Section 14 Limited Dependent Variables

Transcription

1 Sectn 14 Lmted Dependent Varables What s a lmted dependent varable? Our standard assumptn f an errr term that s nrmally dstrbuted cndtnal n the regressrs mples that the dependent varable can be (wth pstve prbablty) any real number, pstve r negatve Lmted dependent varables are dependent varables that have lmted ranges: usually ether dscntnuus r range bunded There are many mdels f LDVs based n what the lmtatns are: 0/1 dependent varables (dummes) by prbt and lgt Ordered dependent varables by rdered prbt and lgt Categrcal dependent varables (wth mre than tw categres) by multnmal lgt Truncated dependent varables by Heckman s prcedure Censred dependent varables by tbt Cunt (nteger) dependent varables by Pssn regressn Hazard (length) dependent varables by hazard mdels Because f the lmted ranges f the dependent varable, the standard addtve nrmal errr s nt tenable fr these mdels Instead we must mdel the prbablty f varus dscrete utcmes LDV mdels are usually estmated by maxmum lkelhd, gven the assumed dstrbutn f the cndtnal prbabltes f varus utcmes Bnary dependent varables Fr bnary dependent varable: E y x y x Pr 1 Lnear prbablty mdel: usng OLS wth a bnary dependent varable E y x x x We can mdel, as usual n OLS, 1 Shw graph f Pry 1 x K K as lnear functn f x Hwever, we can t ust stck a nrmal errr term nt ths functn If we wrte 1 y E y x e x x e, then snce y s ether zer r ne, K K e can nly take n the values x x x x 1 K K and 1 K K ~ 139 ~

2 E y x Pr y 1 x x x, then the errr term must If 1 K K have a cndtnal Bernull dstrbutn wth Pr e x 1 x Pr e x x, and Sums f randm varables wth the Bernull dstrbutn d cnverge t nrmal, s the ceffcent estmates wll stll be asympttcally nrmal Hwever, the mmedate prblem wth ths s that the lnear functn 1 x KxK wll nt le n the range [0, 1] that s requred fr prbabltes fr all values f x Ths prblem s mrrred by the fact that the predcted values f y fr sme bservatns s lkely t be utsde [0, 1], whch des nt make sense as a predctn f Pr[y = 1 x] Shw dagram f straght-lne predctn f prbablty and pssblty f predctns utsde f [0, 1] Fnally, there s heterskedastcty n the mdel as the varance f the Bernull errr term s x 1 x, whch vares wth x Ths s easly accmmdated wth rbust standard errrs Bttm lne n lnear prbablty mdel: Smple Prbably OK as lng as x s clse t sample means, s that predcted Pr[y = 1 x] stays n [0, 1] Nt the best mdel when the dependent varable s bnary Lgt (lgstc) and prbt regressn These are the standard mdels when the dependent varable s bnary They dffer nly n the assumed dstrbutn f the errr term and are n practce vrtually equvalent Structure f the mdels: E y x y x G x x Pr[ 1 ], 1 K K where G s a cumulatve prbablty functn that s ether z 1 e Lgstc: Gzz fr the lgt mdel r z z 1 e 1 e z z 1 1 Nrmal: Gz z d e d fr prbt Draw graph f cumulatve dstrbutn functn and shw nterpretatn f z and mpled prbablty f y = 1 Cmpare t lnear prbablty mdel s assumptn f lnear relatnshp between z and prbablty Shw hw actual data pnts wuld lk n ths graph Estmatn f prbt and lgt mdels: ~ 140 ~

