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

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## 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

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