A Diagnostic Test for the Multinomial Logit Model

I propose a Lagrange multiplier test for the multinomial logit model against the dogit model (Gaudry and Dagenais 1979) as the alternative hypothesis. In view of the well-known drawback of the restrictive property of independence from irrelevant alternatives implied by the multinomial logit model, a specification test has much to recommend it. Finite sample properties of the test are studied using a Monte Carlo experiment, and the test's power against the nested multinomial logit model and the multinomial probit model is investigated. The test is found to be sensitive to the values of the regression parameters of the linear random utility function.     

Kernel-based semiparametric multinomial logit modelling of political party preferences

Conventional, parametric multinomial logit models are in general not sufficient for capturing the complex structures of electorates. In this paper, we use a semiparametric multinomial logit model to give an analysis of party preferences along individuals’ characteristics using a sample of the German electorate in 2006. Germany is a particularly strong case for more flexible nonparametric approaches in this context, since due to the reunification and the preceding different political histories the composition of the electorate is very complex and nuanced. Our analysis reveals strong interactions of the covariates age and income, and highly nonlinear shapes of the factor impacts for each party’s likelihood to be supported. Notably, we develop and provide a smoothed likelihood estimator for semiparametric multinomial logit models, which can be applied also in other application fields, such as, e.g., marketing.     

A comparison of artificial neural network and multinomial logit models in predicting mergers

A merger proposal discloses a bidder firm's desire to purchase the control rights in a target firm. Predicting who will propose (bidder candidacy) and who will receive (target candidacy) merger bids is important to investigate why firms merge and to measure the price impact of mergers. This study investigates the performance of artificial neural networks and multinomial logit models in predicting bidder and target candidacy. We use a comprehensive data set that covers the years 1979–2004 and includes all deals with publicly listed bidders and targets. We find that both models perform similarly while predicting target and non-merger firms. The multinomial logit model performs slightly better in predicting bidder firms.     

Simple inference in multinomial and ordered logit

This paper provides some simple methods of interpreting the coefficients in multinomial logit and ordered logit models. These methods are summarized in Propositions concerning the magnitudes, signs, and patterns of partial derivatives of the outcome probabilities with respect to the exogenousvariables. The paper also provides an empirical example illustrating the use of these Propositions.     

Simple inference in multinomial and ordered logit

This paper provides some simple methods of interpreting the coefficients in multinomial logit and ordered logit models. These methods are summarized in Propositions concerning the magnitudes, signs, and patterns of partial derivatives of the outcome probabilities with respect to the exogenousvariables. The paper also provides an empirical example illustrating the use of these Propositions.     

Using compositional and Dirichlet models for market share regression

When the aim is to model market shares, the marketing literature proposes some regression models which can be qualified as attraction models. They are generally derived from an aggregated version of the multinomial logit model. But aggregated multinomial logit models (MNL) and the so-called generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross effect parameters. In this paper, we consider alternative models: the Dirichlet model (DIR) and the compositional model (CODA). DIR allows to introduce brand-specific parameters and CODA allows additionally to consider cross effect parameters. We show that these two models can be written in a similar fashion, called attraction form, as the MNL and the GMCI models. As market share models are usually interpreted in terms of elasticities, we also use this notion to interpret the DIR and CODA models. We compare the properties of the models in order to explain why CODA and DIR models can outperform traditional market share models. An application to the automobile market is presented where we model brands market shares as a function of media investments, controlling for the brands price and scrapping incentive. We compare the quality of the models using measures adapted to shares.     

Response shrinkage estimation in multinomial logit models

A shrinkage estimation method for multinomial logit models is developed. The proposed method is based on shrinking the responses for each category towards the underlying probabilities. The estimator is also used in combination with Pregibon's resistant fitting. The resulting estimator can also be used to control the over-estimation of Pregibon's resistant estimator. The proposed method handles not only the problem of separation in multinomial logit models but estimates also exist when the number of covariates is large relative to the sample size. Estimates exist even when the MLE does not exist. Estimates can be easily computed with all commonly used statistical packages supporting the fitting procedures with weights. Estimates are compared with the usual MLE and Firth's bias reduction technique in a simulation study and an application.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

Regression Analysis of Multivariate Fractional Data

The present article discusses alternative regression models and estimation methods for dealing with multivariate fractional response variables. Both conditional mean models, estimable by quasi-maximum likelihood, and fully parametric models (Dirichlet and Dirichlet-multinomial), estimable by maximum likelihood, are considered. A new parameterization is proposed for the parametric models, which accommodates the most common specifications for the conditional mean (e.g., multinomial logit, nested logit, random parameters logit, dogit). The text also discusses at some length the specification analysis of fractional regression models, proposing several tests that can be performed through artificial regressions. Finally, an extensive Monte Carlo study evaluates the finite sample properties of most of the estimators and tests considered.     

Properties of Bayes Factors Based on Test Statistics

Abstract.  This article examines the consistency, interpretation and application of Bayes factors constructed from standard test statistics. Primary conclusions are that Bayes factors based on multinomial and normal test statistics are consistent for suitable choices of the hyperparameters used to specify alternative hypotheses, and that such constructions can be extended to obtain consistent Bayes factors based on likelihood ratio statistics. A connection between Bayes factors based on likelihood ratio statistics and the Bayesian information criterion is exposed, as is a connection between Bayes factors based on F statistics and parametric Bayes factors based on normal-inverse gamma models. Similarly, Bayes factors based on chi-squared statistics for multinomial data are shown to provide accurate approximations to Bayes factors based on multinomial/Dirichlet models. An illustration of how the simple form of these Bayes factors can be exploited to generate easily interpretable summaries of the experimental 'weight of evidence' is provided.     

