Transportation Systems

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1 Transortation Systms Working Par Sris Addrssing Endognity in Discrt Choic Modls: Assssing Control-Function and Latnt- Variabl Mthods Par# TSI-SOTUR Jun 009 Cristian Anglo Guvara & Mosh Bn-Akiva Massachustts Institut of Tchnology

2 Addrssing Endognity in Discrt Choic Modls: Assssing Control-Function and Latnt-Variabl Mthods MIT Portugal Program Transortation Systms Focus Ara Rsarch Domain: Transortation Systms Intgration Rsarch Projct: Stratgic Otions for Intgrating Transortation Innovations and Urban Rvitalization (SOTUR) Par#: TSI-SOTUR Jun 009 Cristian Anglo Guvara Rsarch Assistant ITS Lab, Massachustts Institut of Tchnology Dartmnt of Civil & Environmntal Enginring 77 Massachustts Av., 1-49 Cambridg, MA 0139 Mosh Bn-Akiva Edmund K. Turnr Profssor of Civil and Environmntal Enginring Massachustts Institut of Tchnology Dartmnt of Civil & Environmntal Enginring 77 Massachustts Av., Cambridg, MA 0139 This ublication was mad ossibl by th gnrous suort of th Govrnmnt of Portugal through th Portugus Foundation for Intrnational Cooration in Scinc, Tchnology and Highr Education and was undrtakn in th MIT-Portugal Program.

3 ABSTRACT Endogty or non-orthogonality in discrt choic modls occurs whn th systmatic art of th utility is corrlatd with th rror trm. Undr this misscification, th modl s stimators ar inconsistnt. This roblm is virtually unavoidabl, for xaml, in discrt choic modls of rsidntial choic whr ndognity occurs at th lvl of ach obsrvation mainly bcaus of th omission of attributs. In such a cas, th rincial tchqu to trat for ndognity is th control-function mthod. This mthod consists in th construction of a function that accounts for th ndognous art of th rror trm which is thn includd as an additional variabl in th modl. Altrnativly, th latnt-variabl mthod can also b viwd as a rocdur to addrss th ndognity roblm in discrt choic modls. In this cas, th omittd quality attribut which is causing th ndognity can b considrd as a latnt-variabl and modld, in a structural quation, as a function of obsrvd variabls and otntially nhancd through indicators. Th main objctiv of this ar is to analyz similaritis and diffrncs among control-function and latnt-variabl tchqus and th xloration of ways by mans of which both mthods would nhanc ach othr in addrssing ndognity in discrt choic modls. This objctiv is achivd by analyzing th rortis of both mthods and by tsting thir rformanc in th corrction of th ndogty roblm in a Mont Carlo xrimnt. Th ar concluds with th analysis of otntial futur lins of rsarch in this ara. Jun 009

4 1 INTRODUCTION Th main goal of dvloing dmand modls is to forcast usrs bhavior using availabl information which is usually vry limitd. To achiv such a goal, a rang of assumtions ar ndd. First, rgarding th bhavior of th individual as a function of that limitd information and, scond, rgarding th statistical rortis of th information itslf. Whn ths assumtions do not hold, forcasting caabilitis of th modl ar invalid or, at last, challngd. On critical assumtion to attain consistnt stimators of modl aramtrs is known as xognity. It assums that obsrvd modl variabls ar uncorrlatd to non-obsrvd ons. Th analysis of mthods to corrct for ndognity in modls of discrt choic is an ara of currnt dvlomnt in conomtrics (Louvir t al, 005). On of thos tchqus corrsonds to th control-function mthod which is articularly suitabl whn ndognity occurs at th lvl of ach obsrvation. Th uros of this ar is to xlor ossibl nhancmnts of th two stag control-function mthod to corrct for ndognity in discrt choic modls that was alid by Guvara and Bn-akiva (006), in th light of th latnt variabls aroach. This ar continus as follows. Th nxt sction dscribs th roblm of ndognity in discrt choic modls and in th subsqunt th basics of th control-function and th latnt variabls mthods ar rvisd. Thn, both mthods rortis ar contrastd and otntial quivalncs and dissimilaritis ar studid. In sction 5, roosd formulations ar tstd using an omittd attribut ndognous modl of discrt choic constructd with synthtic data. Th final sction summarizs th rincial findings, draws conclusions and rviws otntial nhancmnts of this rsarch. THE PROBLEM: ENDOGENEITY IN DISCRETE CHOICE MODELS Considr a grou of individuals who fac th slction of on altrnativ among a choic st. Assum that ach individual mthodically chooss th altrnativ from which a largr amount of utility can b rtrivd. Considr that th utility of ach altrnativ dnds on its attributs, which ar conditional on th individual s charactristics lus a scific rror trm. Considr now an analyst who wants to dict individuals bhavior givn som limitd information. If th saml usd for th stimation is infitly larg and all th attributs ar obsrvd by th analyst, th tru modl cofficints would b obtaind. Howvr, if som attributs ar not obsrvd th tru modl cofficints would b obtaind if and only if thos attributs ar not corrlatd with th obsrvd ons. This fact can b shown with th following xaml. Considr that w ar intrstd in modling th choic of a car mak and modl. Potntial buyr n considrs in his or hr slction of mak and modl i th following variabls: ric, siz, ful fficincy, safty faturs and whthr th car is rd or not (color). Considr that th marginal indirct utility rcivd by th individual from ach of ths attributs corrsonds to! k and that th utility U is comltd by som Jun 009 3

