Why does Stata show different results when I use -mlogit- versus -logit-?

[Gate Nucht]

I have an outcome variable with three categories, and I noticed that when I run a multinomial logistic regression Stata shows slightly different results than when I run two binary logistic regressions. Specifically, when I compare outcome "1" and outcome "2" to base outcome "0" using mlogit, I get slightly different results than when I compare outcome "1" with outcome "0" and outcome "2" with outcome "0" in two separate binary logistic regression models (using the logit command).

Why is that?

My understanding is that both approaches should get me the same results, since both approaches compare outcomes "1" and "2" to baseline outcome "0".

[Andrew Musau]

The only way you get the same results is if you combine the two categories in the multinomial logit, in doing so you get back to the binary case. Otherwise, multinomial logit places constraints on the estimated coefficients whereas binary logit does not, hence the differences.

[Gate Nucht]

The categories are exactly the same in both approaches. That is, the multinomial logistic regression compares, separately, categories "1" and "2" with category "0."

The two binary logistic regressions respectively compare category "1" with category "0" (category "2" is excluded from that analysis) and category "2" with category "0" (category "1" is excluded from that analysis).

[Andrew Musau]

The model is the same: You have \(j=1, \cdots, m\) choices and the probabilities are defined as:

$$p_{ij}= \frac{e^{\beta^{\prime}x_{ij}}}{\sum_{k=1}^{m} e^{\beta^{\prime}x_{ik}}}$$

In one case, you are estimating the model using a series of binary logits and in another case, you are estimating the parameters jointly using multinomial logit. Multinomial logit only estimates \(m-1\) equations. With \(m=3\), therefore, only 2 sets of coefficients are estimated. The constraint is such that the third set is a linear combination of the other estimates. You do not take into account this constraint in your binary logits and this explains the observed differences. In any case, refer to the manual entry of mlogit to see how Stata estimates the model.

[Gate Nucht]

Thank you for the explanation, I really appreciate it!