ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for mamum lkelhood estmaton, and how to estmate by mamum lkelhood the two most common formulatons of such models, namely probt and logt models.. General Formulaton of Bnary Dependent Varables Models A conventonal formulaton of bnary dependent varables models relates the observed bnary outcome varable to an unobserved (or latent) dependent varable. he unobserved (or latent) dependent varable s assumed to be generated by a classcal lnear regresson model of the form + u () where: a contnuous real-valued nde varable for observaton that s unobservable, or latent; ( X X2 L Xk ), a K row vector of regressor values for observaton ; 0 2 L k ) (, a K column vector of regresson coeffcents; u an d random error term for observaton. ECO 452 -- ote 4: Flename 452note4.doc Page of 7 pages
ECO 452 -- OE 4: Probt and Logt Models he random error terms u are assumed to have zero condtonal means and constant condtonal varances for any set of regressor values : E ( u ) 0 2 2 ( u ) E( u ) Var (2.) (2.2) In addton, the condtonal dstrbuton of the around ther zero condtonal mean. Symmetry around mean zero means that Pr( u a) Pr(u > a) u s assumed to be symmetrc Snce by defnton Pr( u > a) Pr(u a), symmetry means that Pr( u a) Pr(u a) or Pr( u a) Pr(u a). (2.3) he observable outcomes of the bnary choce problem are represented by a bnary ndcator varable that s related to the unobserved dependent varable as follows: f > 0 (3.) 0 f 0 (3.2) he random ndcator varable represents the observed realzatons of a bnomal process wth the followng probabltes: > Pr( ) Pr( > 0) Pr( + u 0) (5.) Pr( 0) Pr( 0) Pr( + u 0) (5.2) What s requred to estmate the coeffcent vector are analytcal representatons of the bnomal probabltes (5.) and (5.2). ECO 452 -- ote 4: Flename 452note4.doc Page 2 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models Interpretaton of the regresson functon Under the zero condtonal mean error assumpton (2.), equaton () mples that ( ) E( ) + E( u ) E. (4) he regresson functon s thus the condtonal mean value of the latent random varable for gven values of the regressors. he slope coeffcents j (j,, k) are the partal dervatves of the regresson functon (4) wth respect to the ndvdual regressors: E ( ) X j X j ( 0 + X + L+ jx X j j + L+ k X k ) j. 2. Analytcal Representaton of Bnomal Probabltes he bnomal probabltes > Pr( ) Pr( > 0) Pr( + u 0) (5.) Pr( 0) Pr( 0) Pr( + u 0) (5.2) are represented analytcally n terms of the cumulatve dstrbuton functon, or c.d.f., for the random error term u n regresson equaton (): + u () ECO 452 -- ote 4: Flename 452note4.doc Page 3 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models he cumulatve dstrbuton functon (c.d.f.) for the random varable u s denoted n general by G(u) and s defned as G where ( a) Pr( u a) g( u) a du a g(u) du ( ) Pr( u ) g( u) du 0 G ( ) Pr( u ) g( u) du G ( a) G( b) G for a < b he probablty that Pr( u a) Pr( u a) Pr ( u > a) G( ) G( a) G( a) For a < b, the probablty Pr( a u b) Pr ( a u b) G( b) G( a). > s gven n terms of G(a) as s gven as: he frst dervatve of the c.d.f. equals the correspondng probablty densty functon, or p.d.f.: dg ( ) ( u) g u or g( a) du dg u du ( ) dg( a) u a da where g(a) s the value of d G(u) du evaluated at u a. ECO 452 -- ote 4: Flename 452note4.doc Page 4 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models he probablty densty functon (p.d.f.) for the random varable u s the functon g(u) defned over all real values of u such that:. g( u) 0 2. g ( u) du 3. for any real values a and b where < a < b <, b Pr ( a u b) g( u) a du Symmetry Property: In addton to the assumptons that the random varable u 2 has zero mean and constant (fnte) varance, t s assumed that the p.d.f. g(u) s symmetrc about ts zero mean. Symmetry of g(u) around mean zero means that ( a) g(a) and Pr ( u a) Pr( u > a) g Snce by defnton Pr. ( u a) G( a) and Pr( u a) Pr( u a) G(a symmetry of g(u) mples that G > ), ( a) G(a) or equvalently that ( a) G( a G ). Geometrcally, the symmetry property means that the lower tal area probablty that u a s equal to the upper tal area probablty that u > a. lower tal area Pr(u a) upper tal