Overview of multinomial models for ordinal data

The purpose of this paper is to relate a number of multinomial models currently in use for ordinal response data in a unified manner. By studying generalized logit models, proportional generalized odds ratio models and proportional generalized hazard models under different parameterizations, we conclude that there are only four different models and they can be specified genericaUy in a uniform way. These four models all possess the same stochastic ordering property and we compare them graphically in a simple case. Data from the NHLBI TYPE II study (Brensike et al (1984)) is used to illustrate these models. We show that the BMDP programs LE and PR can be employed in computing maximum likelihood estimators for these four models.     

Model-Robust Design of Conjoint Choice Experiments

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for attribute interaction models. In this article, three types of Bayesian D-optimal designs for the multinomial logit model are studied: main-effects designs, interaction-effects designs, and composite designs. Simulation studies are used to show that in situations where a researcher is not sure whether or not attribute interaction effects are present, it is best to take into account interactions in the design stage. In particular, it is shown that a composite design constructed by including an interaction-effects model and a main-effects model in the design criterion is most robust against misspecification of the underlying model when it comes to making precise predictions.     

The multinomial logistic regression model for predicting the discharge status after liver transplantation: estimation and diagnostics analysis

The multinomial logistic regression model (MLRM) can be interpreted as a natural extension of the binomial model with logit link function to situations where the response variable can have three or more possible outcomes. In addition, when the categories of the response variable are nominal, the MLRM can be expressed in terms of two or more logistic models and analyzed in both frequentist and Bayesian approaches. However, few discussions about post modeling in categorical data models are found in the literature, and they mainly use Bayesian inference. The objective of this work is to present classic and Bayesian diagnostic measures for categorical data models. These measures are applied to a dataset (status) of patients undergoing kidney transplantation.     

Stochastic variational inference for large-scale discrete choice models using adaptive batch sizes

Discrete choice models describe the choices made by decision makers among alternatives and play an important role in transportation planning, marketing research and other applications. The mixed multinomial logit (MMNL) model is a popular discrete choice model that captures heterogeneity in the preferences of decision makers through random coefficients. While Markov chain Monte Carlo methods provide the Bayesian analogue to classical procedures for estimating MMNL models, computations can be prohibitively expensive for large datasets. Approximate inference can be obtained using variational methods at a lower computational cost with competitive accuracy. In this paper, we develop variational methods for estimating MMNL models that allow random coefficients to be correlated in the posterior and can be extended easily to large-scale datasets. We explore three alternatives: (1) Laplace variational inference, (2) nonconjugate variational message passing and (3) stochastic linear regression. Their performances are compared using real and simulated data. To accelerate convergence for large datasets, we develop stochastic variational inference for MMNL models using each of the above alternatives. Stochastic variational inference allows data to be processed in minibatches by optimizing global variational parameters using stochastic gradient approximation. A novel strategy for increasing minibatch sizes adaptively within stochastic variational inference is proposed.     

Comparison of Ordinary and Penalized Power-Divergence Test Statistics for Small and Moderate Samples in Three-Way Contingency Tables via Simulation

In this article, size and power properties of the ordinary and penalized power-divergence test statistics have been compared to test a nested sequence of log-linear models for various penalization and λ values. 2 × 2 × 2 contingency tables distributed according to a multinomial distribution are considered. The simulation study results reveal that the penalized power-divergence test statistics with penalization value of one and λ = ?1.5, ?2 are the best choices for small and moderate samples, respectively.     

A Proportional Random Utility Approach to Qualitative Response Models

A new functional form of the response probability for a qualitative response model is proposed. The new model is flexible enough to avoid the constraint of independence from irrelevant alternatives, which is perceived as a weakness of the multinomial logit model in some applications. It is computationally simpler than the multinomial probit model and is promising for analyzing problems with a moderate number of alternatives.     

Small area estimates of labour force participation under a multinomial logit mixed model

Summary.  A new methodology is developed for estimating unemployment or employment characteristics in small areas, based on the assumption that the sample totals of unemployed and employed individuals follow a multinomial logit model with random area effects. The method is illustrated with UK labour force data aggregated by sex–age groups. For these data, the accuracy of direct estimates is poor in comparison with estimates that are derived from the multinomial logit model. Furthermore, two different estimators of the mean-squared errors are given: an analytical approximation obtained by Taylor linearization and an estimator based on bootstrapping. A simulation study for comparison of the two estimators shows the good performance of the bootstrap estimator.     

Flexible Covariance Structures for Categorical Dependent Variables Through Finite Mixtures of Generalized Extreme Value Models

A new class of finite mixture discrete choice models, denoted FinMix (fīn m?ks), is introduced. These arise from the combination of a finite number of core Generalized Extreme Value (GEV) models to achieve more flexible functional forms, particularly in terms of error covariance structures. Example members of the class include combinations of (1) Multinomial Logit (MNL) models with differing scales, (2) multinomial logit with nested MNL models, (3) tree extreme value models with differing preference trees, and so on. Compatibility of FinMix models with utility maximization is easily determined, which permits empirical investigation of the suitability of specific model forms for economic evaluation exercises.     

二元选择面板模型的设定检验

 本文将RESET检验扩展到二元选择面板数据模型的设定,考察了固定效应Probit模型和Logit模型的设定检验,包括异方差、遗漏变量和分布误设的检验。模拟结果表明Logit模型的RESET设定检验显示良好的水平和功效,而Probit模型的RESET检验可能由于估计方法的选择导致在某些方面的功效表现不好。但总体说来,在二元选择面板数据模型的设定检验上,RESET检验仍然是一个较好的选择。     

Considerations for Effective Simulation Study Analysis

This article discusses estimation of several percentiles simultaneously, develops a simple test to compare the sizes of two test statistics, and considers the use of logit models to adjust power curves to have the same null hypothesis level.