The HESSIAN method: Highly efficient simulation smoothing, in a nutshell

doi:10.1016/j.jeconom.2011.12.003

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The HESSIAN method: Highly efficient simulation smoothing, in a nutshell

2012-06-01

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William J. McCausland *

DOI：10.1016/j.jeconom.2011.12.003

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Abstract

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I introduce the HESSIAN (highly efficient simulation smoothing in a nutshell) method for numerically efficient simulation smoothing in state space models with univariate states. Given a vector 0 of parameters, the vector of states alpha = (alpha(1),..., alpha(n)) is Gaussian and the observed vector y = (y(1)(inverted perpendicular), ...., y(n)(inverted perpendicular))(inverted perpendicular) need not be. I describe a procedure to construct a close approximation q(alpha vertical bar theta, y) to the target density p(alpha vertical bar theta, y). It requires code to compute five derivatives of log p(y(t)vertical bar theta,alpha(t)) with respect to alpha(t), t = 1,..., n, and is not otherwise model specific. Since q(alpha vertical bar theta, y) is proper, fully normalised and simulable, it can be used as an importance density for importance sampling (IS) or as a proposal density for Markov chain Monte Carlo (MCMC). HESSIAN is an acronym but it also refers to the (sparse) Hessian matrix of log p(alpha vertical bar theta, y) with respect to alpha-the HESSIAN method is based on sparse matrix operations rather than the Kalman filter. I construct q(alpha vertical bar theta, y) and a related approximation q(theta, alpha vertical bar y) of p(theta, alpha vertical bar y) for two stochastic volatility models, two stochastic count models and a stochastic duration model. I illustrate their use for numerical approximation of likelihood function values and marginal likelihoods, using IS, and for posterior inference, using IS and MCMC. Compared with other simulation smoothing methods, the HESSIAN method is highly numerically efficient. In an IS application featuring a Student's t stochastic volatility model and n = 8851 daily log returns, the efficiency of IS for numerical approximation of the elements of the posterior mean E vertical bar theta vertical bar y vertical bar is between 80% and 100%. (C) 2011 Elsevier B.V. All rights reserved,

Keywords:

State space models

Simulation smoothing

Importance sampling

MCMC

Stochastic volatility

Count models

Duration models