5 attributs that ar scific to ach altrnativ and masurd by th altrnativ-scific-constant ASC i, and an rror trm " which is scific to ach individual an indndnt across altrnativs. U = ASCi + # ric + # ssiz + # fficincy + # ssafty + # ccolor + " (1) Considr now that th analyst can rfctly masur ric, siz, fficincy and safty, but forgts to rgistr th car s color. In that cas, th modl s rror will b # instad of ", as shown in (1). Th analyst s omission of this variabl will not comromis th consistncy of th stimators, if and only if car s color dos not dtrmin th obsrvd attributs (ric, siz, fficincy and safty). Evn though, th scal of th modl would b affctd by this omission, sinc th varianc of # will b largr than th varianc of ", and thrfor th scal will b smallr. In turn, if rtailrs vary car s rics dnding on thir color, th crucial xognity assumtion will b brokn. Considr that, for instanc, rd cars suddnly bcom mor oular and rtailrs quickly adjust thir rics uwards to maximiz rofit. In that cas, th analyst will obsrv that, for smingly qual cars, which only diffr in obsrvd ric (and unobsrvd color), som buyrs chos th mor xnsiv altrnativ. Thrfor th analyst will conclud that! is smallr than it rally is or vn that it is ositiv, what is against common sns and would mak th modl worthlss. This modl misscification is calld ndogty. Th ric variabl frquntly is at th hart of th ndognity roblm in a dmand function. For xaml, in rsidntial choic modling th quasi-uqunss of ach dwlling ut maks unavoidabl th omission of rlvant quality attributs which will ncssarily b corrlatd with th markt ric of th dwlling ut (Guvara and Bn-Akiva 006). Byond th omission of attributs, ndognity in discrt choic modls may also b causd by rrors in variabls (Walkr t al, 008), simultanous dtrmination, or saml slction bias (Vlla, 199; Eklöf and Karlsson, 1997; Mabit and Fosgrau, 009).! 3 THE METHODS UNDER STUDY 3.1 Th Control-Function Mthod Th control-function mthod can b thought of as a two stag rocdur to addrss ndognity in conomtric modls. For a comlt dscrition, th radr is rfrrd to Train (009). This mthod is scially suitabl for discrt modls of rsidntial location choic in which th ndognity is xctd to occur at th lvl of th individual dwlling ut, bcaus of th omission of quality attributs. Thortical basis of th mthod is dscribd in Hckman (1978), Hausman (1978), Ptrin and Train (005) and Blundll and Powll (004). Th basic ida is to construct a variabl or controlfunction, which would account for th art of th xctd valu of th rror trm, conditional on th obsrvd attributs, which is not zro. Thus, if this control-function is addd as an xlanatory variabl in th conomtric modl, th ndognity roblm would b solvd. Jun 009 4

