Binary Model: Presentation

May 14th, 2015
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Non-linear Models ; Consider : P (yi = 1jx) = G (xi β), i = 1, ..., N where G (z ) is a function 0 < G (z ) < 1 At the extremes z Aslanidis (URV & UNSW) ! ∞, G (z ) ! 0 ! +∞, G (z ) !

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Lecture 3Discrete Dependent variables: Presentation of BinaryResponse ModelNektarios Aslanidis (Universitat Rovira i Virgili, UNSW)Aslanidis (URV & UNSW)Binary Model: Presentation1 / 14Non-linear ModelsConsiderP (yi = 1jx) = G (xi ), i = 1, ..., Nwhere G (z ) is a function0 < G (z ) < 1At the extremeszzAslanidis (URV & UNSW)!, G (z ) ! 0! +, G (z ) ! 1Binary Model: Presentation2 / 14Logit ModelsConsiderG (xi ) = (xi ) =exp(xi )1 + exp(xi )where G (z ) is the cumulative distribution function for a standard logistic random variable, z.Logit modelyiyiAslanidis (URV & UNSW)= G (xi ) + iexp(xi )=+1 + exp(xi )Binary Model: Presentationi3 / 14Probit ModelsConsiderG (xi ) = (xi ) =Z xi (v )dvwhere (z ) is the cumulative distribution function for a standard normal random variable, z and (v ) is the standard normal density. (v ) = (2 )Aslanidis (URV & UNSW)1/2exp( v 2 /2)Binary Model: Presentation4 / 14Probit ModelsProbit modelAslanidis (URV & UNSW)yi= (xi ) +yi=Z xi i(v )dv +Binary Model: Presentationi5 / 14Latent Variable ModelsUnderlying latent variable modelyi = xi + ei ,yi = 1[yi > 0]where 1[.] is an indicator function implying= 1,= 0,yiyiwhen yi > 0when yi0We also assumewhere (z ) =Aslanidis (URV & UNSW)eiN (0, 2 ), if Probitei(0, 2 ), if Logitexp (z )(1 +exp (z ))2

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