Multinomial logit

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A multinomial logit model is an econometric model which is a subset of logit models where there are two or more cases.

1 Model
- 1.1 notes on identification
2 New developments
3 See also

[edit] Model

$\Pr(y_{i}=j)=\frac{\exp(X_{i}\beta_{j})}{\sum_{j}^{J}\exp(X_{i}\beta_{j})}$

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 $β j$ is a coefficient, $y i = 0$ is the benchmark case. The coefficients are estimated by maximum likelihood.

The estimates can be obtained by sequentially fitting a logit for the next option versus all other options, but this will yield incorrect standard error estimates.

[edit] notes on identification

As with other overspecified models for reasons of indentification one needs one benchmark/base case in the transformed space. This benchmark ends up being besides the point because any transformation that moved all values an equal amount would not change the precicted results.

[edit] New developments

Random multinomial logit (using the analogous reasoning, a set of multinomial logit models are combined into a random ensemble of classifiers), developed by Anita Prinzie & Dirk Van den Poel in 2006.