[R-sig-finance] [R] Stochastic Volatility

Eric Zivot ezivot at u.washington.edu
Sat Nov 5 21:13:50 CET 2005

Estimating a SV model using the kalman filter a la Harvey, Ruiz and Shephard is straightforward and fast. In my book Modeling Financial Time Series with Splus, 2nd Edition, I give an example of doing this using the state space modeling tools (i.e. ssfpack) available in S+FinMetrics. The main problem with the kalman filter approach is that the resulting estimates are only best linear estimators. The state space representation of the SV model is non-Gaussian and the Kalman filter is only optimal for linear Gaussian models. Neil Shepard wrote a very nice survey article comparing SV and GARCH models about 5 years ago that was published in a Chapman & Hall book. I believe you can still download this paper from his webpage.

In writing the 2nd Edition of my book, I spent a lot of time with various forms of SV models (discrete time and continuous time). The main problem is that efficient estimation via maximum likelihood takes a lot of programming effort, and the techniques have to be tailored to specific models. Since volatility is latent, it has to be integrated out of the likelihood in discrete time models. In continuous-time models the transition density for the observables is only known in closed form for very simple models. The simulated MLE approach of Koopman and his co-authors is reasonably straightforward for simple models and provides much better estimates than the Kalman filter (in terms of MSE). I have an illustration of this approach in my paper (joint with Siem Jan Koopman and Jiahui Wang) "State Space Modeling in Macroeconomics and Finance Using SsfPack in S+FinMetrics", which can be downloaded 
from my webpage. However, for more complicated models and multivariate models the approach is not so straightforward.

The most promising approach for estimating SV models appears to be simulation-based. The Bayesian approach uses MCMC or particle filters, and can be very accurate. However, each model requires a different MCMC algorithm. I have had good success with Gallant and Tauchen's efficient method of moments (EMM) technique. The main advantage of this method is that one does not need to tailor the method for a specific problem: all it requires is that you write a simulator for your proposed SV model. Jiahui Wang and I, with help from George Tauchen, implemented EMM in S+FinMetrics 2.0 and the 2nd Edition of Modeling Financial Time Series describes in detail how to use EMM for estimation of general discrete time and continuous time SV models (as well as other models).

*  Eric Zivot                  			               *
*  Associate Professor         phone:  206-543-6715            *
*  Department of Economics     fax:    206-685-7477            *
*  Box 353330                  email:  ezivot at u.washington.edu *
*  University of Washington                                    *
*  Seattle, WA 98195-3330                                      *
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*  www:  http://faculty.washington.edu/ezivot                  *

On Sat, 5 Nov 2005, Patrick Burns wrote:

> This seems much more appropriate for R-sig-finance than
> for R-help.
> I'm curious why you think garch models are less satisfactory
> than stochastic volatility models.  I'm not aware of any literature
> that shows one dominating the other, and not even very much
> that compares the two.
> Patrick Burns
> patrick at burns-stat.com
> +44 (0)20 8525 0696
> http://www.burns-stat.com
> (home of S Poetry and "A Guide for the Unwilling S User")
> Phineas Campbell wrote:
>> Has anybody implemented or tried to implement a stochastic volatility model
>> using the Kalman filter following a series of papers by Harvey, Ruiz and
>> Shepard?
>> This is a sophisticated approach for estimating an important class of
>> models, so I am surprised that no implementation exists, is this because
>> there are unforeseeable problems?
>> In a related but off topic question, it has been a while since I looked at
>> the non homoskedastic time series literature but back then you couldn't pick
>> up a journal without reading another stochastic volatility paper, does
>> anybody have any ideas why the literature has drifted back toward less
>> satisfactory GARCH and EGARCH models?
>> This question is somewhat moot as if I choose to pursue this I will
>> implement a model myself.
>> Phineas Campbell
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