# Conditional logit or multinomial logit?

I have the following problem I need to use a discrete choice model to determine the time values (VOT) of 3 transport alternatives (car, bus and train). Looking at the literature of the discrete choice models, I logit more precisely and came to the conclusion that I should use a conditional logit given that the specific variables change by mode of transport, but in other bibliographies mention that the conditional logit estimators do not vary by alternative. So I do not know what model to estimate ...

For example (option 1), if I need to estimate $$\beta_{0}^{auto},\beta_{time}^{auto},\beta_{cost}^{auto}\beta_{0}^{bus},\beta_{time}^{bus},\beta_{cost}^{bus},\beta_{time}^{train},\beta_{cost}^{train}$$, without intercept for train ($$\beta_{0}^{train}=0$$).

Or for what I need it is better to estimate (opton 2) $$\beta_{0}^{auto},\beta_{time}^{auto},\beta_{cost}^{auto}\beta_{0}^{bus},\beta_{time}^{bus},\beta_{cost}^{bus}$$ with train as category reference. and my database is in wide format

individual  time.auto  cost.auto   time.bus   cost.bus   time.train   cost.train
1          18.5      1.5       20.86      1.8       30.03      2.35
2          31.3     6.05       67.1       2.23      60.2       1.85


being Pij the probability that the individual i choose the bus mode (for example for option 1):

$$P_{ij}=\cfrac{e^{\beta_{0}^{bus}+\beta_{time}^{bus}X_{time}^{bus}+\beta_{cost}^{bus}X_{cost}^{bus}}}{e^{\beta_{0}^{car}+\beta_{time}^{car}X_{time}^{car}+\beta_{cost}^{car}X_{cost}^{car}}+e^{\beta_{0}^{bus}+\beta_{time}^{bus}X_{time}^{bus}+\beta_{cost}^{bus}X_{cost}^{bus}}+e^{\beta_{time}^{train}X_{time}^{train}+\beta_{cost}^{train}X_{cost}^{train}}}$$

and the probability fo the option 2 is $$P_{ij}=\cfrac{e^{\beta_{0}^{bus}+\beta_{time}^{bus}X_{time}^{bus}+\beta_{cost}^{bus}X_{cost}^{bus}}}{1+e^{\beta_{0}^{car}+\beta_{time}^{car}X_{time}^{car}+\beta_{cost}^{car}X_{cost}^{car}}+e^{\beta_{0}^{bus}+\beta_{time}^{bus}X_{time}^{bus}+\beta_{cost}^{bus}X_{cost}^{bus}}}$$

or I'm wrong?

What model and option do you recommend if I need to calculate the time values for the 3 modes of transport?

Thank you!

• I understand, and if I want to estimate, how do I calculate the probabilities and the log-likelihood? to determine $\beta_{0}^{car}$, $\beta_{time}^{car}$, $\beta_{cost}^{car}$, $\beta_{0}^{bus}$, $\beta_{time}^{bus}$, $\beta_{cost}^{bus}$, $\beta_{time}^{train}$ and $\beta_{cost}^{train}$? Thank u! – lampard811 Feb 5 at 1:39
• In the composition of the probability for each mode is where I have doubts. For example, the probability of the individual "i" is to choose car (m=1) is $P_{i}^{car}=\cfrac{exp(U_{i}^{car}-U_{i}^{train})}{1+exp(U_{i}^{car}-U_{i}^{train})+exp(U_{i}^{bus}-U_{i}^{train})}$ and the probability of choosing train is $P_{i}^{train}=\cfrac{1}{1+exp(U_{i}^{car}-U_{i}^{train})+exp(U_{i}^{bus}-U_{i}^{train})}$ and the log-likelihood is the same as it appears in [pdfs.semanticscholar.org/f941/…? this is correct? help me please! thank u! – lampard811 Feb 5 at 21:44