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

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  • McCausland, William J.

Abstract

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 θ of parameters, the vector of states α=(α1,…,αn) is Gaussian and the observed vector y=(y1⊤,…,yn⊤)⊤ need not be. I describe a procedure to construct a close approximation q(α|θ,y) to the target density p(α|θ,y). It requires code to compute five derivatives of logp(yt|θ,αt) with respect to αt, t=1,…,n, and is not otherwise model specific. Since q(α|θ,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 logp(α|θ,y) with respect to α—the HESSIAN method is based on sparse matrix operations rather than the Kalman filter. I construct q(α|θ,y) and a related approximation q(θ,α|y) of p(θ,α|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[θ|y] is between 80% and 100%.

Suggested Citation

  • McCausland, William J., 2012. "The HESSIAN method: Highly efficient simulation smoothing, in a nutshell," Journal of Econometrics, Elsevier, vol. 168(2), pages 189-206.
  • Handle: RePEc:eee:econom:v:168:y:2012:i:2:p:189-206
    DOI: 10.1016/j.jeconom.2011.12.003
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    References listed on IDEAS

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    Cited by:

    1. Joshua C.C. Chan & Angelia L. Grant, 2014. "Issues in Comparing Stochastic Volatility Models Using the Deviance Information Criterion," CAMA Working Papers 2014-51, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
    2. repec:taf:jnlbes:v:35:y:2017:i:1:p:17-28 is not listed on IDEAS
    3. Kleppe, Tore Selland & Liesenfeld, Roman, 2014. "Efficient importance sampling in mixture frameworks," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 449-463.
    4. Chan, Joshua C.C. & Grant, Angelia L., 2015. "Pitfalls of estimating the marginal likelihood using the modified harmonic mean," Economics Letters, Elsevier, vol. 131(C), pages 29-33.
    5. Chan, Joshua C.C. & Grant, Angelia L., 2016. "Fast computation of the deviance information criterion for latent variable models," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 847-859.
    6. Joshua C. C. Chan, 2017. "The Stochastic Volatility in Mean Model With Time-Varying Parameters: An Application to Inflation Modeling," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(1), pages 17-28, January.
    7. István Barra & Lennart Hoogerheide & Siem Jan Koopman & André Lucas, 2017. "Joint Bayesian Analysis of Parameters and States in Nonlinear non‐Gaussian State Space Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(5), pages 1003-1026, August.
    8. Joshua C.C. Chan & Eric Eisenstat, 2015. "Bayesian model comparison for time-varying parameter VARs with stochastic volatility," CAMA Working Papers 2015-32, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.

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