Using Mixed Integer Nonlinearly Constrained Optimization to Do Penalized Maximum Likelihood Estimation for Garch and Arch Models

Compared to the traditional maximum likelihood regression approach, the penalized maximum likelihood estimation (PMLE) is a more rigorous method because of the adjustment for over fitting is directly built into the model development process, instead of relying on shrinkage afterwards. This paper illustrates the application of a nonlinear programming technique on PMLE to develop a prediction model for a Generalized Autoregressive conditional heteroskedasticity (GARCH) model based on Autoregressive Moving Average time series. Our numerical study demonstrates that the mixed integer nonlinearly constrained optimization method can accurately predict the parameters of time series data.