Divide and Conquer: Recursive Likelihood Function Integration for Hidden Markov Models with Continuous Latent Variables
Operations Research · 2018 · Volume 66 · Issue 6
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. The e-companion is available at https://doi.org/10.1287/opre.2018.1750 .
Keywords
hidden Markov models, maximum likelihood estimation, numerical integration, interpolation
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