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."> 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."> need not ">
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The Hessian Method (Highly Efficient State Smoothing, In a Nutshell)

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  • McCAUSLAND, William
Abstract
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.

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  • McCAUSLAND, William, 2008. "The Hessian Method (Highly Efficient State Smoothing, In a Nutshell)," Cahiers de recherche 03-2008, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    • Handle: RePEc:mtl:montec:03-2008
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    1. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-1339, November.
    2. J. Durbin & S. J. Koopman, 2000. "Time series analysis of non‐Gaussian observations based on state space models from both classical and Bayesian perspectives," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 3-56.
    3. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    4. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 361-393.
    5. Liesenfeld, Roman & Richard, Jean-Francois, 2003. "Univariate and multivariate stochastic volatility models: estimation and diagnostics," Journal of Empirical Finance, Elsevier, vol. 10(4), pages 505-531, September.
    6. Håvard Rue, 2001. "Fast sampling of Gaussian Markov random fields," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 325-338.
    7. Håvard Rue & Ingelin Steinsland & Sveinung Erland, 2004. "Approximating hidden Gaussian Markov random fields," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 877-892, November.
    8. McCAUSLAND, William J. & MILLER, Shirley & PELLETIER, Denis, 2007. "A New Approach to Drawing States in State Space Models," Cahiers de recherche 07-2007, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    9. J. Durbin, 2002. "A simple and efficient simulation smoother for state space time series analysis," Biometrika, Biometrika Trust, vol. 89(3), pages 603-616, August.
    10. Richard, Jean-Francois & Zhang, Wei, 2007. "Efficient high-dimensional importance sampling," Journal of Econometrics, Elsevier, vol. 141(2), pages 1385-1411, December.
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    1. McCausland, William J. & Miller, Shirley & Pelletier, Denis, 2011. "Simulation smoothing for state-space models: A computational efficiency analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 199-212, January.
    2. Chan, Joshua & Strachan, Rodney, 2012. "Estimation in Non-Linear Non-Gaussian State Space Models with Precision-Based Methods," MPRA Paper 39360, University Library of Munich, Germany.
    3. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    4. Kleppe, Tore Selland & Skaug, Hans Julius, 2012. "Fitting general stochastic volatility models using Laplace accelerated sequential importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3105-3119.

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