Maximum Likelihood Estimation of Gaussian and Student’s $ t $ GARCH: A Unified Penalty Method

The generalized autoregressive conditional heteroskedasticity (GARCH) model is widely used to characterize time-varying conditional volatility in time series analysis. This paper studies the maximum likelihood (ML) estimation of GARCH model parameters under the assumption that the conditional distribution of the innovations follows either a Gaussian or a Student’s <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> distribution. The estimation problems are challenging due to the non-convex and recursively coupled nature of the model parameters, which often leads to convergence issues for existing methods. Moreover, many existing methods fail to incorporate the stationarity constraint, a desirable property for GARCH models. To address these challenges, we propose a unified penalty method for ML estimation of GARCH models that effectively handles the parameter coupling and allows flexible incorporation of stationarity constraints. We develop a convergent estimation algorithm based on the block majorization-minimization (BMM) framework, which efficiently exploits the problem structure and updates the model parameters effectively. We establish that the sequence generated by the BMM algorithm converges to the set of Karush-Kuhn-Tucker points. Notably, for the Student’s <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ML estimation problem, we provide the first theoretical characterization of the convexity of the negative conditional log-likelihood function with respect to the shape parameter. Furthermore, our algorithm naturally extends to M-estimation of GARCH, estimation of GARCH variants, and joint estimation of GARCH and conditional mean models. Numerical experiments on synthetic data demonstrate that the proposed algorithm outperforms existing methods in terms of parameter estimation accuracy and objective value. Its effectiveness is further validated on real-world datasets from applications including financial volatility modeling and radar target detection.