3 These mdels are always estmated by (nnlnear) maxmum lkelhd The (dscrete) densty functn f y cndtnal n x s y 1 y f y x, G x 1 G x, y 0,1, whch can be rewrtten less cmpactly (but mre ntutvely) as Pry 1 x, Gx, Pry 0 x, 1 Gx The lkelhd functn, assumng that all bservatns n the sample are N y IID, s 1 y L y x G x G x, 1 1 The lkelhd maxmzatn s always dne n terms f the lglkelhd functn: N ln L y, x y ln G x 1 y ln 1 G x 1 Ths functn can be evaluated fr any chce f By searchng ver the parameter space fr the value f that maxmzes ths value, we can calculate the lgt r prbt ceffcent estmatr as the that leads t the hghest value f the lkelhd functn Maxmum lkelhd estmatrs are knwn t be cnsstent, asympttcally nrmal, and asympttcally effcent under bradly applcable cndtns Calculatng the standard errrs f the ceffcent estmatrs s cmplcated, but s handled by Stata The asympttc cvarance matrx f any MLE s the nverse f the nfrmatn matrx : 1 ˆ ln L ; Y, X cv I E The nfrmatn matrx nvlves the expected values f the matrx f secnd partal dervatves f the lg-lkelhd functn wth respect t the parameters It can be apprxmated fr the sample numercally t get an estmated cvarance matrx fr the parameter vectr Hypthess tests n ths, as n any ML mdel, are easest as lkelhd rat tests: ln Lu ln Lr ~ q Stata test cmmand als wrks and des ˆ c a Wald test: t ~ t ˆ NK where the t dstrbutn s asympttc se Gdness f ft: Fractn predcted crrectly: 1 ~ 141 ~

4 If yu take the predctn f y t be 1 f G x ˆ 05 and zer therwse, then yu get a predctn f zer r ne fr each y The fractn predcted crrectly s ust what t sunds lke Pseud-R : In the sprt f the usual R, ths s ln L ˆ ; x, y 1, Z y,0,0,,0 ln L ; x, y Z [Nte: Ths frmula s very strange and lks upsde dwn, but t s nt The reasn t lks werd s because we are takng the rat f lgs (we usually subtract them) Because (wth a dscrete dependent varable) the lkelhd functn s a prduct f prbabltes, t s always less than ne Ths means that the lgs are negatve, wth the denmnatr beng mre negatve than the numeratr Ths, an mprvement n ft ncreases the lkelhd n the numeratr by decreasng ts abslute value, makng the rat smaller and the R value clser t ne] Ths rat s the lkelhd functn wth the best parameter estmate dvded by the lkelhd functn f we ust predct each y by the sample prprtn f y values that are ne Interpretatn f n prbt and lgt regressns: E y x In the usual OLS mdel,, whch s what we are x nterested n knwng z In prbt r lgt mdel, x s nt n useful unts because z has n drect nterpretatn Use graph t demnstrate as hrzntal mvement What we re nterested n knwng (fr a cntnuus regressr x) s Pry 1 d Pr y 1 z Gz gz, where g s the x dz x prbablty densty functn asscated wth the cumulatve dstrbutn functn G ~ 14 ~

5 Graphcal nterpretatn: measures hrzntal mvement due t unt change n x; G (z) measures effect f unt hrzntal mvement n prbablty f y = 1 They have the same sgn, s tests f = 0 are equvalent t tests Pr y 1 f 0 x z e g z z 1 1 z z G z G z 1 e The results f the lgt mdel s ften expressed n terms f dds rats: x e Pry 1 xx x 1 e x x 1e x e Fr lgt, 1 Pr 1 x y x 1x Pry 0 x x e x e x e e x x x x "dds rat" s the effect f x n the lg dds rat 1 g z z e Fr prbt, 1 z Because they are densty functns, g(z) 0 fr all z, s the Pr y 1 partal effects have the same sgn as x Fr dummy regressrs, we are nterested n Pr y1 x 1 Pr y1 x 0 In Stata: prbt reprts the ceffcents and dprbt reprts the partal effects The regressn s dentcal fr each Nte that the partal effects depend n z and thus n x Yu can specfy the values at whch t evaluate the partal effects n dprbt wth the default beng at the means Partal effects f dummy varables are reprted (by default) as dfference n prbabltes abve, wth ther varables at means In Stata: lgt reprts ceffcents and lgstc reprts the ddsrat e ˆ (Ths s really the prprtnal effect f the varable n the dds rat, nt the dds rat tself) ~ 143 ~

