Normalising Flow Enhanced GARCH Models: A Two-Stage Framework for Flexible Innovation Modelling in Financial Time Series

Abstract

We introduce the Normalising Flow GARCH (NF-GARCH), a two-stage hybrid framework that enhances traditional GARCH models by replacing restrictive parametric innovation distributions with learned densities via normalising flows. Our approach preserves the interpretability of standard variance dynamics while addressing the common issue of innovation misspecification. In the first stage, we estimate standard GARCH variants (sGARCH, TGARCH, and gjrGARCH) to extract standardised residuals. In the second stage, a Masked Autoregressive Flow learns the underlying residual distribution, with samples from the flow subsequently driving the GARCH recursion for out-of-sample forecasting. Evaluated on 13 daily financial series (six FX pairs and seven equities), NF-GARCH demonstrates systematic, statistically significant improvements in forecast accuracy for skewed-t baselines. Wilcoxon signed-rank tests confirm superior performance specifically for gjrGARCH-sstd and sGARCH-sstd specifications. While the framework offers enhanced flexibility and generative realism, we observe that computational overhead is increased, and the log-variance specification of eGARCH exhibits instability when paired with flow-based innovations. These results suggest that while NF-GARCH effectively captures empirical tail behaviour in univariate settings, future research should explore conditional flow architectures and multivariate extensions to account for time-varying innovation shapes. For risk management, gains are most relevant where skewed-t baselines are used and where closer residual realism supports scenario analysis; effect sizes remain modest relative to model risk and implementation cost.