Aguirregabiria and Mira (2010) Dynamic Discrete Choice Structural Models: A Survey

Summary

Comprehensive survey of estimation methods for dynamic discrete choice (DDC) structural models, updating earlier reviews by Eckstein-Wolpin (1989) and Rust (1994a). The paper organizes estimators around whether they require solving the full dynamic programming (DP) problem at each trial parameter vector, contrasting Rust's (1987) nested fixed-point (NFXP) algorithm with conditional-choice-probability (CCP)-based two-step methods (Hotz-Miller 1993) and the nested pseudo-likelihood (NPL) recursive algorithm (Aguirregabiria-Mira 2002). Coverage spans single-agent models, competitive equilibrium models, and dynamic games under Markov perfect equilibrium (MPE).

Key Claims

• Rust's NFXP: outer Berndt-Hall-Hall-Hausman (BHHH) gradient search over the likelihood + inner DP solver (value/policy iteration); requires full DP solution at each trial $\theta$; outer loop converges to partial maximum likelihood estimate (MLE) but inner loop is computationally intensive.
• Hotz-Miller (1993) CCP invertibility: the mapping from CCPs to value function differences is one-to-one; enables a two-step estimator — (1) nonparametrically estimate $\hat{P}(a\mid x)$, (2) substitute into a pseudo-likelihood — fast but finite-sample biased when $\hat{P}$ is imprecise.
• NPL algorithm (Aguirregabiria-Mira 2002): iterative CCP fixed-point update $\hat{P}^{k+1}=\Lambda(\hat{P}^k,\hat{\theta}(\hat{P}^k))$; fixed point coincides with partial MLE; better finite-sample properties than Hotz-Miller two-step; delivers full DP solution upon convergence.
• Simulation-based CCP (Hotz et al. 1994): forward simulation replaces analytical value function inversion for large or continuous state spaces; especially useful for Keane-Wolpin (1994) occupational choice models.
• Three model classes: (1) single-agent (Rust 1987 bus engine replacement); (2) competitive equilibrium (Lee-Wolpin 2006 intersectoral labor mobility); (3) dynamic games under MPE (Ericson-Pakes 1995 entry/exit oligopoly).
• Finite-sample bias of CCP estimators: imprecise nonparametric $\hat{P}$ propagates into the pseudo-likelihood, creating bias absent in NFXP; NPL iteration reduces this iteratively.
• Additional complications: initial conditions problem for serially correlated unobservables; finite mixture expectation-maximization (EM) (Arcidiacono-Jones 2003) for unobserved heterogeneity; Geweke-Hajivassiliou-Keane (GHK) simulator (Keane 1993) for correlated multivariate normal error structures.

Concepts Introduced or Extended

• Dynamic Discrete Choice Model

Entities Mentioned

• Victor Aguirregabiria
• Pedro Mira
• Michael Keane

Quotes

"The Hotz and Miller (1993) inversion theorem is a key result that enables the construction of estimators that avoid the repeated solution of the dynamic programming problem."

"The NPL algorithm provides estimators with better finite-sample properties than the two-step CCP estimator while maintaining the computational advantages of not requiring a full DP solution at each iteration."

My Take

The survey's central contribution is clarifying the computational–statistical trade-off landscape: NFXP is asymptotically gold-standard but expensive; CCP two-step is fast but loses finite-sample precision; NPL bridges the gap iteratively. The treatment of dynamic games is valuable but acknowledges that MPE multiplicity and identification are genuinely hard open problems. The paper predates Markov chain Monte Carlo (MCMC) approaches to DDC estimation (Imai-Jain-Ching 2009; Norets 2009), which would become a productive strand, and barely treats Bayesian methods. Applications are heavily weighted toward labor and industrial organization (IO); the methods have since spread to health, education, and marketing.