Common mistake 1 with Multinomial logit

Reading a coefficient as an effect on a probability. It is an effect on the log-odds against the base; the probability effect depends on where you evaluate it.

Common mistake 2 with Multinomial logit

Ignoring independence of irrelevant alternatives when the options include near-substitutes (bus and tram, two similar brands).

Common mistake 3 with Multinomial logit

Using it on an ordered outcome. It will work but throws away the ordering and spends many extra parameters to do so.

Common mistake 4 with Multinomial logit

Reading a relative risk ratio as a probability. It multiplies the odds against the base category; the effect on the probability of choosing an alternative can even have the opposite sign, because raising one alternative's utility lowers every other share.

EconStatsCraft › Multinomial logit

Multinomial logit

For a categorical outcome with three or more options that have no natural order - occupation, travel mode, brand. Reports one equation per category against a base, relative risk ratios, and the average marginal effect on each probability.

What it needs

Unordered categorical outcome - Three or more categories, no natural ranking.
Independent variables

The intuition

When the outcome is a choice among unordered alternatives - bus, car, train - there is no latent scale to cut up, because there is no sense in which train is "more" than bus. Multinomial logit instead gives each alternative its own utility index and lets the probabilities be the softmax of them. One alternative must be the reference, because only differences in utility are identified: adding a constant to every alternative changes nothing anyone chooses.

When to use it

The outcome is categorical with three or more values and ranking them would be meaningless. If they can be ranked, an ordered model is more parsimonious and more powerful.

Where the estimator comes from

With J alternatives and a base category J, P(y_i=j\mid x)=\frac{\exp(x_i'\beta_j)}{1+\sum_{k=1}^{J-1}\exp(x_i'\beta_k)},\qquad \beta_J\equiv0. The normalisation \beta_J=0 is what makes the model identified rather than an arbitrary choice of convenience. Taking logs of the ratio of any two probabilities gives \ln\frac{P_j}{P_k}=x'(\beta_j-\beta_k), which depends on j and k ALONE - no other alternative appears. That is independence of irrelevant alternatives, and it is a consequence of the functional form rather than an extra assumption bolted on.

The model

P(y=j\mid \mathbf{x})=\frac{\exp(\mathbf{x}^{\prime}\boldsymbol{\beta}_{j})}{1+\sum_{k=2}^{J}\exp(\mathbf{x}^{\prime}\boldsymbol{\beta}_{k})},\qquad \boldsymbol{\beta}_{1}=\mathbf{0}

What it assumes

Independence of irrelevant alternatives: the relative odds of two options do not change when a third is added or removed. This is the famous weakness - with close substitutes among the options it fails.
The categories are exhaustive and mutually exclusive.
Enough observations in every category. A category with a handful of cases will produce enormous standard errors.
The alternatives are exhaustive and mutually exclusive: every observation picks exactly one, and the set on offer is the same for everyone. If some respondents never had an option available, their choice was made from a different set and pooling them imposes a comparison that did not exist.

Statistical properties

IIA is the model's central weakness. The red-bus blue-bus problem: introduce a bus identical to an existing one except for colour, and the model predicts it steals share equally from cars and from the original bus, which is absurd.
The Hausman-McFadden test checks IIA by dropping an alternative and seeing whether the remaining coefficients move. It has poor power and can return a negative test statistic, at which point it is uninformative rather than passing.
Coefficients are relative to the base category and change entirely if the base changes. Only the probability predictions and marginal effects are invariant, so report those.
The number of parameters grows as (J-1) times K. With five alternatives and eight regressors that is 32 coefficients, and interpretation collapses under its own weight.
A marginal effect can have the opposite sign to its coefficient, because raising one alternative's index also changes the denominator for every other. Never read the sign off the coefficient.
Nested logit and mixed logit relax IIA - the first by grouping similar alternatives, the second by letting coefficients vary randomly across people. Mixed logit can approximate any random-utility model (McFadden and Train, 2000).

How to read the output

Every coefficient is relative to the base category. Changing the base changes all the numbers without changing the model, so always say which base you used.
A relative risk ratio above 1 means that variable pushes the choice towards that category and away from the base.
Marginal effects are easier to talk about than coefficients, and they sum to zero across categories by construction - a variable that raises one probability must lower others.
Coefficients are relative to the BASE category and change entirely if you change the base - the fit, the predicted probabilities and the marginal effects do not. If two write-ups of the same data disagree about a coefficient, check the base before checking anything else.

A worked reading

Occupation for 900 people - Clerical (371), Manual (228) and Professional (301) - with Clerical as the base, so every coefficient is read against it. The model is jointly significant (LR χ² = 175.33 on the intercept-only model, p < 0.0001) but explains a modest share of the variation: McFadden's pseudo-R² is 0.0902, which for a three-way choice is ordinary and is NOT comparable to an R² from a regression. Now consider what independence of irrelevant alternatives asserts here: the odds of Manual against Professional are fixed by their own coefficients, regardless of whether Clerical is available at all. If Clerical and Manual are close substitutes while Professional is quite different - which is exactly what these three categories look like - that is false. No amount of data will reveal it from inside the model, because the restriction lives in the functional form rather than in the estimates, and this is the reason the nested and mixed logits exist.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Coefficients flipping when the base changes	Correct and expected	Report marginal effects, which do not
Hausman-McFadden statistic negative	A small-sample artefact	Uninformative, not a pass
Two alternatives that are near-substitutes	IIA will fail on exactly this pair	Use nested logit with them in one nest
One alternative chosen by very few	Its coefficients are barely identified	Consider merging it
Perfect prediction for a category	Separation	A regressor determines the outcome; drop it or use a penalty
Marginal effect sign opposing the coefficient	Normal, via the denominator	Not an error

Common mistakes

Reading a coefficient as an effect on a probability. It is an effect on the log-odds against the base; the probability effect depends on where you evaluate it.
Ignoring independence of irrelevant alternatives when the options include near-substitutes (bus and tram, two similar brands).
Using it on an ordered outcome. It will work but throws away the ordering and spends many extra parameters to do so.
Reading a relative risk ratio as a probability. It multiplies the odds against the base category; the effect on the probability of choosing an alternative can even have the opposite sign, because raising one alternative's utility lowers every other share.

Where to read this properly

McFadden (1974), "Conditional Logit Analysis of Qualitative Choice Behavior", in Zarembka (ed.).
Hausman and McFadden (1984), "Specification Tests for the Multinomial Logit Model", Econometrica 52(5).
Train, Discrete Choice Methods with Simulation (2009), ch. 3 to 6.
McFadden and Train (2000), "Mixed MNL Models for Discrete Response", Journal of Applied Econometrics 15(5).

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Multinomial logit

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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