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

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.