6 To illustrat th intuition bhind th construction of th control-function, assum, without loss of gnrality, that only on obsrvd variabl P is corrlatd with th rror trm. Considr also a ror instrumntal variabl Z. This instrumntal variabl has to b corrlatd with th ndognous variabl but, at th sam tim not corrlatd with th rror trm of th modl. Undr thos assumtions, th control-function will simly corrsond to th fittd rror of th ordinary last squars (OLS) rgrssion of P as a function of Z. This hans bcaus th OLS stimator is a rojction of th lft hand sid variabl onto th sac sannd by th right hand sid variabl and th fittd rrors ar, thrfor, orthogonal to th instrumnts (Grn, 003). Thn, sinc th instrumnt is not corrlatd with th original rror trm, th fittd rror of th ric quation caturs th art of P which is corrlatd with th rror in th original modl, and thrfor, srvs as a control for it. 3. Th Latnt Variabls Aroach A comlt rviw of th latnt variabls aroach for modls of discrt choic can b found in Walkr and Bn-Akiva (00). Th basic ida in this cas is that, togthr within th choic modl, som latnt or unobsrvabl variabls may lay a rlvant rol in th choic bhavior. Ths latnt variabls can b ithr dtrmind though structural quations as a function of obsrvd variabls, or accountd through masurmnt quations within which som obsrvabl quantity or indicator is assumd to b a function of th latnt variabls. For xaml, in th cas of a mod choic modl, a latnt variabl would corrsond to th unobsrvd quality of th mod. Thn, structural quations can b statd within which this quality attribut can b writtn as a function of, for xaml, th numbr of assngrs r ut of sac and th xistnc of air-conditiong. Additionally, if th assngrs ar askd to valuat thir arciation of th mod s quality on a scal from 1 to 10, this survy qustion could b usd as an indicator in a masurmnt quation of th tru unobsrvd quality attribut. Th latnt variabl aroach can b stimatd ithr squntially or simultanously. In cas th simultanous stimation is considrd, th joint liklihood of th modl should b accountd for, and th latnt variabl should b intgratd out, making som assumtion on its distribution. 4 COMBINING CONTROL-FUNCTION AND LATENT VARIABLES TO CORRECT FOR ENDOGENITY 4.1 Ascts to b addrssd Th latnt variabls and th control-function mthods wr originally concivd with diffrnt uross. Th formr was scifically cratd to addrss ndognity which is not th cas of th latnt-variabl mthod. Howvr, som quivalncs and diffrncs among both mthods can b clarly idntifid. Jun 009 5

7 Th first issu is rlatd to th fact that th control-function is stimatd in two stags, whras th latnt-variabls mthod is stimatd, in gnral, simultanously. In th nxt sction, insird by th latnt variabls aroach, diffrnt altrnativs ar xlord to adat th control-function mthod so that it could b stimatd simultanously. Th scond concrn is to study how th comonnts of th latnt variabls aroach (th structural quations, th masurmnt quations and th latnt variabls thmslvs) find thir rsctiv countrart, if any, in th control-function aroach. Conctually, th controlfunction mthod focuss on th statistical rortis of th variabls whil th latnt-variabl aroach is rimarily bhaviorally basd. Morovr, dsit thir otntial similarity, it is not clar th rlationshi btwn th statistical rortis of th instrumntal variabls and, for xaml, thos of th right hand sid variabls of th masurmnt quations. 4. Simultanous Estimation of th Control-Function Mthod Th first asct to addrss in th liaison btwn th control-function mthod and th latnt variabls aroach is rlatd to th simultanous stimation of th formr. This issu can b addrssd using a Full Information Maximum Liklihood (FIML) aroach whr th liklihood of both th choic modl and th quation usd to build th control-function, ar stimatd simultanously. Sinc data is shard by both modls, this simultanous rocdur will ncssarily incras fficincy although it is not clar u to what xtnt. Howvr, this otntial incras in fficincy is not fr sinc th stimation using FIML imlis considring assumtions about th joint distribution of th rrors in both modls. This aroach to th control-function mthod has bn rviously alid, with variations, by Villas-Boas and Winnr (1999) and Park and Guta (009). To xlain th rocdur roosd in this sction, w rsnt first th two stag control-function mthod as it was alid by Guvara and Bn-Akiva (006). Considr that an individual n rcivs a crtain utility U from an altrnativ j that is a linar function of a st of attributs X and th ric, a vctor of aramtrs!, a aramtr! and an rror trm!, as it is shown in (). U = " + X " +! () Assuming that th rror trm! is distributd Extrm Valu 1 (0,µ) th choic modl rsulting is th Logit modl (Bn-Akiva and Lrman, 1985) and th liklihood of an obsrvation (L n ) corrsonds to xrssion (3), whr C n is th choic st of individual n and i corrsonds to th chosn altrnativ in for that individual. L Choic n = µ (# + X # ) µ (# + X # )! j" C n (3) Jun 009 6

