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REAL ANALYTIC DISCRETE CHOICE MODELS OF DEMAND: THEORY AND IMPLICATIONS

Published online by Cambridge University Press:  20 May 2024

Alessandro Iaria Open the ORCID record for Alessandro Iaria [Opens in a new window]  and
Ao Wang Open the ORCID record for Ao Wang [Opens in a new window]
Alessandro Iaria
Affiliation:
University of Bristol and CEPR
Ao Wang*
Affiliation:
University of Warwick and CAGE Research Centre
*
Address correspondence to Ao Wang, Department of Economics, University of Warwick and CAGE Research Centre, Coventry, United Kingdom, e-mail: ao.wang@warwick.ac.uk
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Abstract

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We demonstrate that a large class of discrete choice models of demand can be approximated by real analytic demand models. We obtain this result by combining (i) a novel real analytic property of the mixed logit and the mixed probit models with any distribution of random coefficients and (ii) an approximation property of finite mixtures of Gumbel and Gaussian distributions. To illustrate some of the implications of this result, we discuss how real analyticity facilitates nonparametric and semi-nonparametric identification, extrapolation to hypothetical counterfactuals, numerical implementation of demand inverses, and numerical implementation of the maximum likelihood estimator.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 49
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

We are grateful to the Editor (Peter C. B. Phillips), the Co-Editor (Simon Lee), and two anonymous referees for comments and suggestions which greatly improved the article. An early version of this paper circulated under the title “The Mixed Logit and Mixed Probit are Real Analytic.”

