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!

1 Answer

You need to use a multinomial logit (or probit) model. Then you can decide whether you want to estimate "mode-specific effects" or "generic effects" (Basically mode-specific effects consist in allowing for interaction effects between the mode of transport and the features - For example, people might care more about COST when it comes to travel by train). As in any regression with categorical variables, you will need to omit some levels/modalities to ensure model identification (So one of the model will serve as reference).

A multinomial logit (MNL) model is a good starting point to analyse your data, but then you also need to consider other - more flexible - models, such as nested logit to relax the assumption of independent errors (which gives rise to the so-called "Independence of Irrelevant Alternatives" property).

• 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 '19 at 1:39
• Once you have estimated the model parameters, you simply apply your proba formula and you will obtain choice probabilities for the different options (depending on their composition) - The likelihood of the model is the product of the observed choices over the sample (see en.wikipedia.org/wiki/Discrete_choice) - In your case, the key feature of your MNL model is that the vector of betas will differ across the options - But nothing really challenging – Umka Feb 5 '19 at 10:08
• 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 '19 at 21:44
• Yes but you don't need to write "U_Train" in your equations as utility of the baseline/reference option is implicitly normalized to 0. – Umka Feb 15 '19 at 10:50