6 x x x If x ncreases by ne, e ncreases t e e e, s ˆ e measures the estmated prprtn by whch a neunt change n x changes the dds rat Interpretatn can be trcky: All e values are pstve A zer effect means that = 0 and e = 1 A varable that reduces the dds rat has a < 1 A varable that ncreases the dds rat has a > 1 Example: If e = and the ntal prbablty p f y = 1 fr ths bservatn s, (s the ntal ddsrat p/(1 p) s () / (8) = 05), then a ne-unt ncrease n x multples the dds rat by e =, makng t 05, whch means that the prbablty f y = 1 has ncreased frm 0 t 0333 = 05/(1 + 05) If we d the same example fr an bservatn wth an ntal p = 05, then the ntal dds rat s 1, the unt ncrease n x multples t by, makng the new dds rat, and thus the prbablty has ncreased frm 05 t /(1 + ) = 0667 Relablty f prbt and lgt estmatrs Omtted-varable bas Ths s mre f a prblem n prbt and lgt mdels because a ceffcent f an ncluded varable can be ncnsstent even when t s uncrrelated wth the mtted varable Heterskedastcty Agan, mre f a prblem n prbt and lgt because the standard MLE based n an assumptn f hmskedastcty s ncnsstent Yu can use the Whte rbust estmatr fr the cvarance ( rbust standard errrs ), but yu are calculatng a vald standard errr fr a ceffcent that des nt cnverge t the true parameter value, s t s f less utlty than n OLS, Hw t deal wth these ssues? Be careful abut mtted varables Try t specfy the mdel n a scaled way that makes varance as cnstant as pssble ~ 144 ~

7 Dscrete-chce dependent varables What f there are mre than tw chces? Instead f the dependent varable beng whether smene attends Reed r nt, t culd be whether smene attends Reed (y = 3), attends anther prvate cllege (), attends a publc cllege (1), r desn t attend cllege at all (y = 0) Ths wuld be fur chces rather than tw Ths s an unrdered-chce mdel: There s n bvus rder t these chces If we defne y as abve and say that changes n characterstcs f the ndvdual (nt f the chces) x (say, hgher SAT) that make y mre lkely t mve frm 0 t 1, we can t als be cnfdent that these changes n x are mre lkely t make y mve frm 1 t r frm t 3 Multnmal (plytmus) lgt mdel (Greene 6/e, sectn 311) x e Pr y x, where there are M dstnct chces Ths mdel has M x m e m 1 M(k + 1) parameters, but nly (M 1)(k + 1) f them are unque because the sum f the prbabltes must be ne (If an ncrease n famly ncme rases the prbabltes that yu wll chse y =, 3, and 4, t must lwer the prbablty f chsng y = 1 by an equvalent amunt Thus,, 1 can be determned frm,,, 3, and, 4 Where the secnd subscrpt refers t the chce and the frst t the ndependent varable) We usually nrmalze by settng the vectr 1 = 0, whch makes the numeratr f the prbablty fractn 1 fr chce 1 Pr y x In the multnmal lgt mdel, ln x Pry 1 x The ceffcents thus can be nterpreted as the effect f x n the lg dds rat Independence f rrelevant alternatves assumptn s mplct n multnmal lgt mdel It shuldn t matter fr the ceffcents f the attendng-reed equatn whether ne adds attendng Lews & Clark as a specal case f attendng a prvate cllege (makng 5 alternatves) r nt Ths assumptn may nt be reasnable n sme cases, makng the mdel napprprate Multnmal lgt mdels can be estmated by maxmum lkelhd methds In Stata, use mlgt Related mdels: Cndtnal lgt mdel: The x varables relate t prpertes f the chces nstead f r n addtn t the ndvdual (Nt clgt n Stata; that s smethng else) ~ 145 ~

8 Nested lgt mdel: Decsns are nested Fr example, decsn whether t attend cllege, then f attendng whether t attend Reed, anther prvate cllege, r a publc In Stata, use nlgt Multnmal prbt: Same thng wth nrmal rather than lgstc functn Very tme-cnsumng t estmate, s t s nt used ften Ordered dependent varables Many varables are rdnal n nature: we knw that 4 s bgger than 3 and that 3 s bgger than, but we dn t knw the 4 s the same amunt bgger than 3 as 3 s bgger than Examples wuld nclude bnd ratngs, pnn-survey respnses, academc actns, and perhaps grades and SAT scres We can thnk f the rdnal dependent varable y as representng levels f the utcmes f sme underlyng latent varable y * * We assume that y x e, and that we bserve the rdnal chce y : * 1 f y 1, * f 1 y, * y 3 f y 3, * M f M 1 y If the errr term s nrmal, then we can use rdered prbt t estmate the vectr and the threshlds crrespndng t the dfferent levels f the varables Ordered lgt s used when the errr term fllws the lgstc dstrbutn Ordered prbt/lgt nvlves estmatng the vectr and the threshld values 1 thrugh M 1 by maxmum lkelhd If we nrmalze the mdel t gve the errr term unt varance (dvde y and x by the standard devatn), then we have y x1 x y x x1x y x x x Pr 0 Pr 1 Pr 3 y M x x Pr 1 M1 M, ;, Pr,, where The lkelhd functn s L y x Iy m y m x I(y = m) s an ndcatr functn that s ne f the cndtn s true and the prbablty s gven by the frmulas abve The lkelhd functn s maxmzed by searchng ver alternatve values f and t fnd thse that maxmze m1 ~ 146 ~