8 From th stimation of (3) only µ! and µ! can b rtrivd but not µ nithr! nor! saratly. Thrfor, normalization is rquird to attain idntification. This is usually don by stting th scal cofficint to b qual to on. Undr this normalization th scal µ disaars from xrssion (3). Considr now that ric is ndognous bcaus it is corrlatd with som variabl which is rlvant to th choic rocss. As xlaind bfor, if th liklihood function (3) is maximizd, th stimatd cofficints obtaind by such rocdur would not b consistnt. Howvr, considr that can also b writtn as a function of xognous instrumnts Z, a vctor of aramtrs $, and an rror trm % as it is shown in (4). W will call this xrssion th ric quation modl. = " +! (4) Z If th instrumnts ar aroriat, th fittd rrors of th ric quation will account for th art of th ric which is corrlatd with th rror trm! in quation (). This can b shown by E # = E # Z " +! and, if th instrumnts ar not corrlatd with!, it noting that ( ) ( ) follows dirctly that E (" ) E( "! ) E (# ) = "!! =. Thn, if! and! ar assumd to b jointly normal,, trm which will thrfor account for th conditional man of th rror which is not qual to zro and, thrfor, corrcts for th ndognity roblm if it is includd in th utility. Thrfor, th first stag of th traditional control-function corrction corrsonds to th stimation of th ric quation using Ordinary Last Squars (OLS) to obtain th fittd rrors!ˆ, which ar thn usd as an auxiliary variabl of th utility function in th scond stag U = Z = $ % + # + X OLS!!" $ + $ # # ˆ ˆ # + (5) Th nhancmnt of th two stag control-function mthod into a on stag rocdur follows dirctly by rcalling that, if it is assumd that th rror! in (4) is distributd Normal(0, & %I), th liklihood of th ric quation will corrsond to th following xrssion. ( ) 1 "! " Z $ " q. 1 % & j# Cn = % & L n (6) Thrfor, if th ric quation wr to b stimatd by maximum liklihood, th rsult would b xactly th sam as if it wr stimatd using OLS sinc, if w tak th logarithm of xrssion (6), what lasts is rcisly th sum of squard rsiduals lus a multilicativ and an additiv constant. As a rsult, if it is assumd that th rrors in th ric quation ar normally distributd, in ordr to achiv th simultanous stimation of th control-function mthod it would only b ndd to Jun 009 7

9 considr as th objctiv function to b maximizd, th roduct of th liklihood of th ric quation and th liklihood of th choic modl whr in th scond, th rror of th ric quation is considrd as an additional variabl in th utility function, as it is shown in (7). This rocdur is calld Full Information Maximum Liklihood (FIML) in conomtrics litratur (Grn, 003). L FIML. n = ( + X ( + ( ( $ Z %) ( $ Z %)! ( + X ( + ( " j# C j# C n n ( Z ) 1 $ " $ % 1 & j# C n )& (7) Th FIML stimation of th control-function mthod can b dirctly obtaind by maximizing xrssion (7). Howvr, a rocdur that may facilitat th rformanc of th stimation is to considr an itrativ rocss in which, for a givn itration k, roblm (7) is solvd conditional on a givn varianc of " and thn its valu is calculatd in th nxt itration (until ˆ! _ k convrgnc) as it is shown in xrssion (8) whr N is th saml siz and J is th siz of th choic st, which is assumd to b qual across th saml. Th xtnsion to th cas of diffrnt choic st sizs is obvious. C n ( ˆ ) 1 % ˆ $ _ k+ 1 =! $ ˆ # k (8) JN n, i" 4.3 Using th Latnt-Variabls Mthod to Corrct for Endognity Two Stags On way of using th latnt variabls mthod to addrss ndognity would b to maintain th two stags rocdur of th control-function mthod but with a shift. Th roblm of ndognity coms from th fact that som quality attribut q was omittd in th scification of th choic modl utility. Thrfor, instad of using dirctly th fittd rror of th ric quation as th omittd attribut, on can considr a structural quation whr th omittd quality attribut is a latnt variabl writtn, in an structural quation, as th sum of this fittd rror and an additional rror trm, as it is shown in xrssion (9). q = " ˆ +! (9) Th sam rsult may b attaind if th fittd rrors ar usd instad as indicators of th latnt variabl in a masurmnt quation. This can b asily notd by rvrsing xrssion (9). In both cass th choic utility is scifid as including th omittd quality attribut q, as it is shown in xrssion (10). U =! + X! +! q + q (10) Jun 009 8

