~ * AC. ( E 1 vector), where 0 AC is a matrix of zeros of
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1 Ole spleety ote to e ppe ettled A opehesve Dwell Ut hoe Model Aoodt Psyholol ostuts w Seh Sttey fo osdeto Set Foto Model Syste Estto et E y y y E γ γ 0 A [ E A tx] d ε ε ε E veto whee 0 A s tx of zeos of. Defe [ veto] d d d [ E tx] deso A. et be e olleto of petes to be estted. W e deftos bove we obt e fol edued fo syste fo y d U : y γx dz Ξ ε γx d αw η ε γx dαw dη ε V ε Σ 0 0 IDE U bx z bx αw η bx αw η ow osde e [ E ] B B B γx dαw bx αw d veto yu y U Ω Ω Ω. Defe Ω dγd Σ Ω Γd dγ Γ Λ The yu MV B Ω. E 3 Bht 05 hs detfed suffet detfto odtos fo e DM odel whh e suzed hee detls e Bht 05: ee e t lest two ltet vbles w eh ltet vble oelted w t lest oe oe ltet vble e oelto tx Γ dolty s ted oss e eleets of e eo te veto ε t s Σ s dol 3 blo-dolty s ted fo e tx Λ s Euto of e ppe 4 The eo te vetos ε d ς e depedet of eh oe 5 fo eh ltet vble ee e t lest two outoe vbles t lod oly o t ltet vble d o oe ltet vble t s ee s t lest two fto oplexty oe outoe vbles fo eh ltet vble 6 f spef vble e veto x lods oto eleet of e o-ol outoe veto y e t eleet does ot deped o y ltet vble t ots e spef vble s ovte e stutul euto syste 7 f spef vble e veto x ffets e
2 utlty of ltetve of ol vble e e utlty of ltetve does ot deped o y ltet vble t ots t spef vble s ovte e stutul euto syste d 8 edoeous vble effets be spefed oly sle deto s dsussed e footote Seto. of e ppe; ddto whe otuous obseved vble sy vble A ppes s ht sde vble e eesso fo oe otuous obseved vble o s ht sde vble e ltet eesso udely oe out o odl vble eh ltet vble ppe e eesso/ltet eesso fo e oe otuous/out/odl vble sy vble B should hve two fto oplexty oe outoe vbles fte exlud e euto fo vble B. Ths ltte odto s ot eeded whe o-otuous obseved vble ppes s ht sde vble e eesso of y oe obseved vble beuse of e o-le tue of e eltoshp betwee e ltet eessos d e obseved o-otuous vbles. To estte e odel ote t ude e utlty xzto pd U U ust be less zeo fo ll oespod to e ol vble se e dvdul hose ltetve. et u U U d st e ltet utlty u u u... u I ;. Also defe dffeetls to veto u u u... u yu y U d develop e dstbuto of e veto yu y u. To do so defe tx M of sze E E fo t of. Fll s tx w vlues of zeo. The set detty tx of sze E to e fst E ows d E olus of e tx M. ext osde e ows fo E to E I d olus fo E to E I. These ows d olus oespod to e fst ol vble. Iset detty tx of sze I fte spleet w olu of - vlues e olu oespod to e hose ltetve. ext ows E I ouh E I I d olus E I ouh E I I oespod to e seod ol vble. A posto detty tx of sze I fte spleet w olu of - vlues e olu oespod to e hose ltetve fo e seod ol vble. otue s poedue fo ll ol
3 vbles. W e tx M s defed we wte yu MV B Ω whee B MB E d Ω MΩM. ext defe eshold vetos s fols: ψ ψ ψ [ ] 0 veto d ψ ψ [ ] ψ veto whee s -olu veto of etve ftes d 0 s oe -olu veto of zeos. The e lelhood futo y be wtte s: P ψ yu ψ f B Ω d D whee e teto do : ψ ψ } s sply e ultvte eo pled D { by e obseved o-ol dto outoes d e e fo e utlty dffeees te w espet to e utlty of e obseved hoe ltetve fo e ol outoe. f B Ω s e ultvte ol desty futo of deso w e of B d ove of Ω d evluted t. The lelhood futo fo sple of households s obted s e podut of e household-level lelhood futos. The bove lelhood futo volves e evluto of -desol etul tel fo eh deso-e whh be oputtolly expesve. A ltetve estto tehue s Bht s 0 Mxu Appoxte oposte Ml elhood MAM ppoh. 0 4 The Jot Mxed Model Syste d e MAM Estto Appoh I e MAM poedue we develop e fol pwse oposte l lelhood futo foed by t e poduts oss e oed vbles e out vbles d ol vbles of e ot pwse pobblty of e hose ltetves fo household: 3
