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I'm studying the logistic regression for estimate the Probability of Default of SME's. Fortunately the event (firm's default) is a rare event.

King and Zeng tell us that "logistic regression can sharply underestimate the probability of rare events" (Logistic regression in rare events data, 2001). This is because the logistic regression coefficient is biased in these situations.

Could someone tell me in which paper is proved that the logistic regression intercept is biased when the event (Y=1) is rare.

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    $\begingroup$ Link for those interested: gking.harvard.edu/files/abs/0s-abs.shtml $\endgroup$ – Tim Dec 5 '14 at 18:24
  • $\begingroup$ Don't the McCullagh & Nelder GLM book cited in @Tim's link and the appendices in King et al's paper contain the proofs? $\endgroup$ – Dimitriy V. Masterov Dec 5 '14 at 19:31
  • $\begingroup$ Thanks @Dimitriy, I saw the proof on the appendices but I hoped to find another one simpler. I'll check the one on the McCullagh&Nelder book cited in King and Zeng paper. $\endgroup$ – Luca Dibo Dec 6 '14 at 13:03
  • $\begingroup$ I checked, but the main problem for me is conceptual: since the logistic regression belong to the GLM's family, we know that the estimation of the parameters is done through the Fisher scoring algorithm and that there isn't a closed formula for the estimator. So if this is true, how could I now that the intercept is biased? $\endgroup$ – Luca Dibo Dec 7 '14 at 20:55
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    $\begingroup$ @LucaDibo Take a look at pp. 703-704 of King and Zheng paper for the intuition on the single regressor case. $\endgroup$ – Dimitriy V. Masterov Dec 7 '14 at 22:42

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