Binary or multinomial logit?

Question

I am having some doubts on whether I should use binary or multinomial logit. I am modelling product choice. The dependent variable is the choice of either shampoo or toothpaste. Although my dependent variable has two outcomes, I am wondering whether it is appropriate to use multinomial logit instead of binary.

The case is that the customer does not choose among these two products, but among the whole assortment. And I have taken a sample for only 2 products. Do you think I should use the multinomial logit?

Answer 1

In addition to @HongOoi 's point, could a person buy neither ? Then you;d have four choices:

Both
Toothpaste
Shampoo
Neither

It's not exactly clear what your sample is, but it seems to me you have to have these four choices. In that case, multinomial logit is correct.

However, if your sample includes only people who bought one or the other but not both, you could use (ordinary) logistic. Then you would be comparing people who bought toothpaste to people who bought shampoo.

Answer 2

Late answer, but maybe useful for others:

In a binary Choice Model, we estimate $$\text{Pr}(y_i = 1 \mid x_i) = \frac{\text{exp}(\bf{x_i^\prime \beta_j})}{1+\text{exp}(\bf{x_i^\prime \beta_j})}$$ and in a Multinomial Logit we estimate $$\text{Pr}(y_i = j \mid x_i) = \frac{\text{exp}(\bf{x_i^\prime \beta_j})}{1+\sum_{h=2}^J \text{exp}(\bf{x_i^\prime \beta_h})}, j = 2\text{,...,}J$$ The first choice $j = 1$ is omitted, because it is taken as the baseline (as is the $0$ choice in the binary logit). When you now set $J=2$ as it is in your case (disregarding the valid objections made by Peter Flom) you get exactly the same as in the binary choice model: $$\frac{\text{exp}(\bf{x_i^\prime \beta_j})}{1+\sum_{h=2}^{J=2} \text{exp}(\bf{x_i^\prime \beta_h})} = \frac{\text{exp}(\bf{x_i^\prime \beta_j})}{1+\text{exp}(\bf{x_i^\prime \beta_j})}$$

Just be aware that by the definitions above, you have the choice between $0,1$ in the binary choice model and between $1,2$ in the multinomial choice model (with two choices).