Leak-safe functional forecasting of daily variance: window-to-curve predictors for DAX Rogers–Satchell variance

We study one-step-ahead forecasting of daily DAX equity-index variance under an explicit information-availability design. The target is the Rogers–Satchell range-based variance proxy computed from open–high–low–close (OHLC) data, and predictors are daily log returns, with rate changes for yield variables, from a multi-asset panel. Each predictor is represented by a smoothed window curve: The most recent W = 20 W = 20 trading days are mapped through penalized B-splines and evaluated on a common grid, then summarized by functional partial least squares (PLS). The resulting functional variance models are compared with persistence, autoregressive log-variance, heterogeneous autoregressive (HAR), HAR-X(logVIX), an Elastic Net lag-stack benchmark, generalized autoregressive conditional heteroskedasticity (GARCH), exponential GARCH (EGARCH), and Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) specifications. All preprocessing and design choices are fixed on the development sample only; recursive fitting, component selection, and Elastic Net penalty selection use only information available at each forecast origin. Final comparisons use a set under the conservative DAX-Close convention, with Global-Close retained as supplementary robustness. Accuracy is evaluated primarily by quasi-likelihood (QLIKE), with mean absolute error (MAE), root mean squared error (RMSE), Diebold–Mariano tests with Newey–West heteroskedasticity and autocorrelation consistent corrections, and moving-block bootstrap intervals as complements. Under DAX-Close , HAR-X(logVIX) has the lowest average QLIKE, followed closely by HAR. The Elastic Net lag-stack ranks third, while fVAR-X(+logv ₜ t ) is the strongest functional specification. The supplementary Global-Close design improves several forecasts but does not change the main interpretation. Recent multi-asset histories are informative, but the smoothed functional representation is competitive rather than dominant in this application.