Abstract
This study develops and compares performance of Hamiltonian Monte Carlo (HMC) and Riemann manifold Hamiltonian Monte Carlo (RMHMC) samplers with that of multi-move Metropolis-Hastings sampler to estimate stochastic volatility (SV) and realized SV models with asymmetry effect. In terms of inefficiency factor, empirical results show that the RMHMC sampler give the best performance for estimating parameters, followed by multi-move Metropolis-Hastings sampler. In particular, it is also shown that RMHMC sampler offers a greater advantage in the mixing property of latent volatility chains and in the computational time than HMC sampler.
Similar content being viewed by others
References
Andersen TG, Bollerslev T, Diebold FX, Labys P (2001) The distribution of realized exchange rate volatility. J Am Stat Assoc 96:42–55
Asai M (2005) Comparison of MCMC methods for estimating stochastic volatility models. Comput Econ 25:281–301
Black F (1976) Studies of stock market volatility changes. Am Stat Assoc, Bus Econ Stat Sect 1:177–181
Broto C, Ruiz E (2004) Estimation methods for stochastic volatility models: a survey. J Econ Surv 18:613–649
Dobrev D, Szerszen P (2010) The information content of high-frequency data for estimating equity return models and forecasting risk. Working Paper, Finance and Economics Discussion Series. Retrieved from http://www.federalreserve.gov/pubs/feds/2010/201045/
Duane S, Kennedy AD, Pendleton B, Roweth D (1987) Hybrid Monte Carlo. Phys Lett B 195:216–222
Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 4. Clarendon Press, Oxford, pp 169–194 with discussion
Girolami M, Calderhead B (2011) Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J R Stat Soc, Ser B 73:1–37
Hansen PR, Lunde A (2005) A forecast comparison of volatility models: does anything beat a GARCH(1,1). J Appl Econ 20:873–889
Harvey AC, Shephard N (1996) The estimation of an asymmetric stochastic volatility model for asset returns. J Bus Econ Stat 14:429–434
Ishwaran H (1999) Applications of Hybrid Monte Carlo to Bayesian generalized linear models: quasicomplete separation and neural networks. J Comput Graph Stat 8:779–799
Jacquier E, Polson NG (2011) Bayesian methods in finance. In: Geweke J, Koop G, van Dijk H (eds) Handbook of Bayesian econometrics. Oxford University Press, Oxford, pp 439–512
Jacquier E, Polson NG, Rossi PE (1994) Bayesian analysis of stochastic volatility models. In: Shephard N (ed) Stochastic volatility: selected readings. Oxford University Press, New York, pp 247–282
Jacquier E, Polson NG, Rossi PE (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J Econ 122:185–212
Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with ARCH models. In: Shephard N (ed) Stochastic volatility: selected readings. Oxford University Press, New York
Koopman SJ, Scharth M (2013) The analysis of stochastic volatility in the presence of daily realized measures. J Financ Econ 11:76–115
Leimkuhler B, Reich S (2004) Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge
Marwala T (2012) Condition monitoring using computational intelligence methods. Springer, Berlin
McLachlan RI, Atela P (1992) The accuracy of symplectic integrators. Nonlinearity 5:541–562
Neal RM (2010) MCMC using Hamiltonian dynamics. In: Brooks S, Gelman A, Jones G, Meng X-L (eds) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC Press, Boca Raton, pp 113–162
Omori Y, Chib S, Shephard N, Nakajima J (2007) Stochastic volatility model with leverage: fast likelihood inference. J Econ 140:425–449
Omori Y, Watanabe T (2008) Block sampler and posterior mode estimation for asymmetric stochastic volatility models. Comput Stat Data Anal 52:2892–2910
Pasarica C, Gelman A (2008) Adaptive scaling the metropolis algorithm using expected squared jumped distance. Stat Sin 20:343–364
Shephard N (1993) Fitting non-linear time series models, with applications to stochastic variance models. J Appl Econ 8:135–152
Shephard N, Pitt MK (1997) Likelihood analysis of non-Gaussian measurement timeseries. Biometrika 84:653–667
Takahashi M, Omori Y, Watanabe T (2009) Estimating stochastic volatility models using daily returns and realized volatility simultaneously. Comput Stat Data Anal 53:2404–2426
Takahashi M, Omori Y, Watanabe T (2014) Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution. Working Paper Series CIRJE-F-921, CIRJE, Faculty of Economics, University of Tokyo. Retrieved from http://www.cirje.e.u-tokyo.ac.jp/research/dp/2014/2014cf921
Takaishi T (2009) Bayesian inference of stochastic volatility by Hybrid Monte Carlo. J Circuits Syst Comput 18:1381–1396
Taylor SJ (1982) Financial returns modelled by the product of two stochastic processes–a study of the daily sugar prices 1961–75. In: Shephard N (ed) Stochastic volatility: selected readings. Oxford University Press, New York, pp 60–82
Watanabe T, Omori Y (2004) A multi-move sampler for estimating non-Gaussian times series models: comments on Shephard and Pitt. Biometrika 91:246–248
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Estimation of RASV model
Let us take the logarithm of the full conditional posterior density defined in Eq. (3):
where
The following subsections describe how to sample latent variable \(\varvec{\alpha }\) and parameters in \(\theta _1=(c,\sigma _u,\sigma _\epsilon ,\phi ,\sigma _\eta ,\rho )\).
