Publication Type

Working Paper

Publication Date



The Laplace approximation is used to perform maximum likelihood estimation of univariate and multivariate stochastic volatility (SV) models. It is shown that the implementation of the Laplace approximation is greatly simplified by the use of a numerical technique known as automatic differentiation (AD). Several algorithms are proposed and compared with some existing maximum likelihood methods using both simulated data and actual data. It is found that the new methods match the statistical efficiency of the existing methods while significantly reducing the coding effort. Also proposed are simple methods for obtaining the filtered, smoothed and predictive values for the latent variable. The new methods are implemented using the open source software AD Model Builder, which with its latent variable module (ADMB-RE) facilitates the formulation and fitting of SV models. To illustrate the flexibility of the new algorithms, several univariate and multivariate SV models are fitted using exchange rate and equity data.


Empirical Bayes; Laplace approximation; Automatic differentiation; AD Model Builder; Simulated maximum likelihood; Importance sampling


Applied Statistics | Econometrics

Research Areas


Last Page


Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.


Computational Statistics & Data Analysis