Maximum Likelihood Estimation of Latent Markov Models Using Closed-Form Approximations

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. The method makes it possible to estimate parameters of multivariate Markov models with latent factors 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 prove the convergence of our method as the frequency of observation increases and conduct Monte Carlo simulations to demonstrate its performance.

Short Bio: Chen Xu Li is a postdoctoral research associate and a lecturer of the Bendheim Center for Finance, Princeton University. He obtained his Ph.D. from Peking University in July, 2018. His research interests include but are not limited to financial econometrics, financial engineering, stochastic modeling, applied probability, and statistics. His current work centers on the inference of models underlying financial derivatives and the estimation of models with latent factors.