3.1 Introduction to Multinomial Logit and Probit

Transcription

1 ES3008 Ecooetrcs Lecture 3 robt ad Logt - Multoal 3. Itroducto to Multoal Logt ad robt 3. Estato of β 3. Itroducto to Multoal Logt ad robt The ultoal Logt odel s used whe there are several optos (ad therefore possble outcoes) that caot logcall be ordered, ad the odellg of ths s based o the utlt theor wor of McFadde. A exaple would be how people travel to the Uverst wth the choces beg walg, cclg, b bus, b car. Let: x be a vector of attrbutes of a alteratve to dvdual, e.g. how far the are fro the Uverst ad hece how log the have to wal, ccle, or how frequet ad the cost of the bus; s be a vector of persoal characterstcs of dvdual ; ad z be a fucto of x ad s : z z x, s. The utlt of the th dvdual choosg alteratve s the assued to be: u u z Vz Fgure. Where V z s the explaed (or ea) ter, ad s a stochastc ter. The dvdual wll choose over all other alteratves f: Fgure. V uz uz z V z V z V z E.g. V z = 6+ =8 ad Vz 3+4=7. The dvdual wll choose opto because the dfferece betwee the predcted utlt (6-) s ot outweghed b the dfferece the rado copoets. Now f stead V V 3+4=7, the the rado eleet would z z = 6- =4 ad doate the predcted eleet ad the dvdual would choose. I the terolog of the last le of (.): -(-) > 6-3; the equalt (.) does NOT hold.

2 Thus the probablt for the th alteratve to be chose s: r Vz Vz Fgure.3 Let there be ust two alteratves, ad let G be the cuulatve dstrbuto fucto of the dfferece. The: Fgure.4 Assue that: Fgure.5 r V G V z V z V z V z z z Assue also that s ot a fucto of z [sees ocuous] ad that ad [the error relatg to choces ad ] are statstcall depedet [ore questoable]. If G s logstc we have: exp z z Fgure.6 Ths gves rse (or s equvalet) to the Logt odel. What probablt dstrbuto do we eed to assue for ad to geerate the Logt odel? ad have to be depedet Webull. For ore o the Webull dstrbuto see p.58 Chow (...) ad also for the oe paraeter Webull dstrbuto. Oe of ts plcatos s that the geeral case of alteratves [ote the th subscrpt has bee otted, t referred to the dvdual ad V has replaced V(Z ) ]:

3 r V V exp V exp V,...,, Fgure.7 where s the sole paraeter of the relevat Webull dstrbuto. Defg V z so that V z [.e. lettg the costat α becoe absorbed wth the costat ter β z], we ca use Fgures.3 ad.7 to get: Fgure.8 exp z exp z Ths gves the probablt that a dvdual wll choose aog alteratves, accordg to the ultoal Logt odel. For, ths s reduced to: exp exp z z exp z exp z z Fgure.9 Ths, aga, correspods to the Logt odel, as also show Fgure.6. A plcato of the Logt odel s that: Fgure.0 exp z exp z For exaple, f a cosuer s choosg betwee a rado, a T.V. set, or a stereo, the regresso ca cocetrate o the frst two probabltes to the excluso of the thrd. 3. Estato of β See p.36 Maddala 3

4 Let be a du varable, tag a value of f dvdual chooses alteratve ad 0 otherwse. It s what we call a dcator varable. The the lelhood fucto for the ultoal logt odel ca be wrtte as: L... Fgure. Ths ght loo le we are cludg the lelhood fucto ot ol the probablt that the dvdual waled to the Uverst [whch the dd], but also the probablt the ccled, cae b car etc. Yet surel the lelhood fucto wors b axsg the probablt of people dog what the do, ot what the do ad also what the do ot do. But f walg s opto the = ad all the other s wll equal zero. Hece ths product ter ol pacts o the lelhood, all the other probabltes are rased to the power 0 ad hece gored. Therefore axu lelhood chooses paraeters to axse the lelhood of the dvduals havg ade the choces that the actuall ae. Recallg that. ca be wrtte as: L [ote there are ow alteratves] The tag logarths: Fgure. log L log log z exp z exp z log exp z Tag the frst dervatve of ths: 4

5 log L z exp z z exp z Fgure.3 Ths s doe b usg the rules lx/ x = x - ad e x / x = e x. Set ths equal to zero ad chec the secod order dervatve to esure t s a axu pot (see p.6 Chow () or p.37 Maddala ()). Useful refereces: McFadde: Chow: Ecooetrcs (McGraw-Hll Boo Copa, New Yor, 983); Maddala: tatter.pdf 5