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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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Bibliographic Info

Paper provided by Centre interuniversitaire de recherche en économie quantitative, CIREQ in its series Cahiers de recherche with number 03-2008.

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Length: 27 pages
Date of creation: 2008
Date of revision:
Handle: RePEc:mtl:montec:03-2008

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  1. Sangjoon Kim, Neil Shephard & Siddhartha Chib, . "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers W26, revised version of W, Economics Group, Nuffield College, University of Oxford.
  2. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-39, November.
  3. 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.
  4. McCAUSLAND, William J. & MILLER, Shirley & PELLETIER, Denis, 2007. "A New Approach to Drawing States in State Space Models," Cahiers de recherche 2007-06, Universite de Montreal, Departement de sciences economiques.
  5. Jean-Francois Richard, 2007. "Efficient High-Dimensional Importance Sampling," Working Papers 321, University of Pittsburgh, Department of Economics, revised Jan 2007.
  6. H�vard 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.
  7. 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.
  8. 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.
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Cited by:
  1. Joshua Chan & Rodney Strachan, 2012. "Estimation in Non-Linear Non-Gaussian State Space Models with Precision-Based Methods," CAMA Working Papers 2012-13, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
  2. 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.

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