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A stochastic volatility model with random level shifts and its applications to S&P 500 and NASDAQ return indices



This paper proposes a framework for the modelling, inference and forecasting of volatility in the presence of level shifts of unknown timing, magnitude and frequency. First, we consider a stochastic volatility model comprising both a level shift and a short-memory component, with the former modelled as a compound binomial process and the latter as an AR(1). Next, we adopt a Bayesian approach for inference and develop algorithms to obtain posterior distributions of the parameters and the two latent components. Then, we apply the model to daily S&P 500 and NASDAQ returns over the period 1980.1–2010.12. The results show that although the occurrence of a level shift is rare, about once every 2 years, this component clearly contributes most to the variation in the volatility. The half-life of a typical shock from the AR(1) component is short, on average between 9 and 15 days. Interestingly, isolating the level shift component from the overall volatility reveals a stronger relationship between volatility and business cycle movements. Although the paper focuses on daily index returns, the methods developed can potentially be used to study the low-frequency variation in realized volatility or the volatility of other financial or macroeconomic variables.