Advertisement

Quantitative Marketing and Economics

, Volume 14, Issue 4, pp 271–323 | Cite as

Identification and semiparametric estimation of a finite horizon dynamic discrete choice model with a terminating action

  • Patrick Bajari
  • Chenghuan Sean Chu
  • Denis Nekipelov
  • Minjung ParkEmail author
Article

Abstract

We study identification and estimation of finite-horizon dynamic discrete choice models with a terminal action. We first demonstrate a new set of conditions for the identification of agents’ time preferences. Then we prove conditions under which the per-period utilities are identified for all actions in the agent’s choice-set, without having to normalize the utility for one of the actions. Finally, we develop a computationally tractable semiparametric estimator. The estimator uses a two-step approach that does not use either backward induction or forward simulation. Our methodology can be implemented using standard statistical packages without the need to write specialized computational routines, as it involves linear (or nonlinear) projections only. Monte Carlo studies demonstrate the superior performance of our estimator compared with existing two-step estimation methods. Monte Carlo studies further demonstrate that the ability to identify the per-period utilities for all actions is crucial for counterfactual predictions. As an empirical illustration, we apply the estimator to the optimal default behavior of subprime mortgage borrowers, and the results show that the ability to identify the discount factor, rather than assuming an arbitrary number as typically done in the literature, is also crucial for obtaining correct counterfactual predictions. These findings highlight the empirical relevance of key identification results of the paper.

Keywords

Finite horizon optimal stopping problem Time preferences Semiparametric estimation 

JEL Classification

C14 C18 C50 

Notes

Acknowledgments

We are grateful to the editor and anonymous reviewers for their insightful comments and constructive suggestions. The paper has also benefited from helpful comments by seminar participants at Chicago Booth, Olin Business School, Stanford GSB, Berkeley ARE, IO fest, Cirpée Conference on Industrial Organization, and Conference on “Recent Contributions to Inference in Game Theoretic Models” at University College London. All remaining errors are our own.

Supplementary material

11129_2016_9176_MOESM1_ESM.pdf (48 kb)
(PDF 47.8 KB)

