly heoreically preferred in erm of allowing for a more complee model including everal oher iue uch a aving, appeal, reurn o work, raniion from SSDI ino reiremen, ec. All our model undereimae he rik of an immediae applicaion following he one. Thi ugge ha for a ignifican ube of our ample unoberved everiy of a work limiaion may be limiing heir choice of iming over and above he oberved healh meaure. However, our reul alo how ha while he everiy of a work limiing healh condiion ignificanly affec he iming of SSDI applicaion, a do policy variable uch a employer accommodaion and he relaive value of income in he applicaion ae o income in he work ae. We conclude ha while he road o diabiliy benefi au begin wih a healh condiion, he SSDI 29