Monotonic Polynomial GARCH Models for Conditional Higher Moments

Density expansions such as the Gram-Charlier (GC) expansion allow for the modeling of time-varying higher moments. However, they can suffer from spurious multimodality, negative densities, and asymptotically light tails if truncated. This paper introduces monotonic polynomial generalized autoregressive conditional heteroskedasticity (GARCH) models. They generate conditional skewness and kurtosis via a monotonic polynomial transformation of innovations. By construction, this approach guarantees a valid, unimodal probability density without requiring truncation. It naturally accommodates heavy Weibull-type tails. We provide a theoretical framework proving strict stationarity and ergodicity. In empirical applications to financial returns, the proposed estimator outperforms both GC-based and score-driven benchmarks in out-of-sample density forecasting. It demonstrates superior structural stability and robustness against overfitting.