### The Hessian method for models with leverage-like effects

*Barnabe Djegnene*, William McCausland*

Last modified: 2012-07-11

#### Abstract

We propose a new method for simulation smoothing in state space models with univariate states and leverage-like eects. Given a vector of parameters, the state sequence is Gaussian and the sequence of observed vectors may be conditionally non-Gaussian. By leverage-like effect, we mean conditional dependence between the current observationand the contemporaneous innovation of the state equation, not just the contemporaneous state. We use this term since stochastic volatility models with the leverage effect are a leading example.Our method is an extension of the HESSIAN method described in McCausland[2012], which only works for models without leverage-like effects, models in which the conditional density of observation given state depends only on contemporous state. Like that method, ours is based on a close approximation of the conditional density of the observation given the state. One can use the approximation for importance sampling or Markov chain Monte Carlo (MCMC). With use this approximation to build an importance or proposal density for the joint posterior distribution of parameters and states. Applications include the approximation of likelihood function values and the marginal likelihood, and Bayesian posterior simulation. We construct the approximation for Gaussian and Student's t stochastic volatility models with leverage. For both models, we make a joint proposal of the state and parameter vectors. Unlike Omori et al. [2007] and Nakajima and Omori [2009], we do not augment the data by adding mixture indicators or heavy tail scaling factors. Our generic procedure is more numerically efficient than the model specific procedures of those papers. Using randomised pseudo-Monte Carlo importance sampling, we obtain relative numerical eciencies close to 100%, at least 4 timeshigher than those obtained using the method of Omori et al. [2007]. The highest value of numerical efficiency reported by Nakajima and Omori [2009] is 29.1% forthe ASV-Student model. The lowest efficiency factor reported using the Hessian with randomised pseudo-Monte Carlo importance sampling is 82.61%. Comparingthese two gures suggest that the Hessian procedure is numerically efficient than the one described in Nakajima and Omori [2009].