Comparison of Stochastic Volatility Models Using Integrated Information Criteria

Abstract

Stochastic volatility (SV) models are a family of models that commonly used in the modeling of stock prices. In all SV models, volatility is treated as a stochastic time series. However, SV models are still quite different from each other from the perspective of both underlying principles and parameter layouts. Therefore, selecting the most appropriate SV model for a given set of stock price data is important in making future predictions of stock market. To achieve this goal, leave-one-out cross-validation (LOOCV) methods could be used. However, LOOCV methods are computationally expensive, thus its use is very limited in practice. In our studies of SV models, we proposed two new model-selection approaches, integrated widely applicable information criterion (iWAIC) and integrated importance sampling information criterion (iIS-IC), as alternatives to approximate LOOCV results. In iWAIC and iIS-IC methods, we first calculate the expected likelihood of each observation as an integral with respect to the corresponding latent variable (the current log-volatility parameter). Since the observations are highly correlated with their corresponding latent variable, the integrated likelihood of each t^th observation (y_t^obs) is expected to approximate the expect likelihood of y_t^obs calculated from the model with y_t^obs as its holdout data. Second, the integrated expected likelihood is used, as a replacement of the expected likelihood, in the calculation of information criteria. Since the integration with respect to the latent variable largely reduces the model's bias towards the corresponding observation, the integrated information criteria are expected to approximate LOOCV results. To evaluate the performance of iWAIC and iIS-IC, we first conducted an empirical study using simulated data sets. The results from this study show that iIS-IC method has an improved performance over the traditional IS-IC, but iWAIC does not outperform the non-integrated WAIC method. A further empirical study using real-world stock market return data was subsequently carried out. According to the model-selection results, the best model for the given data is either the SV model with two independent autoregressive processes, or the SV model with nonzero expected returns.