Binary regression

• With this distribution for the error term, and
• a suitable transformation for the dependent variable
  1. the logit: $log(\frac{p}{1-p})$ or ``log of the odds''
  2. or the inverse of the cumulative normal distribution: $\Phi^{-1}$
• we have a form of regression for binary dependent variables

• These two mappings translate $0\Leftrightarrow 1$ into $-\infty\Leftrightarrow\infty$
• They are symmetric, and quite similar in effect
• For extreme probabilities, a bigger change in f(P) is required for the same change in P as for probabilities around 0.5
• Logit and probit regressions have the same form:
  
  $$g(p) = \alpha + \beta X + \epsilon_{b,n}$$
  
  

• The logit transform gives us ``logistic regression''

• This is favoured over probit in recent years, for its mathematical tractability