Modeling Time-Varying Unconditional Variance by Means of a Free-Knot Spline-GARCH Model (Gabler Theses)

The book addresses the problem of a time-varying unconditional variance of return processes utilizing a spline function. The knots of the spline functions are estimated as free parameters within a joined estimation process together with the parameters of the mean, the conditional variance and the spline function. With the help of this method, the knots are placed in regions where the unconditional variance is not smooth. The results are tested within an extensive simulation study and an empirical study employing the S&P500 index. Foreword Acknowledgements Contents List of Figures List of Tables List of Abbreviations List of Symbols 1 Introduction 1.1 Motivation 1.2 Problem statement 1.3 Outline of the thesis 2 Financial time series 2.1 Definitions and properties 2.2 Stylized facts 2.3 Model specification 2.4 Univariate GARCH models 2.5 Long-range dependence and structural breaks 3 Smoothing long term volatility 3.1 Multiplicative decomposition of the conditional variance function 3.2 Spline functions 3.2.1 Truncated power spline function 3.2.2 B-spline functions 3.3 Model review 3.3.1 Spline volatility models 3.3.2 Spline-GARCH model 3.3.3 B-spline-GARCH model 3.3.4 P-spline GARCH model 4 Free-knot spline-GARCH model 4.1 Optimization 4.2 Estimation methods 4.2.1 Least-squares 4.2.2 Least-squares with free-knots 4.2.3 Jupp transformation 4.2.4 Quasi-maximum-likelihood 4.3 Model selection 4.4 Forecast evaluation 4.5 Starting vector 5 Simulation study 5.1 Previous studies 5.2 Simulation setup 5.2.1 Data generating process 5.2.2 Computational aspects 5.2.3 Sample statistics 5.2.4 Asymptotic statistics 5.2.5 Specification 5.2.6 Starting vectors 5.3 Model selection 5.4 Finite sample properties 6 Empirical study 6.1 Previous studies 6.2 In-sample analysis 6.3 Out-of-sample forecast 7 Conclusion 7.1 Research problems and contributions 7.2 Research questions 7.3 Limitations and future research 7.4 Concluding remarks References Appendices A Standardized Student’s t-distribution B Derivatives B.1 Free-knot spline-GARCH model B.2 P-spline-GARCH model C Tables C.1 Simulation study: knots C.2 Simulation study: Finite sample properties C.3 Empirical study D Figures D.1 Simulation study: distribution of knot selection D.2 Simulation study: asymptotic distribution estimators D.3 Emprical study