# mlogit: Multinomial logit model In mlogit: multinomial logit model

## Description

Estimation by maximum likelihood of the multinomial logit model, with alternative-specific and/or individual specific variables.

## Usage

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 mlogit(formula, data, subset, weights, na.action, start = NULL, alt.subset = NULL, reflevel = NULL, nests = NULL, un.nest.el = FALSE, unscaled = FALSE, heterosc = FALSE, rpar = NULL, probit = FALSE, R = 40, correlation = FALSE, halton = NULL, random.nb = NULL, panel = FALSE, estimate = TRUE, seed = 10, ...) ## S3 method for class 'mlogit' print(x, digits = max(3, getOption("digits") - 2), width = getOption("width"), ...) ## S3 method for class 'mlogit' summary(object, ...) ## S3 method for class 'summary.mlogit' print(x, digits = max(3, getOption("digits") - 2), width = getOption("width"), ...) ## S3 method for class 'mlogit' print(x, digits = max(3, getOption("digits") - 2), width = getOption("width"), ...) ## S3 method for class 'mlogit' logLik(object, ...) ## S3 method for class 'mlogit' residuals(object, outcome = TRUE, ...) ## S3 method for class 'mlogit' fitted(object, outcome = TRUE, ...) ## S3 method for class 'mlogit' predict(object, newdata, returnData = FALSE, ...) ## S3 method for class 'mlogit' df.residual(object, ...) ## S3 method for class 'mlogit' terms(x, ...) ## S3 method for class 'mlogit' model.matrix(object, ...) ## S3 method for class 'mlogit' update(object, new, ...)

## Arguments

 x, object an object of class mlogit formula a symbolic description of the model to be estimated, new an updated formula for the update method, newdata a data.frame for the predict method, returnData if TRUE, the data is returned as an attribute, data the data: an mlogit.data object or an ordinary data.frame, subset an optional vector specifying a subset of observations, weights an optional vector of weights, na.action a function which indicates what should happen when the data contains 'NA's, start a vector of starting values, alt.subset a vector of character strings containing the subset of alternative on which the model should be estimated, reflevel the base alternative (the one for which the coefficients of individual-specific variables are normalized to 0), nests a named list of characters vectors, each names being a nest, the corresponding vector being the set of alternatives that belong to this nest, un.nest.el a boolean, if TRUE, the hypothesis of unique elasticity is imposed for nested logit models, unscaled a boolean, if TRUE, the unscaled version of the nested logit model is estimated, heterosc a boolean, if TRUE, the heteroscedastic logit model is estimated, rpar a named vector whose names are the random parameters and values the distribution : 'n' for normal, 'l' for log-normal, 't' for truncated normal, 'u' for uniform, probit if TRUE, a multinomial porbit model is estimated, R the number of function evaluation for the gaussian quadrature method used if heterosc=TRUE, the number of draws of pseudo-random numbers if rpar is not NULL, correlation only relevant if rpar is not NULL, if true, the correlation between random parameters is taken into account, halton only relevant if rpar is not NULL, if not NULL, halton sequence is used instead of pseudo-random numbers. If halton=NA, some default values are used for the prime of the sequence (actually, the primes are used in order) and for the number of elements droped. Otherwise, halton should be a list with elements prime (the primes used) and drop (the number of elements droped). random.nb only relevant if rpar is not NULL, a user-supplied matrix of random, panel only relevant if rpar is not NULL and if the data are repeated observations of the same unit ; if TRUE, the mixed-logit model is estimated using panel techniques, estimate a boolean indicating whether the model should be estimated or not: if not, the model.frame is returned, seed , digits the number of digits, width the width of the printing, outcome a boolean which indicates, for the fitted and the residuals methods whether a matrix (for each choice, one value for each alternative) or a vector (for each choice, only a value for the alternative chosen) should be returned, ... further arguments passed to mlogit.data or mlogit.optim.

## Details

For how to use the formula argument, see mFormula.

The data argument may be an ordinary data.frame. In this case, some supplementary arguments should be provided and are passed to mlogit.data. Note that it is not necessary to indicate the choice argument as it is deduced from the formula.

The model is estimated using the mlogit.optim function.

The basic multinomial logit model and three important extentions of this model may be estimated.

If heterosc=TRUE, the heteroscedastic logit model is estimated. J-1 extra coefficients are estimated that represent the scale parameter for J-1 alternatives, the scale parameter for the reference alternative being normalized to 1. The probabilities don't have a closed form, they are estimated using a gaussian quadrature method.

If nests is not NULL, the nested logit model is estimated.

If rpar is not NULL, the random parameter model is estimated. The probabilities are approximated using simulations with R draws and halton sequences are used if halton is not NULL. Pseudo-random numbers are drawns from a standard normal and the relevant transformations are performed to obtain numbers drawns from a normal, log-normal, censored-normal or uniform distribution. If correlation=TRUE, the correlation between the random parameters are taken into account by estimating the components of the cholesky decomposition of the covariance matrix. With G random parameters, without correlation G standard deviations are estimated, with correlation G * (G + 1) /2 coefficients are estimated.

## Value

An object of class "mlogit", a list with elements:

 coefficients the named vector of coefficients, logLik the value of the log-likelihood, hessian the hessian of the log-likelihood at convergence, gradient the gradient of the log-likelihood at convergence, call the matched call, est.stat some information about the estimation (time used, optimisation method), freq the frequency of choice, residuals the residuals, fitted.values the fitted values, formula the formula (a mFormula object), expanded.formula the formula (a formula object), model the model frame used, index the index of the choice and of the alternatives.

Yves Croissant

## References

McFadden, D. (1973) Conditional Logit Analysis of Qualitative Choice Behavior, in P. Zarembka ed., Frontiers in Econometrics, New-York: Academic Press.

McFadden, D. (1974) “The Measurement of Urban Travel Demand”, Journal of Public Economics, 3, pp. 303-328.

Train, K. (2004) Discrete Choice Modelling, with Simulations, Cambridge University Press.