THE GENERALISED HERMITE REGRESSION MODEL: A ROBUST FRAMEWORK FOR EXTREME NONLINEAR DATASETS

Abstract

Nonlinear regression modelling is a fundamental problem in 
econometrics and applied statistics, particularly for datasets exhibiting heavy tails, skewness, and volatility clustering. Such features are prevalent in many empirical applications and frequently violate the assumptions underlying classical linear regression methods. The central hypothesis of this study is that a regression framework based on orthogonal polynomial expansions can provide improved stability and predictive performance in the presence of pronounced nonlinear behaviour. The motivation for this work arises from the limitations of Ordinary Least Squares regression, which relies on linearity and distributional regularity, and from the instability of standard polynomial regression when applied to heavy
tailed data. To address these issues, the Generalised Hermite Regression Model (GHRM) is proposed by embedding Hermite polynomial expansions into a regression structure, enabling higher-order nonlinear dependencies to be modelled while preserving key theoretical properties. A numerical experiment based on a simulated dataset of 10,000 observations exhibiting nonlinear and heavy-tailed behaviour is conducted to evaluate the proposed model. The GHRM is assessed in 
comparison with Ordinary Least Squares and cubic polynomial regression using forecasting accuracy measures and information criteria. The results show that Ordinary Least Squares performs poorly under nonlinear conditions, while both polynomial regression and the GHRM achieve substantial improvements in predictive accuracy. Although the two nonlinear models produce comparable numerical results under the cubic specification, the GHRM demonstrates superior structural stability due to the orthogonality of the Hermite basis. These findings 
establish the GHRM as a robust and scalable framework for modelling nonlinear datasets with complex distributional characteristics.