Re-Parameterization of Multi-regime STAR-GARCH Model

It is well known in the literature that the joint parameter estimation of the Smooth Autoregressive – Generalized Autoregressive Conditional Heteroskedasticity (STAR-GARCH) models poses many numerical challenges with unknown causes. This paper aims to uncover the root of the numerical difficulties in obtaining stable parameter estimates for a class of three-regime STAR-GARCH models using Quasi-Maximum Likelihood Estimator (QMLE). The paper also provides an easy and practical solution to alleviate the difficulties based on the findings. The paper is divided into two parts. The first part investigates the numerical difficulties in maximizing the likelihood function by using computer simulations. Previous studies in the literature have identified that the threshold values and the transition rates are particular difficult to estimate. In light of this view, simulated data based on a pre-defined three-regime STAR-GARCH model will be generated and the values of the associated likelihood functions will be computed against different threshold values and transition rates. The results show some interesting characteristics of the likelihood functions that have not been reported previously. Firstly, the log-likelihood functions of Exponential STAR-GARCH (ESTAR-GARCH) models tend to be flat around the global optimum near the true values of the transition rates. This explains the difficulties in estimating the transition rates by maximizing the log-likelihood functions using conventional gradient-based optimization algorithms. Secondly, the surfaces of the log-likelihood functions of the Logistic STAR-GARCH (LSTAR-GARCH) models tend to be lumpy in addition to being flat around the local optimums. This explains the sensitivity of QMLE relative to initial values. These findings have two implications: (i) the shapes of the loglikelihood functions are determined mostly by the choice of transition functions and (ii) it may be possible to transform the shapes of the log-likelihood functions by re-parameterising the model. This paper proposes a simple re-parameterization of the three-regime STAR-GARCH models by transforming the transition rate parameter. The Monte Carlo simulation results show that the proposed method can alleviate the overall flatness and lumpy flatness of the log-likelihood functions for both LSTAR-GARCH and ESTARGARCH. These show promising signs in reducing estimation difficulties when jointly estimating the model parameters. Moreover, the results also open new channels for uncovering the statistical and structural properties of the three-regime STAR-GARCH model.