Maximum Likelihood Estimation Method for GARCH Models

We utilize the maximum likelihood estimation method for GARCH models based on MATLAB programming to forecast volatility in financial time series. This approach computes future volatility by combining historical volatility patterns with current period information, and has been extensively applied in financial market forecasting. The implementation involves several key steps: first, we identify the target time series for prediction through data preprocessing and stationarity checks. Next, we employ the GARCH model framework to estimate the time series volatility, typically using GARCH(1,1) specification where conditional variance depends on both previous volatility shocks and lagged variances. The core algorithm involves constructing the likelihood function based on the assumption of normally distributed innovations, then optimizing parameters using MATLAB's fmincon or similarly constrained optimization functions to maximize the log-likelihood. Key implementation aspects include handling initial parameter guesses, setting appropriate constraints for non-negative variance parameters, and managing convergence criteria. Finally, we determine optimal model parameters through maximum likelihood estimation, enabling predictive applications. This methodology represents a powerful financial analysis tool that enhances our understanding and forecasting capabilities for market volatility dynamics, with MATLAB providing robust numerical optimization routines for efficient parameter estimation.