Implementing Maximum Likelihood Estimation of Empirical Matching Models
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Implementing Maximum Likelihood Estimation of Empirical Matching Models MPEC versus NFXP Baiyu Dong1 · Yu‑Wei Hsieh1 · Xing Zhang2 Accepted: 2 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose two Mathematical Programming with Equilibrium Constraints (MPEC) formulations: the MPEC-Sparse and the MPEC-Dense to estimate a class of separable matching models. We compare MPEC with the Nested Fixed-Point (NFXP) algorithm—a well-received method in the literature of structural estimation. Using both simulated and actual data, we find that MPEC is more robust than NFXP in terms of convergence and solution quality. In terms of computing time, MPECDense is 9 to 20 times faster than NFXP in simulations. For practitioners, MPEC is considerably simpler to program. Keywords Aggregate matching · Two-sided matching · Separable matching models · Mathematical programming with equilibrium constraints · Nested fixed-point algorithm · Constrained optimization
1 Introduction In this paper, we evaluate the numerical performance of two competing algorithms—the mathematical programming with equilibrium constraints (MPEC) and the nested fixed-point algorithm (NFXP)—to estimate the separable matching models. Pioneered by the work of Choo and Siow (2006) (hereinafter, CS), the separable matching models have become indispensable tools to study matching data. Recent * Yu‑Wei Hsieh [email protected] Baiyu Dong [email protected] Xing Zhang [email protected] 1
Department of Economics, University of Southern California, Los Angeles, USA
2
Microsoft Corporation, Redmond, USA
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developments include the transferable utility (TU) matching models studied by Fox (2010), Chiappori et al. (2010), Galichon and Salanié (2015), and Chiappori et al. (2017); imperfectly transferable utility (ITU) matching models studied by Galichon et al. (2018); and non-transferable utility (NTU) matching models studied by Galichon and Hsieh (2019), among others. Chiappori and Salanié (2016) and Galichon and Salanié (2017) provide an excellent survey. The aforementioned literature primarily focuses on the identification, equilibrium analysis, comparative statics, and extensions to cases under alternative behavioral and distributional assumptions. The numerical issues associated with estimating the separable matching models, however, remain largely unexplored. 1.1 Literature To the best of our knowledge, Galichon and Salanié (2015) is the only paper that proposes a numerical method for estimating the separable matching models via the maximum likelihood estimation (MLE). Their algorithm belongs to the class of NFXP, which is widely deployed in the structural estimation literature, including the dynamic discrete-choice model of Rust (1994) and the demand model of Berry et al. (1995). For the models mentioned above, the likelihood or the moment function depends on a set of equilibrium conditions, which in turn depends on the structural parameters. Since the objective f
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