The multinomial logit model in transport demand

Multinomial Logit Model in Transport Demand

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Introduction

Discrete choice models are presented as a development and a renovation of the classical choice theory. They have overcome the rigidities and inadequacies of consumer behavior study by mentioning the problems of economic agent choices in a random and specific environment for each situation involving the choice between mutually exclusive alternatives. The discrete choice models have been the subject of several theoretical developments and empirical validations. Their manipulation has become easier thanks to the availability of increasingly disaggregated data and advances in econometric techniques and software. They were applied for the first time to estimate transport demand. They were subsequently generalized and applied to deal with all the problems of choice concerning mutually exclusive alternatives or also to assess the subjective value of an event.

The Multinomial Logit Model.

Among the discrete choice models the multinomial logit model is the most widespread and used in many different fields. This disaggregated model seeks to study the decisions of choice or the perception of the value of an event among a set of mutually exclusive alternatives. The multinomial logit model will therefore allow us to estimate the probability that an individual i chooses an alternative j in given circumstances characterizing the enviroment variables characterizing this environment of choice (Xk). Formally, this probability is written according to the following expressions:

How do we interpret these variables?

Choices for an individual

For a more detailed discussion, consider that an individual i of a sample N (such as i = 1 … N) is in front of a set of choices (modes of transport, port of call, types of equipment, place of residence, etc.) or belongs to a given category of population or appreciation of a psychological value of a given phenomenon (risk of accident, time value, etc.) j (j ∈J/j = {1,2,3…J}). Individual i chooses the alternative j that optimizes (maximizes or minimizes) its objective function (Si). The variable to be explained is expressed as follows:

How do we interpret these variables?

Objective Function

The objective function of the individual i is dependent on the socioeconomic characteristics of the individual i (X_ik), on the technical ones of the option to be remembered (W_jh), and on those of the enviroment of choice (E_jm):

Pieces of the objective function

A specific variable to the individual is a variable that remains the same regardless of the option chosen by the individual, while a specific variable to the alternative j depends on the specific conditions to the choice. As ong as the objetive finctions is random, we can break it down into two parts: one is determinist (V_ij (X_ik, W_jh, E_jm)) and the other is random (∈_ij):

Elasticity

In the multinomial logit model, several categories of explanatory variables of both qualitative and quantitative orders can be integrated. The interpretation of continuous variables of a quantitative nature does not pose any problem. The exponential value of the coefficient associated with this variable measures the unit variation impact of this explanatory variable on the probability of choosing the alternative j rather than the reference alternative J. For qualitative variables, we distinguish between binary ones which will be coded in 0 and 1 and those polytomous which express themselves in several modalities. We can also evaluate the impact of the variation of the explanatory variable on the comparative probability of the individual choice by the elasticity. Elasticity is defined as a percentage change in the probability of choosing alternative j rather than alternative J resulting from a 1% change in one of the characteristics of alternative j (Wj) by keeping the other arguments of the probability function constant.

Elasticity calculation

The elasticity calculation constitutes a very indispensable information base for decision makers to learn the most influential factors in the individual behavior and determine their optimal action plan in order to achieve their goals. The elasticity can be calculated with respect to all the arguments of the pribability function. We speak of direct elasticity when it is calculated with respect to the arguments relating to the chosen alternative j and of the cross elasticity, when it is calculated with respect to the arguments relative to the other alternatives I # j. This direct individual elasticity is written:

How do we interpret these variables?

What we learned

Discrete choice models are a valuable tool for analyzing the behavior of individuals when faced with a choice between mutually exclusive alternatives. They are based on the logic of economic rationality which aims at optimizing an objective function while taking into account both the socioeconomic characteristics of individuals and the technical-economic characteristics of the alternative to be chosen, as well as the uncertainty of the environment where the choice reigns. The multinomial logit model is the most used in empirical studies. It has the advantage of being able to treat the individual choice between a multitude of options and seeks to estimate the probability of having chosen a given alternative that better meets the requirements of the individual and the specific conditions characterizing the environment of choice.

Conclusions

Multinomial logit model predicts the effects of modifying one of the characteristics of the alternative to choose or the individual’s socioeconomic variables on the probability of making a relative decision of choice. It allows better analysis of economic phenomena in relation to human behavior as a decision making unit such as transport demand, accidentology, and valuation of non market goods (transport time, membership of a given category population, etc.). They constitute an important information base which guides these economic actors to the best choices of preventive actions and the orientation of the transport policy as well as in the matter of investment, pricing, road safety, etc. They offer us the possibility to calculate a specific time value to each individual according to their socioeconomic characteristics, their modal choice, and the conditions of travel (reason for travel, zone origin destination, time of departure, etc.), to propose the best preventive actions to accidentology, etc.

Bibliographic Source

Aloulou, F. (2018). The Application of Discrete Choice Models in Transport. InTech https://www.intechopen.com/chapters/61268

Pij is the probability that an individual i establishes the choice j. The parameters αk are unknown that we seek to estimate. They, respectively, reflect the weight of each explanatory variable (Xk) in the determination of the probability Pij. Fij is a distribution function of the explanatory variables and the vector of parameters αk.

Yi designates the choice observed and Sij the level of objective function that the choice of the alternative j gives to the individual i

Where Wjh is the hth argument of the vector characterizing the alternative j (Wj), βh being its relative parameter, and Pij is the probability of choice of the eventuality j by the individual i.