4 M '. 5 To expltly wte out e M futo tes of e stdd d bvte stdd ol desty d uultve dstbuto futo defe ω s e dol tx of stdd devtos of tx Δ.; Δ fo e ultvte stdd ol desty futo R of deso R d oelto tx Δ Δ ω Δω d.; Δ fo e ultvte stdd ol uultve dstbuto futo of deso E d oelto tx Δ. osde two seleto tes s fols: D seleto tx w v I E ety of e fst ow d e v olu detty tx of sze I oy e lst I ows d e I ouh I olus w 0 e oveto t I 0 d etes of 0 eveywhee else R I I seleto tx w detty tx of sze I oy e fst I ows d e I ouh I olus 0 w e oveto t I 0 d oe detty tx of sze I oy e lst I ows d e olus; ll oe eleets of I ouh I R te vlue of zeo. Also let Ω D ΩD Ω ψ v v RΩR B v Ω vv ψ B v v Ω v Ω vv vv whee ψ vv Ω vv Ω v v v v v v 4
5 epesets e v eleet of ψ eleet of e tx Ω. The d slly fo oe vetos d vv Ω epesets e vv M v ω Dv ψ ; Ωv I ω v B D ψ B I I v v ω R ; B Ω v v v vv v v vv v v I Ωv whee ψ ψ ψ Ω. vv Ωv vv vv ; Ω v 6 I e MAM ppoh ll MVVD futo evluto ete two desos e expesso bove e evluted us lyt ppoxto eod e sulto eod see Bht 0. As hs bee deostted by Bht d Sdh 0 e MAM eod hs e vtue of oputtol obustess t e ppoxte M sufe s sooe d ese to xze tdtol sulto-bsed lelhood sufes. Wte e esult euvlet of Euto 6 oputed us e lyt ppoxto fo e MVD futo s fte todu e dex fo households. The MAM MAM estto s e obted by xz e fol futo: lo lo. 7 MAM MAM I e tul epl lyss e ppe lot sze s ot defed fo dwell uts ptet oplexes. Thus e lelhood futo Euto 4 d e M futo Euto 5 hve to be odfed o wy. Speflly Euto 4 e lelhood futo beoes e podut of two opoets: oe opoet fo dwell uts ptet oplexes whee e lelhood oespods to e pobblty of e ultdesol set of hose ttbutes but ss e lot sze deso s edues e desolty of e tel d seod opoet fo o-ptet oplexes t tes e ext fo s e uet Euto 4. I e M futo of Euto 5 ee e two ultpltve opoets oe : oe opoet fo dwell uts ptet oplexes whee pwse obtos of oe desos w e lot sze 5
6 6 deso do ot ppe d seod opoet fo o-ptet oplexes whee e M futo s extly s s e uet Euto 5. The ove tx of e petes y be estted by e vese of odbe s 960 sdwh foto tx see Zho d Joe 005. V MAM J 8 w MAM lo MAM MAM J lo lo MAM MAM 9 A ltetve estto fo Ĥ y be obted by oput e utty be fo eh household d ve oss households: lo lo ] lo[ lo lo lo lo ] lo[ lo ] lo[ lo lo eh fo ' ' Postve Defteess The tx Ω fo eh household hs to be postve defte. The splest wy to utee s ou xed odel syste s to esue t e oelto tx Γ s postve defte
7 d eh tx Λ... s lso postve defte. A esy wy to esue e postvedefteess of ese tes s to use holesy-deoposto d peteze e M futo tes of e holesy petes. Fue beuse e tx Γ s oelto tx we wte eh dol eleet sy e eleet of e e tul holesy tx of Γ s p whee e p eleets e e holesy ftos t e to be estted. I ddto ote t e top dol eleet of eh Λ tx hs to be olzed to oe s dsussed ele Seto. of e ppe whh ples t e fst eleet of e holesy tx of eh Λ s fxed to e vlue of oe. Refeees Bht.R. 0. The xu ppoxte oposte l lelhood MAM estto of ultol pobt-bsed uodeed espose hoe odels. Tspotto Reseh Pt B Bht.R. 05. A ew eelzed heteoeeous dt odel DM to otly odel xed types of depedet vbles. Tspotto Reseh Pt B foo. Bht.R. Sdh R. 0. A sulto evluto of e xu ppoxte oposte l lelhood MAM estto fo xed ultol pobt odels. Tspotto Reseh Pt B odbe V.P A optu popety of eul xu lelhood estto. The Als of Metl Sttsts Zho Y. Joe oposte lelhood estto ultvte dt lyss. d Joul of Sttsts
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