Specification and estimation of discrete time quadratic stochastic volatility models

I propose a new class of stochastic volatility models that nests the commonly used log normal autoregressive specification. As with the eigenfunction specification of Meddahi (Meddahi, Nour, 2001. An eigenfunction approach for volatility modeling. Unpublished.), the log-quadratic model can generate high kurtosis, a key feature of asset returns, even with Gaussian innovations. I discuss maximum likelihood estimation based on numerical integration of the log-quadratic specification that allows for leverage effects. A small Monte Carlo simulation experiment demonstrates the feasibility of maximum likelihood estimation and the importance of allowing for leverage effects. I fit the log-quadratic specification to the daily S&P 500 index return series and find that it provides a better fit than the commonly used log autoregressive specification with Gaussian and Student-t mean equation innovations.