References

REFERENCES

Aguirregabiria, V., & Mira, P. (2007). Sequential estimation of dynamic discrete games. Econometrica , 75(1), 1–53.CrossRefGoogle Scholar
Allen, R., & Rehbeck, J. (2020). Identification of random coefficient latent utility models. Available at SSRN 3545696.CrossRefGoogle Scholar
Berry, S., Levinsohn, J., & Pakes, A. (1995). Automobile prices in market equilibrium. Econometrica , 63, 841–890.CrossRefGoogle Scholar
Berry, S., Levinsohn, J., & Pakes, A. (2004). Differentiated products demand systems from a combination of micro and macro data: The new car market. Journal of Political Economy , 112(1), 68–105.CrossRefGoogle Scholar
Berry, S. T., & Haile, P. A. (2014). Identification in differentiated products markets using market level data. Econometrica , 82(5), 1749–1797.Google Scholar
Berry, S. T., & Haile, P. A. (2018). Identification of nonparametric simultaneous equations models with a residual index structure. Econometrica , 86(1), 289–315.CrossRefGoogle Scholar
Compiani, G. (2022). Market counterfactuals and the specification of multiproduct demand: A nonparametric approach. Quantitative Economics , 13(2), 545–591.CrossRefGoogle Scholar
Conlon, C., & Gortmaker, J. (2020). Best practices for differentiated products demand estimation with PyBLP. RAND Journal of Economics , 51(4), 1108–1161.CrossRefGoogle Scholar
Dubé, J.-P., Fox, J. T., & Su, C.-L. (2012a). Improving the numerical performance of static and dynamic aggregate discrete choice random coefficients demand estimation. Econometrica , 80(5), 2231–2267.Google Scholar
Dubois, P., Griffith, R., & O’Connell, M. (2020). How well targeted are soda taxes? American Economic Review , 110(11), 3661–3704.CrossRefGoogle Scholar
Fox, J. T., & Gandhi, A. (2016). Nonparametric identification and estimation of random coefficients in multinomial choice models. RAND Journal of Economics , 47(1), 118–139.CrossRefGoogle Scholar
Fox, J. T., Kim, K., Ryan, S. P., & Bajari, P. (2012). The random coefficients logit model is identified. Journal of Econometrics , 166(2), 204–212.CrossRefGoogle Scholar
Goolsbee, A., & Petrin, A. (2004). The consumer gains from direct broadcast satellites and the competition with cable TV. Econometrica , 72(2), 351–381.CrossRefGoogle Scholar
Gowrisankaran, G., & Rysman, M. (2012). Dynamics of consumer demand for new durable goods. Journal of Political Economy , 120(6), 1173–1219.CrossRefGoogle Scholar
Hausman, J. A., & Wise, D. A. (1978). A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences. Econometrica , 46, 403–426.CrossRefGoogle Scholar
Iskhakov, F., Lee, J., Rust, J., Schjerning, B., & Seo, K. (2016). Comment on “constrained optimization approaches to estimation of structural models”. Econometrica , 84(1), 365–370.CrossRefGoogle Scholar
Knittel, C. R., & Metaxoglou, K. (2014). Estimation of random-coefficient demand models: Two empiricists’ perspective. Review of Economics and Statistics , 96(1), 34–59.CrossRefGoogle Scholar
Krasikov, I. (2004). New bounds on the Hermite polynomials. East Journal on Approximations , 10(3), 355–362.Google Scholar
Lang, S. (2012). Real and functional analysis (Vol. 142). Springer Science & Business Media.Google Scholar
Lee, J., & Seo, K. (2016). Revisiting the nested fixed-point algorithm in BLP random coefficients demand estimation. Economics Letters , 149, 67–70.CrossRefGoogle Scholar
Masten, M. A. (2018). Random coefficients on endogenous variables in simultaneous equations models. Review of Economic Studies , 85(2), 1193–1250.CrossRefGoogle Scholar
McFadden, D., & Train, K. (2000). Mixed MNL models for discrete response. Journal of Applied Econometrics , 15, 447–470.3.0.CO;2-1>CrossRefGoogle Scholar
Mugnier, M., & Wang, A. (2022). Identification and (fast) estimation of large nonlinear panel models with two-way fixed effects. Available at SSRN 4186349.CrossRefGoogle Scholar
Nevo, A. (2000). Mergers with differentiated products: The case of the ready-to-eat cereal industry. RAND Journal of Economics , 31, 395–421.CrossRefGoogle Scholar
Nguyen, T. T., Nguyen, H. D., Chamroukhi, F., & McLachlan, G. J. (2020). Approximation by finite mixtures of continuous density functions that vanish at infinity. Cogent Mathematics & Statistics , 7(1), 1750861.CrossRefGoogle Scholar
Petrin, A. (2002). Quantifying the benefits of new products: The case of the minivan. Journal of Political Economy , 110(4), 705–729.CrossRefGoogle Scholar
Rheinboldt, W. C. (1988). On a theorem of S. Smale about Newton’s method for analytic mappings. Applied Mathematics Letters , 1(1), 69–72.CrossRefGoogle Scholar
Rudin, W. (1976). Principles of mathematical analysis . (3rd ed.) McGraw-Hill.Google Scholar
Rust, J. (1987). Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica , 55, 999–1033.CrossRefGoogle Scholar
Salanié, B., & Wolak, F. A. (2019). Fast, “robust,” and approximately correct: Estimating mixed demand systems. Discussion paper, National Bureau of Economic Research.Google Scholar
Seim, K. (2006). An empirical model of firm entry with endogenous product-type choices. RAND Journal of Economics , 37(3), 619–640.CrossRefGoogle Scholar
Smale, S. (1986). Newton’s method estimates from data at one point. In The merging of disciplines: New directions in pure, applied, and computational mathematics: Proceedings of a Symposium Held in Honor of Gail S. Young at the University of Wyoming, August 8–10, 1985. Sponsored by the Sloan Foundation, the National Science Foundation, and Air Force Office of Scientific Research (pp. 185–196). Springer.Google Scholar
Smith, H. (1935). Discontinuous demand curves and monopolistic competition: A special case. Quarterly Journal of Economics , 49(3), 542–550.CrossRefGoogle Scholar
Stinchcombe, M. B., & White, H. (1998). Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory , 14(3), 295–325.CrossRefGoogle Scholar
Su, C.-L., & Judd, K. L. (2012). Constrained optimization approaches to estimation of structural models. Econometrica , 80(5), 2213–2230.Google Scholar
Train, K. E., & Winston, C. (2007). Vehicle choice behavior and the declining market share of US automakers. International Economic Review , 48(4), 1469–1496.CrossRefGoogle Scholar
Wang, A. (2023). Sieve BLP: A semi-nonparametric model of demand for differentiated products. Journal of Econometrics , 235(2), 325–351.CrossRefGoogle Scholar