Censored Models. Li Gan. April, Examples of censored regression: (1) stadium attendance= min(attendance *, capacity)

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1 Censore Moels L Gn Aprl 8 mples o ensore regresson: stm ttenne mn(ttenne pt) Here gropng (those oservtons wth ttenne rtes less thn one vs those oservtons wth ttenne rtes t one) re lerl se on oserve tors. Top-ong: welth mn (welth mllon) We re ntereste n nerstnng how welth s etermne: welth where ~ log-norml Corner solton ensore regresson (no t oservles) (3) Testng sores: The Tes Assessment o Knowlege n Sklls (TAKS) tpll hs 4 or 43 mltple hoe qestons. I stent gets ll orret then heshe gets mmm sores. We re ntereste n how testng sore or stent t shool j s ete mong others per stent spenng t shool j: sore γ PerStentSpenng j j Oserve sores re: sore mn{ sore mmm possle sore} j j j j mple (hrtle ontrtons): M q ( q) log(q ) s.t. p q m n q where s the nnl onsmpton q s the nnl hrtle gvng n α s the mrgnl tlt rom gvng. In ton m s the ml nome n the p s the ollr pre o hrtle ontrton epenng on the mrgnl t rte o the person. For emple or person wth mrgnl t rte o 3% hs p s.7. Plg the get onstrnt n the tlt nton tke the ervtves wth respet to q the rst orer onton:

2 p α q The solton to ths prolem s: q p q p p I we re ntereste n wht hrtersts wol etermne the hrtle gvng we moel ep( γ ) we hve or estmton moel: log(q ) m( γ log(p ) ) Now we re ntereste n nerstnng hrtle gvng mong lt t the Tes A&M Unverst. There re two ws to otn smple. We rnoml rw N people rom the nverst s ontng oe. The oe hs etle normton o ll lt memers on mps. We rnoml rw N lt n ll them. For eh lt we sk the mont o ther hrtle gvng. Inevtl some lt wol rese to nswer ths qeston (reore s RF ). In generl sppose we hve set o normton ( ). The poplton moel s: n re nstrments. Let s e: s elong to one smple; n s elong to nother smple. The ke erene etween Hekmn s smple seleton n ensore regresson s tht the seleton s rnom. More spell or the ensore moel: (s). Or s s nepenent o n. I so then: Qeston: (s)? (s). Conser the lssl emple: n the wge regresson or women. log(wge ) (3) Cse : s or those women who work (we onl oserve those women who work). Cse : s women s wge s hgher thn mnmm wge wge.

3 In Cse : t s lkel tht ( work) ( not work). In other wors t s lkel tht Cov( s ).e. those wth lrger re more lkel to hve s. In ths se ovosl (s). So the ollowng regresson sng log(wge ) on s lkel to e se. In Cse : (s ) (wge wge ) ( wge ) ( wge -). However (s) vs ( wge -) re erent. stmtng ensore moels oes not reqre nstrmentl vrles whle estmtng ensore moels oes. To n ot the smple seleton prolem ests n mportnt qeston to sk s: I person (or rm or hosehol) m hoose tsel nto grop or the person s selete others to e n the grop? I the nswer s es then t s most lkel smple seleton prolem. Otherwse t s ensore moel. A generl ensore regresson moel: m( ) or: s ( ). A lssl smple seleton moel: s Oserve [ v ] [ s ] Cov( v ) Note tht there mst e nstrmentl vrles n the s -eqton. In generl there re two methos to estmte sh moels.. The regresson metho: To onstrt regresson moels t s neessr to n ot ontonl epetton: ( s ). It s sel to work ot how the ontonl epetton or the stnr norml. Sppose ~ N(). 3

4 ep ep ep Pr π π π Gven ths we hve: Smlrl one n otn the epetton:. Thereore: Prevos eqton sggests nonlner regresson metho. For those oservtons tht :. (4) Note n ths nonlner eqton moel the prmeter set ppers t oth the lner prt o the moel n the nonlner prt o the moel:. In ton onl the nensore prt o the normton ( ) s se. I we re ntereste n pplng ll oservtons n sppose tht s ensore t n we oserve or ll : 4

5 m( ). Thereore Pr Pr So nonlner regresson metho tht pples ll t s gven (ssme ): (5) Aorng to eqtons (4) n (5) sng OLS o on n ether the s-smple or the whole smple wol le to the se estmtes. Alterntvel one m ppl Hekmn-tpe two step lest sqres.. stmte Prot o vs : Let the oeent e γˆ.. stmte lner regresson For the s-smple : v γ γ λ ˆ ˆ Dsssons: Aorng to (4) the oeents n γ shol e the sme. Ths proere oes not grntee tht the two oeents to e the sme. A speton test H : γ n e perorme here.. The Mmm Lkelhoo stmton metho: Agn onser ensore regresson moel: m( ). The enst s gven : 5

6 ( ) ( ) ( ) ` Ths s the Tot moel. In the hrt emple (eqton ) the lkelhoo nton s gven : q s the mont o mone gven to hrtes. ( q p ) γ log( p ) ( q ) q log( q ) γ log( p ) ( ) mple: nogenos eplntor vrle moel: m ( α ) v Note tht n v re orrelte. Rewrte the s: θv Plg t nto the prevos eqton: m ( α θv ) Ths sggests two-step proere (smlr to the srete se) Smth n Blnell (986). Step : estmte the moel v n otn the resl ˆv. Step : estmte stnr Tot moel o m ( α θ ˆv ) (6) Ths two-step proere gves onsstent estmtors oeents. Alterntvel one n ppl the ll mmm lkelhoo: ( ) ( ) ( ) 6

7 The enstes re gven : θ θ n v v Dsssons: wh estmtng wol el onsstent estmte whle estmtng smlr eqton wth eng nr vrle wol not? The ke reson s tht n (6) s ontnos. So the ontnos prt o n e se to gre ot the vrne. For nr ths no longer hols. mple : other-tpes o ensorng (7) The enst nton or (7) s: ) ( ) ( One n ppl ML or ths enst nton. 7