Logit Kernel (or Mixed Logit) Models for Large Multidimensional Choice Problems: Identification and Estimation
J L Bowman, Massachusetts Institute of Technology, US
This paper presents, and demonstrates in a case study, an identification rule for estimation of the class of error component logit kernel models.
Logit kernel (or mixed logit) models for large multidimensional choice problems: identification and estimation
John L. Bowman, Ph. D., Research Affiliate, Massachusetts Institute of Technology, 5 Beals Street Apt. 3, Brookline, MA 02446, USA, voice: +1-617-232-3478, email: John_L_Bowman@alum.mit.edu, website: http://JBowman.net
This paper presents an identification rule and insights for estimation of the class of error component logit kernel (EC) models?logit kernel (or mixed logit) models involving heteroscedasticity and subsets of alternatives with shared unobserved attributes. EC includes analogs of nested logit (NL), cross-nested logit (CNL), paired combinatorial logit (PCL), heteroscedastic logit (HL), their combinations, and other generalized extreme value (GEV) forms. The identification rule is necessary but not sufficient; however, it is simpler to use than the underlying necessary rank condition, and is adequate for complex model specifications. A case study demonstrates the specification, identification and estimation (using maximum simulated likelihood (MSL) with shuffled Halton draws) of the type of model for which EC is useful?one with large choice set and a choice outcome consisting of two or more variables considered simultaneously.
In transportation planning the classic multidimensional problem is joint mode-destination choice. Current modeling problems involve more dimensions, including, for example, simultaneous treatment of activity participation, purpose, travel modes, destinations and timing for travel by all household members. It is almost certain that shared unobserved attributes exist among various subsets of alternatives, and heteroscedasticity is likely, so using EC for such problems is attractive. However, large logit kernel models present issues related to identification and normalization.
As Walker et al point out, identification of logit kernel is an important and non-trivial task, because of the joint presence of the IID Gumbel disturbance and the probit-like covariance matrix. They review the necessary Order condition and sufficient Rank condition (for determining the number of identifiable covariance parameters) and what they call the Equality condition (for choosing restrictions to uniquely identify the parameters while preserving equality of the restricted differenced covariance matrix with its unrestricted counterpart). They then use these conditions to establish simple sufficient rules for correctly identifying and normalizing two kinds of logit kernel models: one-dimensional nesting (like the simplest nested logit) and heteroscedasticity. The strengths of the rules are their simplicity and sufficiency. Unfortunately, they only cover a small subset of the EC models with which we are concerned.
This paper presents a rule and accompanying procedure for identification and normalization of all EC models. The rule is necessary but not sufficient; it can be viewed as an enhancement of the order condition, providing a more refined method of discovering identification mistakes, but it does not in all cases guarantee that a particular specification is correctly identified. However, it correctly handles all the particular models presented in this paper. In most cases it identifies and normalizes them. In a few cases it detects the need to rely on a formal (and more tedious) application of the Rank and Equality conditions.
The case study models a worker?s day activity pattern?the choice among 162 alternatives for completing the non-work portion of the day, demonstrating the identification procedure. The estimated model has 57 utility parameters, and 14 significant covariance parameters for several overlapping subsets of the choice set.
The empirical work faced important practical questions about the estimation of large EC models. How many Shuffled Halton draws are necessary, and how long does it take? The model was estimated from a data set with 6170 observed choices, using Maximum Simulated Likelihood with shuffled Halton draws in ALOGIT 4EC, on a desktop machine with a 3.0 GHz Intel Pentium 4 processor and 1gigabyte of memory, operating under Windows XP Pro. In this case, it was advisable to use 1500 draws or more, and to estimate the model at least twice with different simulation seeds, to gain a sense of the variability caused by the simulation procedure. Estimating with 1500 draws per probability calculation took 80000 times longer than the corresponding MNL, and the run time was 28 days.
The identification rule presented in this paper provides a manageable method of increasing confidence that a large error component logit kernel (EC) model is identified, and the case study demonstrates its use. The case study also demonstrates the statistical advantages of logit kernel for a particular large multidimensional choice problem, identifies some development pitfalls to avoid, and concludes that using EC with maximum simulated likelihood should soon be practical for large real-world choice modeling problems.
Walker, J.. M. Ben-Akiva and D. Bolduc. Identification of the Logit Kernel (or Mixed Logit) Model. 2003. Working paper, available at http://mit.edu/jwalker/www/home.htm
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