area Pr(u > a) ECO 452 -- ote 4: Flename 452note4.doc Page 5 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models Representaton of the Bnomal Probabltes he bnomal probablty Pr( ) Pr( > 0) Pr( + u 0) > can be represented n terms of the c.d.f. for the random varable u as follows: Pr( ) Pr( > 0) Pr + u ( > 0) Pr ( u > Pr( u G( ( G by symmetry of g u (6.) he bnomal probablty Pr( 0) Pr( 0) Pr( + u 0) can be represented n terms of the c.d.f. for the random varable u as follows: Pr( 0) Pr( 0) Pr + u ( 0) Pr ( u G ( G( ( ) ( ) by symmetry of g u (6.2) he probablty densty functon, or p.d.f., for the bnary dependent varable can thus be wrtten as: g ( ) [ G( G( [ for 0,. (7) ECO 452 -- ote 4: Flename 452note4.doc Page 6 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models 3. he Sample Lkelhood and Log-Lkelhood Functons he sample lkelhood functon for a sample of ndependent observatons { :,, } s: L(,, K, ) g ( ) 2 [ G( [ G( (8) G ( ( G( ) 0 he sample log-lkelhood functon for a sample of ndependent observatons { :,, } s: ln L(,, K, ) ln ( L) 2 ln g( ) { ln G( + ( )ln[ G( } [ ln G( + ( )ln G( (9) ln G( + ln[ G( 0 ECO 452 -- ote 4: Flename 452note4.doc Page 7 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models 4. Dstrbutonal Specfcatons of the Model o complete specfcaton of the model, a specfc probablty dstrbuton must be chosen for the random error terms u. he most commonly adopted dstrbutons n econometrc applcatons are the standard normal and the standard logstc.. he standard normal dstrbuton yelds the probt model. 2. he standard logstc dstrbuton yelds the logt model. Probt Model he standard normal dstrbuton has mean μ 0 and varance 2, and s symmetrc around ts zero mean. If the random varable s normally dstrbuted wth mean μ and varance 2, then the standard normal varable z ( μ) s normally dstrbuted wth mean 0 and varance. hat s, 2 f ~ ( μ, ), then ~ (0,) where z z ( μ). he standard normal p.d.f. s φ z 2. 2 ( ) ( π ) z 2 ep 2 he standard normal c.d.f. s Z Z 2 2 z ( ) ( ) ( ) ( ) Z φ π Pr z Z z dz 2 ep dz. 2 Choce of the standard normal for the dstrbuton of the random error terms u leads to the probt model. ECO 452 -- ote 4: Flename 452note4.doc Page 8 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models Logt Model he standard logstc dstrbuton has mean μ 0 and varance π / 3, and s symmetrc around ts zero mean. 2 2 he standard logstc p.d.f. s f ( ) ep( ). 2 ( + ep( )) ( + ep( )) 2 ep( ) he standard logstc c.d.f. s F(X ) [ + ep( X ) ( + ep( X )) ep(x ) ( + ep(x )). Choce of the standard logstc for the dstrbuton of the random error terms u leads to the logt model. ECO 452 -- ote 4: Flename 452note4.doc Page 9 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models 5. he Unvarate Probt Model Probt Representaton of the Bnomal Probabltes In the probt model, the bnomal probabltes Pr( ) and ( 0) represented analytcally n terms of the standard normal c.d.f. ( ): Z Z 2 2 z ( Z ) Pr( z Z ) φ ( z) dz ( 2 π ) he bnomal probablty ( ) represented n the probt model as follows: Pr( ) Pr( 0) > Pr( + u > 0) Pr ( > u ep dz 2 Pr Pr( 0) Pr( + u 0) Pr are Z > s > u Pr > dvdng by > 0 u Pr by defnton u snce ~ (0,) by symmetry of φ(z) (0) ECO 452 -- ote 4: Flename 452note4.doc Page 0 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models he bnomal probablty ( 0) represented n the probt model as follows: ote that Pr( 0) Pr( 0) Pr( + u 0) Pr ( u Pr Pr( 0) Pr( + u 0) s u Pr dvdng by > 0 u snce ~ (0,) by symmetry of φ(z) () Z 2 2 z Z ep dz where 2 ( ) ( 2 π ) Z. he contrbuton to the sample lkelhood functon of the -th sample observaton s: g ( ) 0, for for 0 ECO 452 -- ote 4: Flename 452note4.doc Page of 7 pages
ECO 452 -- OE 4: Probt and Logt Models Probt Lkelhood Functon he probt lkelhood functon for a sample of ndependent observatons { :,, } s: ECO 452 -- ote 4: Flename 452note4.doc Page 2 of 7 pages ) L (, ( ) g (2) 0 Probt Log-lkelhood Functon he probt log-lkelhood functon for a sample of ndependent observatons { :,, } s: ( L, ln ) ( ) [ L ln ( ) ln g + )ln ( ln + )ln ( ln (3) + 0 ln ln A property of the probt log-lkelhood functon s that the coeffcent vector and the scalar parameter are not separately dentfable.
