Parameter Estimation Of Log-GARCH Model

The GARCH model has always been the focus of research in the field of time series analysis.Compared with the non-exponential GARCH model,the exponential GARCH model is more popular because of its advantages in model interpretation and practical application.This paper chooses to study the log-GARCH model,a kind of exponential GARCH model and considers the parameter estimation problem of the model.At present,the parameter estimation of the log-GARCH model can be roughly divided into two methods.One is to directly estimate the parameters,and the other is to write the model in ARMA representation to estimate the parameters of the new model,so as to obtain the estimated values of the original model parameters.In addition,most studies are based on the assumption that the distribution of random terms is known,such as the assumption that it follows a normal distribution,t-distribution,etc.In this paper,the research is carried out on the basis that the density function of conditional density is unknown.We use the ARMA representation of the log-GARCH model to convert the original problem into an ARMA model parameter estimation problem.Since conditional density in the original model is unknown,we apply quasi-maximum likelihood estimation(QMLE)to the innovation of the new ARMA model.According to the parameter relationship between the original model and the new model,we get the estimate of the log-GARCH model parameters.At the same time,we prove the consistency and asymptotic normality of this estimator,and give the concrete expression of asymptotic distribution.Finally,the consistency and asymptotic normality of this estimator are verified by numerical simulation.Combined with the actual data,this paper compares the estimation method with other methods.At the same time,it also uses actual data to fit a variety of GARCH models and compare their prediction effects.