9 Shw Greene (6/e) Fgure 34 frm p 833 Partal effects: what des mean? As n the standard prbt and lgt mdels, s the dervatve f the unbserved y * wth respect t x We can derve margnal effects f x n the prbabltes f y beng each value as Pr y 1 x 1 x, x, Pr y x 1 x x x,, Pr y M x M 1 x x, In Stata, yu can use the margns cmmand after prbt t get margnal effects (I m nt sure exactly hw ths wrks) Predct gves predcted prbabltes f each level fr each bservatn ~ 147 ~

10 Cunt dependent varables Cunt dependent varables can nly take n nn-negatve nteger values Nrmal dstrbutn s nt a plausble chce Pssn dstrbutn s ften used fr cunt mdels: n e Pr y n x n! Pssn dstrbutn has mean and varance bth equal t, s E y x x The lg-lkelhd functn s ln L e yxln y! and we estmate as usual by maxmzng ths functn Interpretatn f ceffcents: Ey x x e x In Pssn regressn we mdel Pssn regressn s mplemented n Stata by the pssn cmmand Lmtatn f Pssn regressn The fact that the cndtnal mean and cndtnal varance f the Pssn dstrbutn are the same s restrctve If t desn t ft yur data well, then a mre general mdel mght be apprprate ~ 148 ~ X e N 1

11 The mst cmmn alternatve s the negatve bnmal regressn mdel, whch s mplemented as nbreg n Stata Tbt, censred, and truncated regressn mdels These three mdels are very easy t cnfuse! All nvlve stuatns where we have n bservatns frm sme regn f the (usually nrmal) dstrbutn Example: Smetmes we have crner slutns n ecnmc decsns: many peple chse t cnsume zer f many cmmdtes (Ths s the tbt mdel) Example: Smetmes surveys are tp-cded wth the maxmum respnse beng 50,001 r smethng lke that (Ths s censred regressn) Example: If the dependent varable s duratn untl death f patents after treatment, sme patents wll nt yet have ded (Anther censred regressn) Example: Sme events sell ut, meanng that the bserved values f demand are censred at the ste capacty (Yet anther censred regressn) Example: Sample cnssts nly f peple wth values f y belw a lmt c (Ths s truncated regressn mdel) Tbt estmatr fr crner slutns Suppse that sme fnte fractn f bservatns chse zer, but thse chsng pstve quanttes fllw the remander f the nrmal dstrbutn (lppng ff the left-end f prbablty) Why can t we ust use OLS? Lke lnear prbablty mdel, we gnre restrctns n the dstrbutn f e and we predct values < 0 Why can t we ust use the bservatns wth y > 0? Ths wuld mply selectn n e because we d be mre lkely t elmnate bservatns wth e < 0 Why can t we use ln(y)? Observatns wth y = 0 wuld have ln(y) = * We can mdel ths wth a latent varable y 1x e as a latent underlyng * * y, f y 0, varable wth a nrmal dstrbutn and y as the bserved 0, therwse utcme Ths varable has a censred dstrbutn wth fnte prbablty f a zer utcme but therwse nrmally dstrbuted ver the pstve values ~ 149 ~