10 This aroach can b sn as an imrovmnt to th two stag control-function mthod sinc it addrsss th fact that th omittd attribut dos not corrsonds xactly to th fittd rror of th first stag. Howvr, it is arguably not idal sinc it still rlis on an indndnt OLS stimation to obtain th fittd rrors, losing th otntial gain in fficincy that may b achivd from a joint stimation. If it is assumd that! in (9) is distributd N(0,& I), th liklihood of ach obsrvation in this cas corrsonds to xrssion (11). L LV % Stag n +$ %$ +$ # + + (* ˆ + ) q % 1 ) =!...! + ( + )" d (11) + X + + q * ˆ j C () % $ & n j& Cn + X 1 j Just as it occurrd with FIML, on disadvantag of this aroach is that it rlis on an assumtion about th whol distribution of th rror trm and not just about its xctd valu. This could b an imortant issu whn th tru distribution diffrs sigficantly from th distribution assumd in th latnt variabls modl. It is worth noting also that th stimation of modl (11) involvs solving a multifold intgral in which th numbr of dimnsions is qual to th numbr of altrnativs in th choic st. Sinc th numbr of altrnativs in, for xaml, rsidntial location, may b hug, th solving algorithm will ncssarily involvs Mont Carlo intgration with otntially imortant costs in accuracy. Thrfor, vn though this aroach suoss a thortical imrovmnt from th two stag control-function stimation sinc it rcogzs that th omittd attribut is a latnt variabl, th comutational burdn involvd in its alication may gloom any imrovmnt in ractic. On Stag In an attmt to achiv th simultanous stimation of th control-function modl within th latnt variabl mthod, th following modl can b roosd with th basic ida of using dirctly th information of th instrumntal variabls instad of th fittd rrors of th ric quation. This can b achivd by combing quations (9) and (4) in th following way. q q = # = +! (1) $ Z " +! Again, if it is assumd that! is distributd N(0,&!I), th liklihood of ach obsrvation corrsonds to th following xrssion. L LV % 1 stag n +$ %$ +$ # + + ( % Z * + ) q % 1 ) =!...! + ( % + )" d (13) + X + + q Z* j C () % $ & n j& Cn + + X 1 j Jun 009 9

11 In th following sction th mthods roosd ar tstd and comard in trms of thir ability to corrct for th ndognity in a Mont Carlo xrimnt. 5 MONTE CARLO EXPERIMENT To study th diffrnt altrnativs roosd to nhanc th corrction for ndognity, w crat synthtic data in way that th omission of a quality attribut will ncssarily caus ndognity. Th xrimnt considrs 000 (N) synthtic individuals who choos btwn thr altrnativs. Each individual (n) maximizs its utility (U ), which was assumd to b a linar function of th attributs (a, b, c, a quality attribut q and th ric ) of ach availabl altrnativ (i) and an rror trm ( ). U = 10a + 10b + 10c + 10q! 10 + (14) Th rror trm is constructd to b distributd iid Extrm Valu (0,1) what imlis a Logit form for th robability that individual n chooss altrnativ i. Additionally, ric is dtrmind by th ric quation shown in (15), which is linar in th attributs c, q, z 1, z, and an rror trm % which was assumd to b distributd Normal (0,0.01). +! = 0.5c + 0.5q + 0.5z z (15) Variabls a, b, c and q wr considrd iid Uform (1,) for ach individual and altrnativ. Instrumnts z 1 and z wr considrd iid Uform (0,1). Variabl was gnratd using q. (15), as a function of c, d and th xognous instrumnts z 1 and z. Within this stting, variabls c and q ar corrlatd with ric but not with ithr a or b. Thrfor if, for xaml variabl q is omittd, ric will b corrlatd wit th rror trm which, in this cas, would b qual to! = 10q +. At th sam tim, variabls z 1 and z ar, by construction, ror instrumnts for ric sinc thy ar corrlatd with it, but not with th rror trm summarizs th synthtic data considrd in ths xrimnts. Tabl 1: Summary Statistics of Synthtic Data Corrlation Variabl Man Standard a B c q z 1 z Error a b c q z z !. Th following tabl Using ths synthtic data, svn modls wr stimatd. Th first fiv wr stimatd using th on sourc softwar R (R Dvlomnt Cor Tam, 008). Th first modl (Modl I) Jun

