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 time, which makes the integration over hundreds and thousands of periods well feasible.

Second, I prove that the numerical error is accumulated sub-linearly over time; consequently, using highly efficient and fast converging numerical quadrature and interpolation methods for low and medium dimensions, such as Gaussian quadrature and Chebyshev polynomials, the numerical error can be well controlled even for very large numbers of periods.

Lastly, I show that the numerical convergence rates of the quadrature and interpolation methods are preserved up to a factor of at least 0.5 under appropriate assumptions.

I apply this method to the bus engine replacement model of Rust [Econometrica, 55 (5): 999–1033, (1987)]: first, I estimate the model using the original dataset; second, I verify the algorithm’s ability to recover the parameters in an extensive Monte Carlo study with simulated datasets.

Keywords: Hidden Markov Models, Numerical Quadrature, Interpolation, Recursion

JEL Classification: C25, C63