12 The cndtnal densty f y s 1 y x f y x fr y > 0, and y x x Pr 0 1 / Ths densty s the bass fr the tbt estmatr f the vectr Tbt maxmzes (ver, ) the lg-lkelhd functn: x 1 y x ln L, ; y, x ln 1 ln y : 0 y : 0 The lmt value (zer here, but t culd be sme value c) must be specfed Can als have dstrbutns that are censred abve, r bth abve and belw (perhaps the share f merlt n ttal wne cnsumptn), where sme peple chse zer and sme chse ne) Interpretng tbt ceffcents There are tw expected values f nterest n the tbt mdel: Cndtnal (n y > 0) expectatns 0 0, E y x y E y y x Draw graph shwng censrshp at 0 and densty functn ver y > 0 = 0 y f y 1 y y Remarkable and useful prperty f standard nrmal c dstrbutn: Ez z c 1 c y > 0 ff e > x and e s (by assumptn) dstrbuted nrmally wth mean 0 and varance Thus e / s e e c standard nrmal and E c 1 c Cndtnal n x, E(x) = x, s E y y 0, x xee e xx e e x xe x x x, x where we use the prpertes that ( z) = (z) and 1 ( z) = (z) ~ 150 ~

13 We defne the nverse Mlls rat as c Then c c x E y y 0, x x s the cndtnal expectatn f y gven that y s pstve Uncndtnal (n y > 0) expectatn (whch s stll cndtnal n x) E y x : E y x 0Pr y 0 x E y y 0, xpry 0 x Ey y 0, x Pry 0 x x e x x Pr x x x x x x x Interpretatn f? E y x In the usual OLS mdel, x Here, / x/ x / E y y 0, x x/ x/ x x x c c c c c c c By qutent rule, But (c) = (c) by defntn and usng the defntn f the nrmal densty functn, (c) = c (c), s c c c c c c c c c c c c c ~ 151 ~

14 Therefre, E y y 0, x x x x x The expressn n braces s between 0 and 1, s the effect f x n the cndtnal expectatn f y s f the same sgn as but smaller magntude Testng = 0 s a vald test fr the partal effect beng zer Gven that E y x E y y x y x 0, Pr 0, 0, E y x E y y x Pry 0 x x x Pr y 0 x E y y 0, x x x Pry 0 x x x E y y 0, x x x x x S (wth all,, and functns evaluated at x /) E y x x x x x x x Dng tbt estmatn n Stata tbt depvar ndvars, ll(0) des tbt wth zer lwer censrshp ul( ) ptn specfes pssble upper pnt f censrshp After estmatn, can use the predct cmmand t generate sme useful seres: predct, pr(0, ) gves the predcted prbablty that each X bservatn s nt censred, PrY 0 X predct, e(0, ) gves the predcted value f each bservatn EY Y 0, X cndtnal n nt beng censred, ~ 15 ~

15 predct, ystar(0, ) gves the uncndtnal predcted value f each E y x bservatn New margns cmmand des what used t be dne by mfx: mfx cmpute, predct( ) calculates the margnal effects f each varable n the tem specfed n the predct ptn: mfx cmpute, predct (pr(0,)) gves the margnal effects f each Pr y 0 x varable n the prbablty f nt beng censred: x mfx cmpute, predct(e(0,)) gves the margnal effects n the E y y 0, x expectatn cndtnal n nn-censrshp: x mfx cmpute, predct (ystar(0,)) gves the margnal effects f Ey x each varable n the uncndtnal expectatn f y: x In mfx, the default s t evaluate at the means f the x varables Yu can verrde ths wth the at( ) ptn (lstng chsen values f all varables n rder) Censred regressn (tp-cdng prblems, unexpred duratn mdels, etc) We have data n the x varables fr all bservatns, but have n bservatns n y fr thse at ne end (r bth ends) f the dstrbutn If y > c, then we bserve c Let y x e, where e s hmskedastc nrmal We dn t bserve y but nstead bserve w y c mn,, where c s a knwn cnstant that can vary wth Nte dfference frm tbt mdel: In tbt a fnte fractn f peple chse the lmt value Here they chse smethng cntnuus utsde the lmt but we smply d nt bserve t Ths means that we dn t have t mdel the censrshp as part f the chce, rather nly accunt fr t n the estmatn based n ur flawed data Fr uncensred bservatns, we have the usual dstrbutn f y: 1 y x f w x f y x ~ 153 ~

