The Hessian Method (Highly Efficient State Smoothing, In a Nutshell)

I introduce the HESSIAN method for semi-Gaussian state space models with univariate states. The vector of states a=(a^1; ... ; a^n) is Gaussian and the observed vector y= (y^1 ; ... ; y^n )> need not be. I describe a close approximation g(a) to the density f(a|y). It is easy and fast to evaluate g(a) and draw from the approximate distribution. In particular, no simulation is required to approximate normalization constants. Applications include likelihood approximation using importance sampling and posterior simulation using Markov chain Monte Carlo (MCMC). HESSIAN is an acronym but it also refers to the Hessian of log f(a|y), which gures prominently in the derivation. I compute my approximation for a basic stochastic volatility model and compare it with the multivariate Gaussian approximation described in Durbin and Koopman (1997) and Shephard and Pitt (1997). For a wide range of plausible parameter values, I estimate the variance of log f(a|y) - log g(a) with respect to the approximate density g(a). For my approximation, this variance ranges from 330 to 39000 times smaller.