ECO 452 -- OE 4: Probt and Logt Models Consequently, only the probt coeffcent vector can be estmated. However, t s conventonal to mpose the normalzaton, n whch case the probt coeffcent vector. Computng Probt Coeffcent Estmates Mamum lkelhood estmates of the probt coeffcent vector or are obtaned by mamzng the probt log-lkelhood functon (3) wth respect to the K elements of or : Ma{ } ln ln [ L( ) L( ) + ln ( )ln where ln ( ) + ( )ln[ ( ) (3.) or ln ) ln[ L(,) Ma{ } L(, ln ( + ( )ln[ ( (3.2) Mamzaton of the probt log-lkelhood functon (3.)/(3.2) wth respect to or requres the use of nonlnear optmzaton algorthms such as ewton's method. he result s an ML estmate ˆ ˆ of the probt coeffcent vector together wth an ML estmate of the covarance matr for ˆ ˆ, Vˆ (ˆ ) Vˆ (ˆ) Vˆ. ˆ ECO 452 -- ote 4: Flename 452note4.doc Page 3 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models 6. he Unvarate Logt Model Logt Representaton of the Bnomal Probabltes In the logt model, the bnomal probabltes Pr( ) and ( 0) Pr are represented analytcally n terms of the standard logstc c.d.f. F(Z ): ( ) F(Z ) Pr z Z ep(z ) ( + ep(z )) he bnomal probablty ( ) represented n the logt model as follows: Pr( ) Pr( 0) >. Pr Pr( 0) Pr( + u 0) Pr( + u > 0) Pr ( u > Pr( u F( ( he bnomal probablty ( 0) > s > by defnton snce ~ f (z) F by symmetry of f (z) (4) represented n the logt model as follows: Pr( 0) Pr( 0) Pr( + u 0) Pr ( u ( F( Pr Pr( 0) Pr( + u 0) u s F by defnton of F(Z) by symmetry of f (z) (5) ECO 452 -- ote 4: Flename 452note4.doc Page 4 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models he contrbuton to the sample lkelhood functon of the -th sample observaton s: g ( ) [ F( F( ( F( [ 0, F for for 0 Logt Lkelhood Functon he logt lkelhood functon for a sample of ndependent observatons { :,, } s: L ( g ( ) [ ( [ F( F (6) F ( [ F( 0 ECO 452 -- ote 4: Flename 452note4.doc Page 5 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models Logt Log-lkelhood Functon he logt log-lkelhood functon for a sample of ndependent observatons { :,, } s: ln L( ln[ L( ln g( ) ln [ F( [ F( { ln F( + ( )ln[ F( } ( ) [ ( ) ln F + ( )ln F (7) ln F( + ln[ F( 0 Computng Logt Coeffcent Estmates by Mamum Lkelhood Mamum lkelhood estmates of the logt coeffcent vector are obtaned by mamzng the logt log-lkelhood functon (7) wth respect to the K elements of : ln ( ) ln[ L( Ma{} L ln F( + ( )ln[ F( (7) ln F( + ln[ F( 0 ECO 452 -- ote 4: Flename 452note4.doc Page 6 of 7 pages
ECO 452 -- OE 4: Probt and Logt Models A convenent property of the logt log-lkelhood functon (7) s that t s globally concave wth respect to the coeffcent vector. L( ( ) [ ( ln ln F + ( )ln F (7) ln F( + ln[ F( 0 hs property makes nonlnear mamzaton of the logt log-lkelhood functon (7) wth respect to farly straghtforward. he most commonly used nonlnear optmzaton algorthm for computng the ML estmates of the logt coeffcents s ewton's method, whch uses analytcal frst and second dervatves of ln L( wth respect to. he result s an ML estmate ˆ L of the logt coeffcent vector together wth an ML estmate of the covarance matr for ˆ L, Vˆ (ˆ L) Vˆ ˆ. ) L ECO 452 -- ote 4: Flename 452note4.doc Page 7 of 7 pages