1.1 Updating latent variable \(\varvec{\alpha }\)
Sampling from the conditional posterior density \(p(\varvec{\alpha }|\mathbf {R},\mathbf {Y},\theta _1)\), is a difficult work because of the high-dimensional log-normal structure. Therefore, we sample latent variable \(\varvec{\alpha }\) using the HMC and RMHMC algorithms, where the expressions required by both algorithms to sample the entire latent volatility at once are evaluated as follows.
On the basis of Eq. (5), the logarithm of the full conditional posterior density of \(\varvec{\alpha }\) has the following expression:
The important aspect of the HMC and RMHMC algorithms is that both algorithms require the calculation of a gradient vector \(\nabla _{\alpha _t}{\mathcal {L}}(\alpha _t)\). These are expressed by
for \(i=2,\ldots ,T-1\), and
To apply the leapfrog-RMHMC algorithm, then we also need expressions for the metric tensor and its partial derivatives. The metric tensor \(\mathbf {M}(\varvec{\alpha })\) is a symmetric tridiagonal matrix, where elements of main diagonal are calculated as follows:
for \(i=2,\ldots ,T-1\), and elements of superdiagonal and subdiagonal are calculated as
for \(i=2,\ldots ,T\). Since the above metric tensor is fixed, that is not a function of the latent volatility \(\varvec{\alpha }\), the associated partial derivatives with respect to the latent volatility are zero. This implies that both metric tensor and its inverse are not recalculated in the generalized leapfrog integration scheme.
1.2 Updating parameter \(\phi \)
As far as \(\phi \) is only concerned to be sampled, the logarithm of conditional posterior density for \(\phi \) on the basis of Eq. (5) has the following expression:
which is in non-standard form, and therefore we cannot sample from it directly. In the following, we evaluate the expressions required by the HMC and RMHMC algorithms to sample \(\phi \).
For dealing with the constraints \(-1<\phi <1\), it is necessary to implement the transformations \(\phi =\tanh (\tilde{\phi })\) and then introduce a Jacobian factor into the acceptance ratio. The derivative of the transformation is \(\hbox {d}\phi /\hbox {d}\tilde{\phi }=1-\tanh ^2(\tilde{\phi })=1-\phi ^2\). Then, the partial derivative of the above log joint posterior with respect to the transformed parameter is as follows:
To apply the leapfrog RMHMC algorithm, then we also need expressions for the metric tensor and its partial derivative with respect to the transformed parameter. The metric tensor \(\mathbf {M}(\phi )\) for the log joint posterior density is calculated as follows:
Thus the partial derivative of metric tensor \(\mathbf {M}(\phi )\) follows as
1.3 Updating parameters \((\sigma _\epsilon ,\sigma _\eta ,\rho )\)
On the basis of Eq. (5), the log joint posterior density of \({\varSigma } = (\sigma _\epsilon ,\sigma _\eta ,\rho )\) has the following expression:
which is in non-standard form. The expressions required by the HMC and RMHMC algorithms to sample \({\varSigma }\) are evaluated as follows.