References

  1. Aguirregabiria, V (2010). Another look at the identification of dynamic discrete decision processes: an application to retirement behavior. Journal of Business & Economic Statistics, 28(2), 201–218.CrossRefGoogle Scholar
  2. Aguirregabiria, V, & Magesan, A (2013). Euler equations for the estimation of dynamic discrete choice structural models. Advances in Econometrics, 31, 3–44.CrossRefGoogle Scholar
  3. Aguirregabiria, V, & Suzuki, J (2014). Identification and counterfactuals in dynamic models of market entry and exit. Quantitative Marketing and Economics, 12(3), 267–304.CrossRefGoogle Scholar
  4. Ai, C, & Chen, X (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica, 71(6), 1795–1843.CrossRefGoogle Scholar
  5. Ahn, H, & Manski, C (1993). Distribution theory for the analysis of binary choice under uncertainty with nonparametric estimation of expectations. Journal of Econometrics, 56(3), 291–321.CrossRefGoogle Scholar
  6. Altug, S, & Miller, R (1998). The effect of work experience on female wages and labour supply. Review of Economic Studies, 65(1), 45–85.CrossRefGoogle Scholar
  7. Andrews, D (1991). Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica, 59(2), 307–345.CrossRefGoogle Scholar
  8. Arcidiacono, P, Bayer, P, Blevins, J, & Ellickson, P (2015). Estimation of dynamic discrete choice models in continuous time with an application to retail competition. forthcoming in Review of Economic Studies.Google Scholar
  9. Arcidiacono, P, & Miller, R (2011). CCP estimation of dynamic discrete choice models with unobserved heterogeneity. Econometrica, 79, 1823–1867.CrossRefGoogle Scholar
  10. Arcidiacono, P, & Miller, R (2015a). Identifying dynamic discrete choice models off short panels. Working paper.Google Scholar
  11. Arcidiacono, P, & Miller, R (2015b). Nonstationary dynamic models with finite dependence. Working paper.Google Scholar
  12. Bajari, P, Benkard, L, & Levin, J (2007). Estimating dynamic models of imperfect competition. Econometrica, 75(5), 1331–1370.CrossRefGoogle Scholar
  13. Bajari, P, Hong, H, & Nekipelov, D (2013). Game theory and econometrics: A survey of some recent research. In Advances in economics and econometrics: tenth world congress of econometric society (Vol. 3). Cambridge University Press.Google Scholar
  14. Beauchamp, A (2015). Regulation, imperfect competition, and the U.S. Abortion Market. International Economic Review, 56(3), 963–996.CrossRefGoogle Scholar
  15. Chen, X (2007). Large sample sieve estimation of semi-nonparametric models. Handbook of Econometrics, 7, 5549–5632.CrossRefGoogle Scholar
  16. Chen, X, Linton, O, & van Keilegom, I (2003). Estimation of semiparametric models when the criterion function is not smooth. Econometrica, 71(5), 1591–1608.Google Scholar
  17. Chen, X, Chernozhukov, V, Lee, S, & Newey, W (2014). Local identification of nonparametric and semiparametric models. Econometrica, 82(2), 785–809.CrossRefGoogle Scholar
  18. Chung, D, Steenburgh, T, & Sudhir, K. (2014). Do bonuses enhance sales productivity? A dynamic structural analysis of bonus-based compensation plans. Marketing Science, 33(2), 165–187.CrossRefGoogle Scholar
  19. Dubé, J-P, Hitsch, G, & Jindal, P (2014). The joint identification of utility and discount functions from stated choice data: an application to durable goods adoption. Quantitative Marketing and Economics, 12, 331–377.CrossRefGoogle Scholar
  20. Duffie, D, & Singleton, KJ (1997). An econometric model of the term structure of interest-rate swap yields. Journal of Finance, 52(4), 1287–1321.CrossRefGoogle Scholar
  21. Eckstein, Z, & Wolpin, K (1999). Why youths drop out of high school: The impact of preferences, opportunities, and abilities. Econometrica, 67(6), 1295–1339.CrossRefGoogle Scholar
  22. Fang, H, & Wang, Y (2015). Estimating dynamic discrete choice models with hyperbolic discounting, with an application to mammography decisions. International Economic Review, 56(2), 565–596.CrossRefGoogle Scholar
  23. Frederick, S, Loewenstein, G, & O’Donohue, T (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, 40, 350–401.CrossRefGoogle Scholar
  24. Härdle, W (1990). Applied nonparametric regression. Cambridge University Press.Google Scholar
  25. Hausman, J (1979). Individual discount rates and the purchase and utilization of energy-using durables. Bell Journal of Economics, 10(1), 33–54.CrossRefGoogle Scholar
