A Semiparametric Real-Time GARCH Model with Nonstationary Variance

We propose a semiparametric real-time GARCH (SRT-GARCH) model that combines a smooth deterministic long-run volatility component with a stationary short-run process driven by contemporaneous return innovations. The model accommodates gradual changes in unconditional variance while allowing short-run volatility to respond directly to current information rather than relying solely on lagged observations. We derive the variance normalization, conditional moments, and closed-form conditional density of the model. For estimation, we develop a two-step procedure that combines kernel smoothing with conditional Gaussian maximum likelihood. Under Gaussian innovations and mild regularity conditions, we establish consistency and asymptotic normality of the short-run parameter estimator. The analysis exploits geometric moment contraction to control serial dependence and shows that estimating the long-run component adds a positive semidefinite rank-one adjustment to the oracle covariance matrix obtained when the long-run component is known. The resulting asymptotic covariance matrix admits a plug-in estimator that requires neither lag truncation nor additional tuning parameters. Monte Carlo simulations show good finite-sample performance of the estimator and the associated inference procedure. Applications to FTSE~100 returns and U.S. natural gas futures show that allowing for a time-varying long-run component improves forecasting relative to the parametric RT-GARCH benchmark, while the additional real-time feedback mechanism provides more pronounced gains in the natural gas application than in the FTSE application. The proposed model also delivers competitive one-step-ahead Value-at-Risk forecasts.