Study Of Semi-parametric GARCH Model Based On Transformed Kernel Density Estimation

Modeling the volatility of financial assets is an important research area of financial time series analysis, and it is significant to assets pricing, financial risk management as well as market microstructure analysis. Volatility of financial assets typically exhibits clustering and long memory, and the influence of positive or negative returns on volatility is asymmetric-the so called "leverage effect". GARCH model is the most commonly used time series model that to capture the characteristics of the volatility of financial assets.For conventional parametric GARCH model, estimates of the model parameters can be obtained by maximizing the likelihood function under a certain assumption of the conditional distribution of returns, and the most commonly used is the quasi-maximum likelihood estimation (QMLE) based on the normality assumption. But a lot of literatures show that the distribution of returns usually presents leptokurtosis and skewness, and its conditional distribution is typically uneven, which does not comply with the normality assumption. Although under certain regularity conditions, quasi-maximum likelihood estimator is asymptotically consistent, but its loss in efficiency cannot be ignored. In addition, parametric models under the assumption of particular distributions tend to have a higher risk of misspecification. In view of this, some authors combine the non-parametric methods with the parametric GARCH settings, and construct semi-parametric GARCH models which do not depend on the assumption of the conditional distribution in order to enhance the relative efficiency of parameter estimation and the accuracy of the model. However, traditional non-parametric methods do not provide a good estimate of the conditional density of returns, especially failing to capture the feature of fat tails.To address the problem stated above, this paper follows the transformed kernel density estimation, proposing a generalized logistic transformation, and Beta kernel estimation is then employed for the transformed data in order to eliminate boundary bias. Simulation experiments show that the proposed method considerably improves the accuracy of the density estimation for peaked and fat-tailed distributions. A new semi-parametric GARCH model is then constructed by combining the approach with the parametric GARCH settings. This semi-parametric model has two advantages: first, it provides a more accurate estimate of the conditional density of returns based on the proposed transformed kernel density estimation; second, it improves the robustness of estimation through iterations. Simulation studies show that, compared with Quasi Maximum Likelihood Estimation and the semi-parametric method based on Discrete Maximum Penalized Likelihood Estimation, the method proposed in this paper is more efficient. Empirical research with CSI300index verifies the validity of the model.