  26. Heckman, J, & Navarro, S (2007). Dynamic discrete choice and dynamic treatment effects. Journal of Econometrics, 136(2), 341–396.CrossRefGoogle Scholar
  27. Hotz, J, & Miller, R (1993). Conditional choice probabilities and the estimation of dynamic models. Review of Economic Studies, 60(3), 497–529.CrossRefGoogle Scholar
  28. Hotz, J, Miller, R, Sanders, S, & Smith, J (1994). A simulation estimator for dynamic models of discrete choice. Review of Economic Studies, 61(2), 265–289.CrossRefGoogle Scholar
  29. Joensen, JS (2009). Academic and labor market success: The impact of student employment, abilities, and preferences. Working paper.Google Scholar
  30. Judd, K (1998). Numerical methods in economics. MIT Press.Google Scholar
  31. Kalouptsidi, M (2014). Time to build and fluctuations in bulk shipping. American Economic Review, 104(2), 564–608.CrossRefGoogle Scholar
  32. Kalouptsidi, M, Scott, P, & Souza-Rodrigues, E (2016). Identification of counterfactuals in dynamic discrete choice models. Working Paper.Google Scholar
  33. Magnac, T, & Thesmar, D (2002). Identifying dynamic discrete decision processes. Econometrica, 70(2), 801–816.CrossRefGoogle Scholar
  34. Mammen, E, Rothe, C, & Schienle, M (2012). Nonparametric regression with nonparametrically generated covariates. Annals of Statistics, 40(2), 1132–1170.CrossRefGoogle Scholar
  35. Manski, C (1991). Nonparametric estimation of expectations in the analysis of discrete choice under uncertainty. In: Nonparametric and semiparametric methods in econometrics and statistics: proceedings of the fifth international symposium in economic theory and econometrics. Cambridge University Press.Google Scholar
  36. Manski, C (1993). Identification of endogenous social effects: The reflection problem. Review of Economic Studies, 60(3), 531–542.CrossRefGoogle Scholar
  37. Manski, C (2000). Identification problems and decisions under ambiguity: Empirical analysis of treatment response and normative analysis of treatment choice. Journal of Econometrics, 95(2), 415–442.CrossRefGoogle Scholar
  38. Newey, W (1997). Convergence rates and asymptotic normality for series estimators. Journal of Econometrics, 79, 147–168.CrossRefGoogle Scholar
  39. Newey, W, & Powell, J (2003). Instrumental variable estimation of nonparametric models. Econometrica, 71(5), 1565–1578.CrossRefGoogle Scholar
  40. Norets, A, & Takahashi, S (2013). On the surjectivity of the mapping between utilities and choice probabilities. Quantitative Economics, 4, 149–155.CrossRefGoogle Scholar
  41. Norets, A, & Tang, X (2014). Semiparametric inference in dynamic binary choice models. Review of Economic Studies, 81(3), 1229–1262.CrossRefGoogle Scholar
  42. Pagan, A, & Ullah, A (1999). Nonparametric econometrics. Cambridge University Press.Google Scholar
  43. Pollard, D (1984). Convergence of stochastic processes. Springer-Verlag.Google Scholar
  44. Pesendorfer, M., & Schmidt-Dengler, P. (2008). Asymptotic least squares estimators for dynamics games. Review of Economic Studies, 75(3), 901–928.CrossRefGoogle Scholar
  45. Rust, J (1987). Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica, 55(5), 999–1033.CrossRefGoogle Scholar
  46. Rust, J (1994). Structural estimation of Markov decision processes. In Engle, R, & McFadden, D (Eds.) Handbook of econometrics, Vol. 4. Amsterdam: North-Holland.Google Scholar
  47. Rust, J, & Phelan, C (1997). How social security and medicare affect retirement behavior in a world of incomplete markets. Econometrica, 65(4), 781–831.CrossRefGoogle Scholar
  48. Scott, P (2013). Dynamic discrete choice estimation of agricultural land use. Working paper.Google Scholar
  49. van der Vaart, A, & Wellner, J (1996). Weak convergence and empirical processes. Springer.Google Scholar
  50. Wong, WH, & Shen, X (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLES. Annals of Statistics, 23(2), 339–362.CrossRefGoogle Scholar
  51. Yao, S, Mela, CF, Chiang, J, & Chen, Y (2012). Determining consumers’ discount rates with field studies. Journal of Marketing Research, 49(6), 822–841.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Patrick Bajari
    • 1
    • 2
  • Chenghuan Sean Chu
    • 3
  • Denis Nekipelov
    • 4
  • Minjung Park
    • 5
    Email author
  1. 1.University of WashingtonSeattleUSA
  2. 2.NBERCambridgeUSA
  3. 3.FacebookMenlo ParkUSA
  4. 4.University of VirginiaCharlottesvilleUSA
  5. 5.University of CaliforniaBerkeleyUSA

Personalised recommendations