Nested Fixed Point Maximum Likelihood Algorithm

Author: John Rust

Reference: John Rust (1987) "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher" Econometrica 55-5 999-1033.

Description: The nested fixed point algorithm is a maximum likelihood estimation algorithm that computes maximum likelihood estimates of structural parameters of a wide class of discrete control processes; i.e. stochastic processes that are solutions to infinite horizon Markovian decision problems with discrete choice or control variables. In general these problems do not possess analytic or closed-form solutions, but must be solved numerically. Under certain assumptions one can show that the numerical solution to such problems is equivalent to the computation of a fixed point to a differentiable contraction mapping. The simple idea behind the nested fixed point algorithm is to solve the stochastic control problem numerically by computing the associated functional fixed point as a subroutine nested within a standard nonlinear maximum likelihood optimization algorithm; hence the name, ``nested fixed point algorithm''. A 1988 paper by John Rust entitled ``Maximum Likelihood Estimation of Discrete Control Processes'' (SIAM Journal on Control and Optimization 26-5 1006-1024) introduced a class of discrete control processes and the nested fixed point algorithm, but concentrated on developing the asymptotic estimation theory for such processes instead of focusing on the computational details of the algorithm itself. In order to provide a concrete application of the theory presented in the SIAM paper, a subsequent paper was written to illustrate the use of the nested fixed point algorithm to estimate the structural parameters of a simple model of optimal replacement of bus engines at the Madison Metropolitan Bus Company. These results are presented in a paper entitled "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher" cited above. This paper subsequently won the 1992 Ragnar Frisch Medal for "best empirical paper in Econometrica in the preceding 5 years".

Keywords: Dynamic Programming, Stochastic Control, Discrete Control Process, Regenerative Optimal Stopping, Discrete Choice, Maximum Likelihood, Contraction Iterations, Newton-Kantorovich Iterations, BHHH Method, Secant Method

Platforms: The NFXP software is written in GAUSS programming language available from Aptech Systems, Inc. (206) 432-7855 GAUSS runs on Intel and Sparc PCs and workstations running Dos/Windows and supported versions of Unix (e.g. Linux or Sun Solaris).

Documentation: Available in PDF NFXP Documentation Manual

Support: Limited. At the discretion of the author, John Rust, who can be contacted by email to

Sample output: Available in Postscript file Appendix 3 of the NFXP documentation manual.

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