16 Fr censred bservatns, Pr w c x Pr y c x Pr e c x x c x 1 S lkelhd functn s same as the tbt mdel, as s estmatn Hwever, n the censred regressn case we dn t need t wrry abut peple chsng the lmt value, we nly wrry abut bservng t Thus, s the effect f x n y, perd We dn t need t hassle wth the margnal effects calculatns as n the tbt mdel Cnsequently, we can use the Stata tbt cmmand and ust neglect the mfx cmmand afterward Truncated regressn mdels Truncated regressn dffers frm censred regressn n that nether y nr x s bserved fr bservatns beynd the lmt pnt Thus, we cannt use these data pnts at all, makng the tbt estmatr mpssble t calculate Ths s a sample prblem agan, but truncatn f the sample (all varables) s mre severe than censrshp f a sngle varable because we have less (n) nfrmatn abut the mssng bservatns In the censred mdel, we can use the x values f the censred bservatns t determne what knds f bservatns wll be n the censred range In the truncated mdel, we dn t have that nfrmatn Truncated regressn mdel y 0 xe, e x ~ N(0, e) IID assumptn s vlated: We bserve (x, y ) nly f y c, where the truncatn threshld can vary wth and can depend n x The cndtnal densty functn f y gven that t s n the sample (> c ) y x f y x,, e e s g y x, c, y c F c x,, e c x e The functn n the denmnatr s the prbablty that bservatn s nt censred, gven x and c We dvde by ths t redstrbute the truncated amunt f prbablty ver the remanng densty functn The lg-lkelhd functn s ust the lg f ths densty summed ver all the bservatns n the sample OLS n ths case wuld gve slpe estmates that are based tward zer Incdental truncatn and sample selectn ~ 154 ~

17 Sample selectn des nt bas OLS estmatrs unless the selectn crtern s related t e S selectn based exclusvely n x r n smethng utsde the mdel that s uncrrelated wth e des nt present a prblem Incdental truncatn ccurs when we bserve y fr nly a subset f the ppulatn that depends nt n y but n anther varable, but the ther varable s crrelated wth e The prmary (nly?) example n the lterature s y = ln(wage ffer), whch s bserved nly fr peple wh wrk But peple wh have unusually lw wage ffers (gven ther ther characterstcs) are less lkely t wrk and therefre mre lkely t be truncated, s the varable determnng truncatn (wrk status) s crrelated wth the errr term f the wage equatn y xe 1, f wu 0, z 0, therwse z s a sample ndcatr that s ne fr bservatns fr whch we bserve y and zer therwse We assume that E e x, w 0 and x s a strct subset f w We als assume that u s a standard nrmal that s ndependent f w, but that t may be crrelated wth e Ey w uxe e w u x Ee u Let Ee u u,, wth beng a parameter f ther nt nrmal dstrbutn (related t the crrelatn) Ths means that Ey w, uxu, E y w, z xe u w, z Snce ur sample s the set f bservatns fr whch z = 1, we need the expected value f y cndtnal n z = 1, and by lgc smlar t that used E u w, z1 w, where s the nverse Mlls n the tbt mdel, rat / Thus, EY Z s X Z, 1 We can t bserve the term unless we knw The Heckt estmatr s a twstep estmatn prcedure fr estmatng frst, then The selectn varable z fllws a prbt mdel: Prz 1 Pr wu Pr u w Pr u w w ~ 155 ~

18 Thus, we estmate the sample-selectn equatn as a prbt f z n w, usng all f the bservatns (because we dn t need t bserve y fr ths equatn and we bserve z fr all bservatns) We then cmpute the estmated nverse Mlls rat fr each bservatn ˆ as ˆ w ˆ w w ˆ We can then estmate by runnng OLS n y ˆ x e usng nly the bservatns fr whch y s bserved The nclusn f the estmated nverse Mlls rat n the rght-hand sde crrects the bas due t sample selectn and makes the estmates cnsstent and apprxmately nrmal Testng = 0 wth a standard t test s a vald test fr whether there was sample selectn Nte that the regular OLS standard errrs are ncrrect because they assume that s exactly knwn There wll be errr n estmatng by ˆ, s ths errr needs t be taken nt accunt n calculatng the relablty f ˆ Stata cmmand heckman cmputes heckt estmatr ether by full maxmum lkelhd r by the tw-step estmatn methd Ths wll crrect the standard errrs In rder t apply ths mdel relably, there must be at least ne varable that determnes sample selectn that des nt affect y In the wage equatn, t s usually assumed that famly varables such as number f chldren wuld nt affect the wage ffer but wuld affect a persn s chce f whether r nt t accept t and wrk ~ 156 ~