For dealing with the constraints \(-1<\rho <1\) and \(\sigma _\epsilon ,\sigma _\eta >0\), it is necessary to implement the transformations \(\rho =\tanh (\tilde{\rho })\), \(\sigma _\epsilon =\exp (\tilde{\sigma }_\epsilon )\), and \(\sigma _\eta =\exp (\tilde{\sigma }_\eta )\) and then introduce a Jacobian factor into the acceptance ratio. Notice that Girolami and Calderhead (2011) did not take a transformation for the parameter \(\sigma _\epsilon \). The derivatives of the transformations are \(\hbox {d}\rho /\hbox {d}\tilde{\rho }=1-\rho ^2\), \(\hbox {d}\sigma _\epsilon /\hbox {d}\tilde{\sigma }_\epsilon =\sigma _\epsilon \), and \(\hbox {d}\sigma _\eta /\hbox {d}\tilde{\sigma }_\eta =\sigma _\eta \). Then, the partial derivatives of the above log joint posterior with respect to the transformed parameters are as follows:
To apply the leapfrog RMHMC algorithm, then we also need expressions for the metric tensor and its partial derivatives with respect to the transformed parameter. The metric tensor \(\mathbf M (\sigma _\epsilon ,\sigma _\eta ,\rho )\) is a symmetric tridiagonal matrix, where the nonzero elements are calculated as follows:
and its partial derivatives follow as
1.4 Updating parameters \(\sigma _u^2\) and \(c\)
In the steps 4 and 5, we sample from the conditional posterior density of \(\sigma _u^2\) and \(c\) directly:
where
Appendix 2: Estimation of RASVt model
Let us take the logarithm of the full conditional posterior density defined in Eq. (4):
where
On the basis of the above density, the full conditional densities of \(\mathbf h \), \(\xi \), \(\sigma _u\), \(\phi \), and \(\sigma _\eta \) are similar to \(\varvec{\alpha }\), \(c\), \(\sigma _u\), \(\phi \), and \(\sigma _\eta \), respectively, for the RASV model. Thus, we follow the same sampling procedure described in the previous section with different specifications of \(\widetilde{R}_t\) and \(\widetilde{h}_t\) defined in Eqs. (7) and (8), respectively. To sample parameters \(\mu \) and \(\nu \) and latent variable \(\mathbf z \), we describe these sampling procedures as follows.
1.1 Updating parameter \(\mu \)
On the basis of Eq. (6), parameter \(\mu \) can be sampled directly from its conditional posterior density:
where
1.2 Updating parameter \(\nu \)
The log posterior of \(\nu \) and its gradient vector required in both leapfrog-HMC and -RMHMC algorithms are given by
and
respectively, where \(\varPsi (x)\) is a digamma function defined by \(\varPsi (x)=d\ln \varGamma (x)/dx\). In the implementation of leapfrog-RMHMC algorithm, the metric tensor and its partial derivatives are calculated as follows:
and
respectively, where \(\varPsi '(x)\) is a trigamma function defined by \(\varPsi '(x)=d\varPsi (x)/dx\) and \(\varPsi ''(x)\) is a tetragamma function defined by \(\varPsi ''(x)=d\varPsi '(x)/dx\).
1.3 Updating the latent variable z
The logarithm of the full conditional posterior density of the latent variable \(\mathbf z \) is
From this conditional density, draws can be obtained by implementing the transformation \(z_t=\exp (\hat{z}_t)\) to ensure constrained sampling for \(z_t>0\). Gradient vector of the log posterior required in both leapfrog-HMC and -RMHMC algorithms is then evaluated as follows:
where \(I\) is an indicator function and
Furthermore, the metric tensor \(\mathbf M (\mathbf z )\) required to apply the leapfrog-RMHMC algorithm is a diagonal matrix whose diagonal entries are given by
Since the above metric tensor is fixed, that is not a function of the latent variable z, the associated partial derivatives with respect to the transformed latent variable are zero. This implies that both metric tensor and its inverse are not recalculated in the generalized leapfrog integration scheme.
Rights and permissions
About this article
Cite this article
Nugroho, D.B., Morimoto, T. Estimation of realized stochastic volatility models using Hamiltonian Monte Carlo-Based methods. Comput Stat 30, 491–516 (2015). https://doi.org/10.1007/s00180-014-0546-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-014-0546-6