Continuous and Jump Variation in GARCH-MIDAS: Component Allocation and Volatility Forecasting

This paper develops a generalized autoregressive conditional heteroskedasticity–mixed data sampling model with continuous and jump components (GARCH-MIDAS-CJ). It aligns the continuous–jump (CJ) decomposition of realized volatility with the long-run–short-run multiplicative structure. The model places block-smoothed daily continuous variation in the long-run mixed-data-sampling term and demeaned jump variation in the short-run Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) recursion. Using high-frequency data for the Standard & Poor’s 500 (S&P 500), we find a positive jump coefficient under both the Corsi–Pirino–Renò (CPR) and Andersen–Bollerslev–Dobrev (ABD) decompositions, supported by one-sided Wald, boundary-adjusted likelihood-ratio, and parametric-bootstrap tests. Relative to return- and realized-measure benchmarks, the forecasting value of the CJ allocation is competitive for multi-day and cumulative variance forecasts and in high-volatility periods, whereas established benchmarks can perform better for one-day and low-volatility forecasts. Under the CPR decomposition, the h=22 average cumulative forecasts reduce the mean squared error (MSE) and QLIKE relative to the aggregate-realized-variance benchmark by 8.2% and 3.3%, respectively. One-day-ahead value-at-risk (VaR) results are strongest at the 1% and 0.5% tails; the 5% forecasts have exceedance rates above the nominal level. Across alternative information-timing conventions, forecast windows, and United States equity assets, the clearest gains remain concentrated at multi-day horizons, although their magnitude varies across settings. The findings support a persistence-based allocation of continuous and jump variation within the GARCH-MIDAS structure.