I consider two filtering algorithms (quadrature and mixture Gaussian) based on numerical integration for maximum likelihood estimation of stochastic volatility models with leverage. These algorithms extend straightforwardly to stochastic volatility models with non-Gaussian innovations. A small Monte Carlo simulation experiment shows that the mixture Gaussian filter performs remarkably well both in terms of accuracy and computation time. As an empirical application, I fit the asymmetric stochastic volatility model to the S&P 500 index daily returns with a Gaussian and skew-t innovation. The estimates from the two filtering algorithms are remarkably similar, suggesting the usefulness of the mixture Gaussian filter for practical use
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