Transcription of Lecture Notes On Binary Choice Models: Logit and Probit
1 Lecture Notes On Binary Choice Models: Logit and Probit Thomas B. Fomby Department of Economic SMU March, 2010 Maximum Likelihood Estimation of Logit and Probit Models iiiPPy-1y probabilitwith 0y probabilitwith 1 Consequently, if N observations are available, then the likelihood function is NiyiyiiiPPL111. (1) The Logit or Probit model arises when iP is specified to be given by the logistic or normal cumulative distribution function evaluated at iX . Let iXF denote either of theses cumulative distribution functions. Then, the likelihood function of both models is NiyiyiiiXFXFL111 . (2) Then, the log-likelihood function is NiiiiiXFyXFylL11ln1lnln . (3) Now, the first order conditions arising from equation (3) are nonlinear and non-analytic.
2 Therefore, we have to obtain the ML estimates using numerical optimization methods, eg, the Newton-Raphson method. This method (which will be explained further later) implies the following recursion. nnllnn ~1~21~~ (4) In equation (4), n ~ is the n-th round estimate and the Hessian and score vectors are evaluated at this estimate. From our previous ML theorem, we know that 12,0~ lENNNasyML (5) whereML ~ represents the last iteration of the Newton-Raphson procedure. For finite samples, the asymptotic distribution of ML ~ can be approximated by 12,MLlN . For the Logit model , iiXFP where tetF 11 (6) is the logistic cdf and the logistic pdf is 21tteetftF (7) Also, note that tFeetFtt 11 (8-1) tFtFtf 1 (8-2) tetFtftf 1 (8-3) Using these results it can be shown for the Logit model , NiiiiiiNiiiiNiiiiXXFyXFyXXyXXyl1111exp11 1exp11 (9) The Hessian can be shown to be NiiiiNiiiiiXXXfXXXXl1122exp1exp (10) Note that this iiXX matrix is for all ~.
3 So, iterate nnllnn ~1~21~~ until nn~~1. For the Probit model , iiXFP where 221exp21ttf (11) is the Probit pdf and the Probit cdf is dvvftFt (12) Also, note that ttftf (13-1) tFtF 1 (13-2) Then, the score vector for the Probit model is iNiiiiiiiXXFXfyXFXfyl 111 (14) The Probit Hessian is then iiNiiiiiiiiiiiiXXXFXFXXfyXFXFXXfyXfl 1222111 Estimation of Marginal Effects in the Logit and Probit Models The analysis of marginal effects requires that we examine KjNiXfXPjiiji,,2,1,,,2,1, . KjNiXfXPjMLMLiXXiji,,2,1,,,2,1,~~, Talk about applications of Logit and Probit : credit scoring, target marketing, bond Rating. Go over example of German on class website.
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