12 corrsonds to a Logit modl in which all th variabls that ar rsnt in th tru modl ar includd. Th stimats of this modl ar shown in th fourth column of Tabl, whr it can b notd that th stimatd cofficints ar statistically qual to th tru cofficints. Th scond modl (Modl II) in Tabl corrsonds to th stimation of modl in which variabl q was omittd from th utility scification. Sinc variabl q is corrlatd with th ric by construction, this modl suffrs of ndognity. As xctd, th cofficint of ric is ositivly biasd. Sinc th scal of th diffrnt modls is not ncssarily th sam, th corrct way to chck that th cofficint of ric is biasd, is by comaring it with th stimatd cofficint of variabls a or b, sinc thos variabls ar indndnt by construction, to all othr variabls in th modl and also to th rror trm. Subsquntly, it can b notd in this cas that th cofficint of is 3 tims smallr (in absolut valu) than th cofficint of a. In th sam way, variabl c is also ushd down (~50%) bcaus it is corrlatd with th ric. Additionally, it can b notd that th log-liklihood of th choic modl L (!ˆ ) is substantially smallr than th on of modl I. Tabl : Mont Carlo Exrimnt. Prformanc of Diffrnt Modl Estimators to Addrss Endognity. Modl I Modl II Modl III Modl IV Modl V Modl VI Modl VII Modl Coff. Tru Includ all Ommit stags Simult Pric q. Lat Vars Lat Vars Valus variabls q C-Funct C-Funct in Utility Stags 1 Stag Choic Modl Pric/Structural Equation ASC1 ASC! a! b! c! q!! z 1! z! µ Intrct " z1 " z " c " d (0.117) (0.0784) (0.116) (0.116) (0.116) (6.01) (0.10) (0.114) (0.0771) (0.113) (0.113) (0.114) (106) (0.115) (0.494) (0.194) (0.484) (0.485) (0.486) (0.3) (0.689) (0.487) (0.19) (0.473) (0.474) (0.475) (0.36) (0.684) (0.504) (0.183) (0.493) (0.51) (0.7) (0.195) (0.0530) (0.51) (0.545) (0.163) (0.539) (0.573) (0.574) (0.164) (0.76) -10. (0.507) (0.539) (1.01) (1.03) (0.084) (0.0660) (0.0107) (0.0107) (0.0107) (0.0064) ( ) (0.0064) (0.173) ( ) ( ) ( ) (0.196) ( ) ( ) ( ) (0.111) N 0 ( ) L ( ) L!ˆ! ˆ! ˆ a! ˆ a! ˆ c (*)Estimator Standard Error in Brackts Th nxt modl (Modl III) is rortd in th sixth column of Tabl and corrsonds to th two stags control-function corrction, as it was dscribd in Guvara and Bn-Akiva (006). In this Jun

13 cas it can b notd that th ric quation cofficints ar statistically qual to th tru valus and th omission of q rsultd in a non-zro intrct. Rgarding th choic modl, it can also b notd that th cofficint of a is again around its tru valu of 10. This occurs just bcaus th rror trm in quation (15) was built to b vry small and bcaus all th tru variabls in (15), but th omittd quality attributs, wr considrd in th stimation of th ric quation. In gnral, w may xct largr rrors in th ric quation and also that som of th tru instrumnts may not b availabl. In that cas, w might obsrv an incras in rror of th choic modl and thrfor, a rduction in th scal (Guvara and Bn-Akiva, 006). Mor imortant than th adjustmnt in scal, it can b notd that th inclusion of th controlfunction as an additional variabl satisfactorily corrctd for th ndognity roblm sinc th absolut valu of th ratio of th cofficints of, both variabls and c with th cofficint of a, ar again around 1 and th cofficints hav th corrct signs. Equally rlvant, th logliklihood of th choic modl is substantially largr than th modl without this corrction and almost qual to th on attaind with th tru modl. Finally it is imortant to rmark that, dsit that variabl c is not corrlatd with q, th omission of q affctd it qually sinc c was corrlatd with. In th sam way, if th ric quation usd to build th control-function dos not includ c as an xlanatory variabl, th mthod will corrct th bias only for but not for c affcting th consistncy of th stimators. Thus, th gnral advic is to us as instrumnts in th ric quation all th modl variabls that ar corrlatd with ric but not with th modl rror. Th following modl stimatd corrsonds to th FIML modl dscribd in xrssion (7), which is labld hr as th simultanous Control-Function (Modl IV). In this alication, th wight btwn th choic modl and th ric quation liklihoods, that is, th invrs of th varianc of th ric quation was calculatd itrativly using xrssion (8). Th itrativ mthod was rfrrd sinc it showd bttr rformanc and stability. Th cost of this otion howvr, is that w had no stimat of th standard dviation of this stimator. First, it can b notd that th log-liklihood of th choic modl in this cas is almost th sam as th on attaind with th modl whr th instrumnts wr just includd as additional variabls in th choic modl. Howvr, in this cas, th ndognity roblm is corrctly solvd sinc th sign of th cofficints of and c ar corrct and thir siz is statistically qual to that of a. Th gain with th joint stimation was a slight imrovmnt of th choic modl liklihood which is accomad with an incras in th standard rrors of th som cofficints of th choic modl. This could b misintrrtd as a otntial rduction, instad of an incras in fficincy rsulting from th simultanous stimation. Th truth howvr is that th stimators of th standard rrors in th two stags control-function wr an incorrct aroximation, sinc thy did not account for th whol variability of th two undrlying modls. Th nxt modl corrsonds to an xaml of simultanous stimation. It can b notd that if th control-function mthod convys th addition of th fittd rror as a variabl in th choic modl, th simultanity could b achivd by just rlacing th ric quation dirctly in th utility function as it is shown in (16). Jun 009 1

