Modeling yes/no outcomes: latent variables, marginal effects, and odds ratios.
Warming up the engine…
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
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.
Never interpret a probit coefficient as a probability change, compute the marginal effect at means or the average marginal effect.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Also in Econometrics
Equilibrium at X 50, p 50.
Equilibrium: X* = 50.0, p* = 50.0
Current equations
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.