Transcription of The Logit Model: Estimation, Testing and Interpretation
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The Logit Model: Estimation, Testing and Interpretation
The Logit Model: estimation , Testing and InterpretationHerman J. BierensOctober 25, 20081 Introduction to maximum likelihood The likelihood functionConsider a random sampleY1, .., Ynfrom the Bernoulli distribution:Pr[Yj=1]=p0Pr[Yj=0]=1 p0,wherep0is unknown. For example, tossntimes a coin for which you suspectthat it is unfair:p06= ,and for each tossingjassignYj=1if the outcomeis heads andYj=0if the outcome is tails. The question is how to estimatep0and how to test the null hypothesis that the coin is fair:p0= probability function involved can be written asf(y|p0)=Pr[Yj=y]=py0(1 p0)1 y=(p0ify=1,1 p0ify= , lety1, .., ynbe a given sequence of zeros and ones. Thus, eachyjis ei-ther0or1. The joint probability function of the random sampleY1,Y2, .., Ynis defined asfn(y1, .., yn|p0)=Pr[Y1=y1andY2= andYn=yn].1 Because the random variablesY1,Y2, .., Ynare independent, we can writePr[Y1=y1andY2= andYn=yn]=Pr[Y1=y1] Pr[Y2=y2].)
2 Motivation for maximum likelihood esti-mation A more formal motivation for ML estimation is based on the fact that for 0 <x<1 and x>1, ln(x) <x−1. This is illustrated in the following picture: 1How to draw such a sample is beyond the scope of this lecture note. 5
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