Hoang Giang Pham, Tien Mai, Minh Ha Hoang
arXiv 1 Sep 2025 · Econometrics
arXiv:2509.01562 · PDF · DOI · OpenAlex · Extracted main text
In this paper, we revisit parameter estimation for multinomial logit (MNL), nested logit (NL), and tree-nested logit (TNL) models through the framework of convex conic optimization. Traditional approaches typically solve the maximum likelihood estimation (MLE) problem using gradient-based methods, which are sensitive to step-size selection and initialization, and may therefore suffer from slow or unstable convergence. In contrast, we propose a novel estimation strategy that reformulates these models as conic optimization problems, enabling more robust and reliable estimation procedures. Specifically, we show that the MLE for MNL admits an equivalent exponential cone program (ECP). For NL and TNL, we prove that when the dissimilarity (scale) parameters are fixed, the estimation problem is convex and likewise reducible to an ECP. Leveraging these results, we design a two-stage procedure: an outer loop that updates the scale parameters and an inner loop that solves the ECP to update the utility coefficients. The inner problems are handled by interior-point methods with iteration counts that grow only logarithmically in the target accuracy, as implemented in off-the-shelf solvers (e.g., MOSEK). Extensive experiments across estimation instances of varying size show that our conic approach attains better MLE solutions, greater robustness to initialization, and substantial speedups compared to standard gradient-based MLE, particularly on large-scale instances with high-dimensional specifications and large choice sets. Our findings establish exponential cone programming as a practical and scalable alternative for estimating a broad class of discrete choice models.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Train, K (2009) Discrete Choice Methods with Simulation | 1.000 | 20 | 6 | 100% |
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| 4 | Boyd, S. and Vandenberghe, L (2004) Convex Optimization | 0.928 | 4 | 3 | 100% |
| 5 | Nesterov, Y. and Nemirovskii, A (1994) Interior-Point Polynomial Algorithms in Convex Programming | 0.874 | 6 | 4 | 67% |
| 6 | McFadden, D (1974) Conditional logit analysis of qualitative choice behavior | 0.843 | 3 | 3 | 100% |
| 7 | Daganzo, C. F. and Kusnic, M (1993) Two properties of the nested logit model | 0.811 | 4 | 2 | 100% |
| 8 | Nesterov, Y. and Nemirovskii, A (1994) Interior-Point Polynomial Algorithms in Convex Programming | 0.794 | 6 | 4 | 50% |
| 9 | Chares, B (2009) Conic representations of the exponential function | 0.737 | 5 | 3 | 40% |
| 10 | Daly, A (1987) tree | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 37 scored citations.