Bilinear Log-GARCH model

In this paper, we propose a new bilinear Log-GARCH model. We establish a sufficient condition ensuring that the proposed model admits a unique, strictly stationary, and ergodic solution. We then apply the quasi-maximum likelihood estimation (QMLE) method to estimate the model parameters. Several theorems are presented to demonstrate the convergence of the estimation procedure and the convergence of the estimated parameters to the true values. Furthermore, we implement a Monte Carlo simulation to generate the time series. The purpose of these simulations is to assess the ability of the proposed model to capture the dynamics of financial volatility, particularly asymmetry and leverage effects, as well as to evaluate the performance of the QMLE estimator in finite samples. Additionally, we present carefully selected numerical examples that produce remarkable results, confirming both the effectiveness of the bilinear log-GARCH model and the reliability of the Monte Carlo simulation framework, as well as the accuracy of the parameter estimation. These results demonstrate that the proposed model is capable of capturing the complex volatility behavior in financial time series, making it a suitable tool for analyzing financial data with seasonal or nonlinear characteristics.