14 U = X = Z # +! &) ( U " + $ & T T U U U = X T = # (# + # ) = ~ = # # + # $ + X T ( % Z ") T T # + # $ T ( % Z ") T +! T T $ + X # % Z"# ~ + X # % Z " +! $ +! +! (16) That is, aarntly, th sam rsult attaind with th control-function modl would b obtaind if th instrumnts ar just addd as additional variabls to th choic modl. Th rsults of this modl ar shown in th ighth column of Tabl (Modl V). First it can b notd that th logliklihood of this modl sigficantly largr than th on of modl II, and vn slightly surior to th on attaind with th control-function corrction. This occurs bcaus th modl is now stimatd simultanously. Howvr, modl stimators ar substantially biasd. Not that th ric cofficint has th incorrct sign and that th ratio btwn th cofficint of a and c is now around 100!. It should b rmarkd that an analyst may b dcivd if h or sh is blindly studying th inclusion of additional variabls to th modl using, for xaml a Liklihood Ratio tsts bcaus h or sh would nd u confidntly (but rronously) including z 1 and z as modl variabls. Th qustion of why, if th control-function modl and th modl including z us xactly th sam information nd u with vry diffrnt rsults, riss naturally. Th answr can b rtrivd from quation (16). It can b notd that th tru cofficints can not b idntifid. For xaml, th cofficint of ric in this cas will corrsond to th sum of th tru cofficint of ric (! ) and th cofficint of th control-function (! % ). It can actually b notd that if thos cofficints ar rtrivd from th two stags control-function stimation, thy sum u to aroximatly 10, th actual stimation rsult for th cofficint of ric in th modl in which z is includd in th utility Th nxt st was to stimat th two latnt variabls modls roosd in th rvious sction. Ths modls wr stimatd using th softwar ICLV (Bolduc, 007). To solv th intgral rquird by th latnt variabls modl, this softwar assums a normal distribution of th rror trms and solvs th intgrals using simulation. Th first latnt variabl modl (Modl VI) corrsonds to th imrovd two stag vrsion of th control-function whr th choic modl considrs a latnt variabl that is a function of th fittd rror of th ric quation and an rror trm (11). Th rsults of this modl ar rortd in th nth column of Tabl. It can immdiatly b notd that this rocdur succssfully corrctd th ndognity roblm sinc th ratio btwn th absolut valu of th aramtrs of a, and c ar again around 1 and hav th corrct sign. Th corrction, howvr, tnd to b blow th lvl of rcision attaind with FIML and th two stags control-function mthod. Rgarding th log-liklihood of th choic modl, it can b notd that it is substantially largr than th on of th ndognous modl but not as good as th on of th two stag control-function. On ossibl xlanation for ths rsults is that this stimator convyd th us of simulation to calculat th intgral and, otntially, th rror rlatd with that rocdur may surass th otntial imrovmnts gaind from th considration of th latnt variabl. Jun

