27 - Efficient Estimation of Nested Logit Models Using Choice-based Samples

Abstract

This paper describes how choice-based samples are often used in transportation and other fields in order to obtain better information about alternatives that are infrequently chosen in the population. For example, when modeling travelers’ mode choice in the United States, it is often desirable to over-sample transit users because of the low frequency with which the mode is chosen relative to other modes. However, standard maximum likelihood estimation (MLE) techniques do not obtain consistent estimated of the underlying parameters. The most widely used approach to obtaining consistent estimators is to use weighted exogenous sampling maximum likelihood (WESML) estimators. However, WESML estimators are not asymptotically efficient. Efficient estimators can be obtained using choice-based samples when using the multinomial logit (MNL) model because, when a full set of alternative specific constants (ASCs) are included in the model, the standard (exogenous sample) maximum likelihood (ESML) estimate can be used with adjustment. This paper shows that the MLE or ESML estimator can also be used for nested logit (NL) models provided that the model has a full set of ASCs and that population share information is known. The ability to recover consistent parameter estimates for NL models using ESML estimators provides practitioners and researchers with a practical method to obtain estimators that are both consistent and asymptotically efficient for NL models using choice-based data. The balance of the paper contains several sections: MNL and NL modes are briefly reviewed; common estimators and their relations to the sample process are described; McFadden’s proof characterizing inconsistency for ESML estimators for choice-based MNL models is summarized; a general procedure for characterizing consistency of ESML estimators is presented; the application of the general procedure to two-level NL models and three0level NL models is illustrated; an explanation for why McFadden’s procedure cannot be directly extended to the class of generalized extreme value (GEV) models is presented; and finally, major conclusions are summarized.