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Divide and Conquer: Recursive Likelihood Function Integration for Hidden Markov Models with Continuous Latent Variables


Reich, Gregor (2018). Divide and Conquer: Recursive Likelihood Function Integration for Hidden Markov Models with Continuous Latent Variables. Operations Research, 66(6):1457-1759.

Abstract

This paper develops a method to efficiently estimate hidden Markov models with continuous latent variables using maximum likelihood estimation. To evaluate the (marginal) likelihood function, I decompose the integral over the unobserved state variables into a series of lower dimensional integrals, and recursively approximate them using numerical quadrature and interpolation. I show that this procedure has very favorable numerical properties: First, the computational complexity grows linearly in the number of periods, making the integration over hundreds and thousands of periods feasible. Second, I prove that the numerical error accumulates sublinearly in the number of time periods integrated, so the total error can be well controlled for a very large number of periods using, for example, Gaussian quadrature and Chebyshev polynomials. I apply this method to the bus engine replacement model of Rust [Econometrica 55(5): 999–1033] to verify the accuracy and speed of the procedure in both actual and simulated data sets.

Abstract

This paper develops a method to efficiently estimate hidden Markov models with continuous latent variables using maximum likelihood estimation. To evaluate the (marginal) likelihood function, I decompose the integral over the unobserved state variables into a series of lower dimensional integrals, and recursively approximate them using numerical quadrature and interpolation. I show that this procedure has very favorable numerical properties: First, the computational complexity grows linearly in the number of periods, making the integration over hundreds and thousands of periods feasible. Second, I prove that the numerical error accumulates sublinearly in the number of time periods integrated, so the total error can be well controlled for a very large number of periods using, for example, Gaussian quadrature and Chebyshev polynomials. I apply this method to the bus engine replacement model of Rust [Econometrica 55(5): 999–1033] to verify the accuracy and speed of the procedure in both actual and simulated data sets.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Business Administration
Dewey Decimal Classification:330 Economics
Scopus Subject Areas:Physical Sciences > Computer Science Applications
Social Sciences & Humanities > Management Science and Operations Research
Language:English
Date:23 November 2018
Deposited On:22 Feb 2019 13:42
Last Modified:29 Jul 2020 10:03
Publisher:Institute for Operations Research and the Management Science
ISSN:0030-364X
OA Status:Green
Publisher DOI:https://doi.org/10.1287/opre.2018.1750
Other Identification Number:merlin-id:17554

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