15 An additional imortant commnt rgarding th latnt variabls modl is that it was dtctd that th modl was not robust to th starting oint usd for th stimation. In som cass th modl was not stimabl and for othrs it attaind convrgnc to incorrct aramtrs. It can b sculatd that this waknss is xlaind by simulation rror. Additionally, rgarding th comarison btwn th two stag latnt variabls control-function, and th siml control-function mthod, it should b notd that th formr rlis mor havily on th normality distribution of th rror trms sinc in that cas, th whol distribution nds to b intgratd out, whras in th scond, only th man nds to b stimatd. Th variabls in this cas wr built uform instad of normal. Dsit that th modl was stimabl, additional xrimnts (not rortd hr) for which th variabls wr normally distributd showd a slightly imrovd bhavior. Th tru normality of th rror trms in th latnt variabls aroach should howvr lay an imortant rol in th stimation of modls with ral data. Th final column of Tabl (Modl VI) corrsonds to th stimatd aramtrs of th latnt variabls modl dscribd in quations (1). It can b sn that, dsit that th log-liklihood of th choic modl in this cas is substantially bttr than th squntial latnt variabls modl; it dos not satisfactorily corrct th ndognity roblm. Similar to what occurrd with th modl in which th instrumnts wr includd as xlanatory variabls in th choic modl, th ric cofficint in this cas is ositiv, making th modl uslss. On ossibl xlanation for this final rsult is that dsit modl (13) is idntifid, th cofficint of $ that maximizs th liklihood is not ncssarily th sam that solvs th ric quation and, thrfor, will not mak th corrct rojction ndd to corrct for ndognity. Thrfor, what may b ndd in this cas is to includ also th liklihood of th ric quation in xrssion (13). Not that this is just quivalnt to formulat a mixtur of th modl considrd to driv (7). Th xloration of this xtnsions ar byond th ossibilitis of th currnt vrsion of stimation softwar usd for this rsarch and thrfor ar lft for furthr rsarch. 6 CONCLUSION This ar xlord diffrnt altrnativs to addrss ndognity in discrt choic modls combing th control-function and th latnt variabls mthods. Its was found that th most aroriat way to combin both mthods is to considr th omittd attribut as a latnt variabl which can b thn writtn ithr as a function of th fittd rrors of th ric quation in a structural quation, or altrnativly, as th right hand sid of a masurmnt quation in which th fittd rrors ar usd as indicators. Undr this framwork, fiv altrnativ mthods wr analyzd by mans of a Mont Carlo xrimnt. In this xrimnt, ndognity was cratd, in a trinomial logit modl, by th omission of an attribut of th systmatic utility that was corrlatd with on of th rmaing attributs. Jun

16 Th first mthod corrsondd to th two stags control-function mthod as it was alid by Guvara and Bn-Akiva (006). Using this mthod it was ossibl to corrct for th ndognity roblm and rcovr also th scal. Howvr it has to b ointd out that scal may not b rcovrd if, for xaml, only on of th instrumnts is usd as an instrumnt. Th scond mthod corrsondd to FIML or simultanous control-function mthod, in which th liklihood of both th choic modl and th ric quation ar maximizd simultanously. Th rsults showd that th ndognity roblm can also solvd in this cas with also a rlativly imrovmnt in fficincy. Th third modl corrsondd to th substitution of th ric quation in th choic utility. Th xrimnt showd that, if ndognity xists, whn th instrumnts ar dirctly includd in th utility function th roblm of ndognity is not solvd at all, but th liklihood of th modl can b substantially imrovd what may dciv th analyst. Th fourth mthod corrsondd to th two stags latnt variabl modl in which a latnt variabl in th choic modl utility was considrd to b a function, through a structural quation, of th fittd rror of th ric quation. Dsit this modl corrctd for th ndognity roblm, its rformanc was blow th two stags control-function and th FIML, arguably, bcaus th comutational burdn associatd with th calculation of th multifold intgral. Th last mthod considrd corrsondd to th stimation of th latnt variabls aroach in a singl stag whr instad of considring th fittd rrors from an OLS stimation of th ric quation, its cofficints ar stimatd simultanously as art of th structural quation. As with th cas whr th instrumnts wr dirctly includd in th utility, this modl rsultd in a sigficant imrovmnt of th liklihood but, did not corrct at all for th ndognity roblm. Finally, som rlvant futur lins of rsarch can b idntifid. Th first xtnsion should considr th analysis of th rlativ rformanc of th diffrnt mthods undr diffrnt simulatd data including diffrnt saml sizs and numbr of altrnativs. A scond ara of rsarch corrsonds to th analysis of mthods to rduc th comutation burdn associatd with th stimation of th intgrals in th latnt variabls aroach is also anothr otntial lin of rsarch. Finally, th stimation of th on stag latnt variabl mthod rsntd in this ar whr, additionally, th liklihood of th ric quation is considrd simultanously, aars a rasonabl xtnsion which may otntially addrss th ndognity roblm. Jun

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