Numerical methods for parameter estimation of Generalized Autoregressive Conditional Heteroskedasticity models of financial time series
Abstract
Numerical methods for parameter estimation of Generalized Autoregressive Conditional Heteroskedasticity models of financial time series
Incoming article date: 01.02.2025This article discusses numerical methods used to estimate the parameters of a family of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, which are widely used for analyzing and predicting financial time series with variable variance. The paper provides a comparative analysis of numerical methods for estimating GARCH effects, which are based on the gradient descent method of adaptive algorithms, various variations of quadratic methods based on the Newton method, as well as alternative methods based on the simplex method, linear and quadratic interpolation. The analysis is carried out on the basis of synthetic data and on real data on quotations of the Moscow Exchange stock index using the Python 3 programming language and libraries scipy, numpy, matplotlib and others. The results of the study show that the specifics of the financial time series problem are sensitive to the choice of numerical methods for solving the optimization problem of maximizing the likelihood function. Numerical experiment has shown that using the Nelder-Meade method to evaluate GARCH effects gives the best results for solving the problem of maximizing the likelihood function.
Keywords: mathematical modeling, numerical methods, maximum likelihood method, gradient descent, Newton's method, mathematical modeling, GARCH, time series, stock market, news flows