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Let the mode of transport be a categorical random variable with three categories, representing Car, Bus and Train respectively. The goal is to predict for the $n$-th person, what mode of transport they use to commute to work.

Suppose that I think that when making the choice of transport to take, a person first decides whether they will catch public transport or not. Then, if they catch public transport, they have the choice of going by bus or train. This hierarchical/nested relationship can be understood using a modification of Figure 4.1 (page 90) from Chapter 4: GEV of Train:

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This leads to a model that produces a vector of probabilities $(p_{0}, p_{1}, p_{2}) = (p_{n0}, p_{n1}, p_{n2})$ for the modes Car, Bus, and Train respectively via the regression equations (I am going to abuse notation and drop subscripts involving $n$ from now on):

$$ \begin{split} \operatorname{ln}(\frac{p_0}{1 - p_0}) &= y_0 \\ \operatorname{ln}(\frac{p_1}{p_2}) &= y_1\\ p_0 + p_1 + p_2 &= 1 \end{split} $$ where $y_0 = \alpha_0 + \alpha_1 x_{01} + \alpha_2 x_{02} + \dots $ and $y_1 = \beta_0 + \beta_1 x_{11} + \beta_2 x_{12} + \dots $ are linear combination of covariates, possibly shared. The expression on the LHS of the first equation is the usual logit and the LHS of equation two is the generalised logit as seen in the Wikipedia article on multinomial logistic regression.

My regression equations describe two logistic regressions of sorts, one nested within the other. The first equation mimics a standard logistic regression problem for predicting whether a person catches public transport or not, and the second equation is a logistic regression (adapter so that probabilities sum to one) for whether a person catches a bus or a train, conditional on them catching public transport.

My question is whether this model is a useful discrete choice model for the question at hand, as it doesn't seem like it is a nested logit (GEV) model, nor does it seem like a multinomial regression model. It doesn't seem like a GEV model for the following reasons. Following Chapter 4.2 of Train, linking my first equation for the "upper" model to the "lower" model seems to require $\lambda$, a parameter for the cumulative GEV distribution (Equation 4.1, reproduced below), to be set to zero, which does not seem to make sense:

$$\exp\left(-\sum_{k=1}^K\left(\sum_{j \in B_k} e^{-\epsilon_{nj}/\lambda_k}\right)^{\lambda_k}\right)$$

It also is not a multinomial logit as the relationship between the covariates and the predicted probabilities are different, when covariates are shared, i.e. $x_{0j} = x_{1j}, \forall j$. My first equation is subtly different to the one given by multinomial logistic regression, when assuming that Train is the reference class:

$$ \operatorname{ln}(\frac{p_0}{p_2}) = y_0 $$

To simplify notation, let $Y_0 = \exp(y_0)$ and $Y_1 = \exp(y_1)$. Then, solving for the probabilities from my equations:

$$ \begin{split} p_0 &= \frac{Y_0}{1+Y_0}\\ p_1 &= \frac{(1 - p_0) Y_1}{1+Y_1}\\ p_2 &= 1 - \frac{p_0}{1+Y_1} - \frac{Y_1}{1 + Y_1} \end{split} $$

In contrast, with multinomial logit one obtains:

$$ \begin{split} p_0 &= \frac{Y_0}{1+Y_0 + Y_1}\\ p_1 &= \frac{Y_1}{1+Y_0 + Y_1}\\ p_2 &= \frac{1}{1+Y_0 + Y_1} \end{split} $$

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If you are using maximum likelihood and are performing no model selection, they are equivalent modeling strategies. However, in most other settings the multinomial model is considerably more constrained. For instance, if you include the effects of various covariates (income/work schedule/residential setting/etc.) the multinomial model requires each parameter to predict each multinomial outcome, however the hierarchical model can use non-overlapping subsets of the candidate predictors in each step leading to considerably sparser model.

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