<![CDATA[This study compares the Log-linear Realized GARCH (LRG) and its extension with Continuous and Jump components (LRG-CJ) in modeling the volatility of financial assets, using daily data from the Tokyo Stock Price Index (TOPIX) over 2004–2011. The urgency arises from the need for more accurate volatility models during turbulent periods such as the 2008 Global Financial Crisis and the 2011 Great East Japan Earthquake, where markets exhibit both smooth fluctuations and abrupt jumps. Methodologically, the LRG-CJ framework introduces a novel integration of continuous and jump decomposition into the LRG structure, offering an applied innovation to high-frequency volatility modeling. Realized Volatility (RV) was calculated from 1-, 5-, and 10-minute intraday data and decomposed into continuous and jump components. Parameter estimation employed the Adaptive Random Walk Metropolis (ARWM) within a Markov Chain Monte Carlo algorithm, while model performance was assessed using multiple information criteria and out-of-sample forecast evaluations. The empirical results reveal that incorporating continuous and jump components improves volatility modeling accuracy, forecasting, and Value-at-Risk estimation. However, these benefits are frequency-dependent: the LRG-CJ model shows superior in-sample fit for 1-minute RV but provides the strongest out-of-sample forecasting and risk prediction at lower frequencies (5- and 10-minute intervals). This highlights that while jumps are best identified at ultra-high frequencies, their predictive value is most effectively captured in slightly aggregated data. The originality of this study lies in being the first empirical application of LRG-CJ, demonstrating how continuous–jump decomposition interacts with the dual-equation structure of LRG, which has not been examined in TGARCH or APARCH contexts. Limitations include sensitivity to microstructure noise in very high-frequency data and computational challenges in parameter convergence. Overall, the findings underscore the novelty and practical importance of the LRG-CJ framework for risk management, offering actionable guidance for aligning volatility models with data frequency]]>
