Maximum likelihood estimation of latent Markov models using closed-form approximations

Abstract

This paper proposes and implements an efficient and flexible method to compute maximum likelihood estimators of continuous-time models when part of the state vector is latent. Stochastic volatility and term structure models are typical examples. Existing methods integrate out the latent variables using either simulations as in MCMC, or replace the latent variables by observable proxies. By contrast, our approach relies on closed-form approximations to estimate parameters and simultaneously infer the distribution of filters, i.e., that of the latent states conditioning on observations. Without any particular assumption on the filtered distribution, we approximate in closed form a coupled iteration system for updating the likelihood function and filters based on the transition density of the state vector. Our procedure has a linear computational cost with respect to the number of observations, as opposed to the exponential cost implied by the high dimensional integral nature of the likelihood function. We establish the theoretical convergence of our method as the frequency of observation increases and conduct Monte Carlo simulations to demonstrate its performance.