Interactive modelEconometricsadvanced

Probit & Logit Models

Modeling yes/no outcomes: latent variables, marginal effects, and odds ratios.

Equilibrium: X* = 50.0, p* = 50.0

Current equations

Logit S-curve: p = 100/(1 + e^(−0.1·(X−50)))Linear probability model: p = -10 + 1.2·XEvaluation point: X = 50

Some outcomes are yes or no: employed or not, default or not, pass or fail. The S-curve maps a score onto a probability, squashing everything into the 0 to 100 percent range. The dashed straight line shows the naive alternative, which happily predicts probabilities below zero and above one.

Shift the curves

Steepness (how sharply X separates outcomes)0

A steep S means X strongly predicts the outcome; flat means it barely matters.

Midpoint (where the odds are 50:50)0

Shift the whole S-curve: the same score now means different odds.

The Probit & Logit Models model, in writing

Definition

Models for yes/no outcomes: a latent index passed through a normal (probit) or logistic (logit) link, keeping probabilities inside [0,1] where OLS strays.

P(y=1|x) = Φ(x′β) or Λ(x′β); logit odds: e^β = odds ratio

The intuition

Raw coefficients only give signs, effects on probability depend on where you stand on the S-curve, so always convert to marginal effects. The linear probability model remains respectable for quick average effects, mis-predictions at the extremes aside.

Exam tip

Never interpret a probit coefficient as a probability change, compute the marginal effect at means or the average marginal effect.

Related models in Econometrics

Linear Regression & OLS →Difference-in-Differences →