Multinomial logistic regression

A Multinomial logit model is an econometric model which is an extension of the logit model for many cases (more than two). For reasons of indentification one needs one benchmark/base case.

Model

$\Pr(y_{i}=j)={\frac {\exp(X_{i}\beta _{j})}{\sum _{j}^{J}\exp(X_{i}\beta _{j})}}\forall j$

and $\Pr(y_{i}=0)={\frac {1}{\sum _{j}^{J}\exp(X_{i}\beta _{j})}}$,

where $y_{i}$ is the observed outcome (e.g. $y_{i}=1$), $X$ is a vector of explanatory variables and $\beta _{j}$ is a coefficient, $y_{i}=0$ is the benchmark case. The coefficients are estimated by maximum likelihood.

See also

Multinomial probit