Modeling and Forecasting of Realized Volatility: Evidence from Brazil
Brazilian Review of Econometrics
Using intraday data for the most actively traded stocks on the São Paulo Stock Market (BOVESPA) index, this study considers two recently developed models from the literature on the estimation and prediction of realized volatility: the Heterogeneous Autoregressive Model of Realized Volatility (HAR-RV), developed by Corsi (2009), and the Mixed Data Sampling model (MIDAS-RV), developed by Ghysels et al. (2004) . Using measurements to compare in-sample and out-of-sample forecasts, better results
... , better results were obtained with the MIDAS-RV model for in-sample forecasts. For out-of-sample forecasts, however, there was no statistically significant difference between the models. We also found evidence that the use of realized volatility induces distributions of standardized returns that are closer to normal. five stocks from the São Paulo Stock Market (BOVESPA). The models that will be considered in this study are the Heterogeneous Autoregressive Model of Realized Volatility (HAR-RV), developed by Corsi (2009) , and the Mixed Data Sampling model (MIDAS-RV), developed by Ghysels et al. (2004). The models' ability to fit and predict data is compared using in-sample and out-of-sample comparison measurements. For in-sample forecasts, we compare the in-sample adjusted R 2 and the mean squared errors (MSE). For out-of-sample comparisons, we use the out-of-sample MSE and a modified Diebold-Mariano test. This study will contribute to the literature on the prediction of realized volatility with high frequency data for emerging countries, similarly to the study by Chung et al. (2008) for the Taiwan market and to the studies by Carvalho et al. (2006) and Sá Mota and Fernandes (2004) for Brazil. In general, these studies compared predictions from realized volatility models with GARCH class models. None of these studies, however, used any form of control for the effects of the microstructure of the market. To control for this possible bias, this study uses an M A(q) model to filter data, as suggested by Hansen et al. (2008) . In addition, this study is also the first to apply the MIDAS methodology to Brazilian data. As asserted by Carvalho et al. (2006) , forecasting returns is not an easy task. Modeling the volatility of returns, however, is easier and therefore generates more reliable predictions. Thus, the literature on volatility modeling has expanded in recent years. Volatility modeling began with the estimation of ARCH-GARCH models and with the use of stochastic volatility models. However, as discussed in Corsi (2009), these models suffer from a "double weakness." First, they do not represent certain specific empirical characteristics of financial data 1 and second, they tend to have nontrivial estimates, particularly when utilizing models of stochastic volatility. 2 The next phase in the literature on volatility modeling was an attempt to 1 Most models are able to replicate some of the stylized facts such as: heavy tails, volatility clustering, high persistence, long memory, but only some models replicate the leverage effect and arrival of news in the market. 2 The exact estimation of the stochastic volatility model is computationally intensive; however, the estimation by quasi maximum likelihood is easy to implement using the Kalman Filter, but the estimates will be biased (see Harvey et al., 1994) .