Handbook of Volatility Models and Their Applications -- Contents -- Preface -- Contributors -- 1 Volatility Models -- 1.1 Introduction -- 1.2 GARCH -- 1.2.1 Univariate GARCH -- 1.2.1.1 Structure of GARCH Models -- 1.2.1.2 Early GARCH Models -- 1.2.1.3 Probability Distributions for zt -- 1.2.1.4 New GARCH Models -- 1.2.1.5 Explanation of Volatility Clustering -- 1.2.1.6 Literature and Software -- 1.2.1.7 Applications of Univariate GARCH -- 1.2.2 Multivariate GARCH -- 1.2.2.1 Structure of MGARCH Models -- 1.2.2.2 Conditional Correlations -- 1.2.2.3 Factor Models -- 1.3 Stochastic Volatility -- 1.3.1 Leverage Effect -- 1.3.2 Estimation -- 1.3.3 Multivariate SV Models -- 1.3.4 Model Selection -- 1.3.5 Empirical Example: S&P 500 -- 1.3.6 Literature -- 1.4 Realized Volatility -- 1.4.1 Realized Variance -- 1.4.1.1 Empirical Application -- 1.4.2 Realized Covariance -- 1.4.2.1 Realized Quadratic Covariation -- 1.4.2.2 Realized Bipower Covariation -- Acknowledgments -- part one Autoregressive Conditional Heteroskedasticity and Stochastic Volatility -- 2 Nonlinear Models for Autoregressive Conditional Heteroskedasticity -- 2.1 Introduction -- 2.2 The Standard GARCH Model -- 2.3 Predecessors to Nonlinear GARCH Models -- 2.4 Nonlinear ARCH and GARCH Models -- 2.4.1 Engle's Nonlinear GARCH Model -- 2.4.2 Nonlinear ARCH Model -- 2.4.3 Asymmetric Power GARCH Model -- 2.4.4 Smooth Transition GARCH Model -- 2.4.5 Double Threshold ARCH Model -- 2.4.6 Neural Network ARCH and GARCH Models -- 2.4.7 Time-Varying GARCH -- 2.4.8 Families of GARCH Models and their Probabilistic Properties -- 2.5 Testing Standard GARCH Against Nonlinear GARCH -- 2.5.1 Size and Sign Bias Tests -- 2.5.2 Testing GARCH Against Smooth Transition GARCH -- 2.5.3 Testing GARCH Against Artificial Neural Network GARCH -- 2.6 Estimation of Parameters in Nonlinear GARCH Models.

2.6.1 Smooth Transition GARCH -- 2.6.2 Neural Network GARCH -- 2.7 Forecasting with Nonlinear GARCH Models -- 2.7.1 Smooth Transition GARCH -- 2.7.2 Asymmetric Power GARCH -- 2.8 Models Based on Multiplicative Decomposition of the Variance -- 2.9 Conclusion -- Acknowledgments -- 3 Mixture and Regime-Switching GARCH Models -- 3.1 Introduction -- 3.2 Regime-Switching GARCH Models for Asset Returns -- 3.2.1 The Regime-Switching Framework -- 3.2.2 Modeling the Mixing Weights -- 3.2.3 Regime-Switching GARCH Specifications -- 3.3 Stationarity and Moment Structure -- 3.3.1 Stationarity -- 3.3.2 Moment Structure -- 3.4 Regime Inference, Likelihood Function, and Volatility Forecasting -- 3.4.1 Determining the Number of Regimes -- 3.4.2 Volatility Forecasts -- 3.4.3 Application of MS-GARCH Models to Stock Return Indices -- 3.5 Application of Mixture GARCH Models to Density Prediction and Value-at-Risk Estimation -- 3.5.1 Value-at-Risk -- 3.5.2 Data and Models -- 3.5.3 Empirical Results -- 3.6 Conclusion -- Acknowledgments -- 4 Forecasting High Dimensional Covariance Matrices -- 4.1 Introduction -- 4.2 Notation -- 4.3 Rolling Window Forecasts -- 4.3.1 Sample Covariance -- 4.3.2 Observable Factor Covariance -- 4.3.3 Statistical Factor Covariance -- 4.3.4 Equicorrelation -- 4.3.5 Shrinkage Estimators -- 4.4 Dynamic Models -- 4.4.1 Covariance Targeting Scalar VEC -- 4.4.2 Flexible Multivariate GARCH -- 4.4.3 Conditional Correlation GARCH Models -- 4.4.4 Orthogonal GARCH -- 4.4.5 RiskMetrics -- 4.4.6 Alternative Estimators for Multivariate GARCH Models -- 4.5 High Frequency Based Forecasts -- 4.5.1 Realized Covariance -- 4.5.2 Mixed-Frequency Factor Model Covariance -- 4.5.3 Regularization and Blocking Covariance -- 4.6 Forecast Evaluation -- 4.6.1 Portfolio Constraints -- 4.7 Conclusion -- Acknowledgments.

5 Mean, Volatility, and Skewness Spillovers in Equity Markets -- 5.1 Introduction -- 5.2 Data and Summary Statistics -- 5.2.1 Data -- 5.2.2 Time-Varying Skewness (Univariate Analysis) -- 5.2.3 Spillover Models -- 5.3 Empirical Results -- 5.3.1 Parameter Estimates -- 5.3.2 Spillover Effects in Variance and Skewness -- 5.3.2.1 Variance Ratios -- 5.3.2.2 Pattern and Size of Skewness Spillovers -- 5.4 Conclusion -- Acknowledgments -- 6 Relating Stochastic Volatility Estimation Methods -- 6.1 Introduction -- 6.2 Theory and Methodology -- 6.2.1 Quasi-Maximum Likelihood Estimation -- 6.2.2 Gaussian Mixture Sampling -- 6.2.3 Simulated Method of Moments -- 6.2.4 Methods Based on Importance Sampling -- 6.2.4.1 Approximating in the Basic IS Approach -- 6.2.4.2 Improving on IS with IIS -- 6.2.4.3 Alternative Efficiency Gains with EIS -- 6.2.5 Alternative Sampling Methods: SSS and MMS -- 6.3 Comparison of Methods -- 6.3.1 Setup of Data-Generating Process and Estimation Procedures -- 6.3.2 Parameter Estimates for the Simulation -- 6.3.3 Precision of IS -- 6.3.4 Precision of Bayesian Methods -- 6.4 Estimating Volatility Models in Practice -- 6.4.1 Describing Return Data of Goldman Sachs and IBM Stock -- 6.4.2 Estimating SV Models -- 6.4.3 Extracting Underlying Volatility -- 6.4.4 Relating the Returns in a Bivariate Model -- 6.5 Conclusion -- 7 Multivariate Stochastic Volatility Models -- 7.1 Introduction -- 7.2 MSV Model -- 7.2.1 Model -- 7.2.1.1 Likelihood Function -- 7.2.1.2 Prior Distribution -- 7.2.1.3 Posterior Distribution -- 7.2.2 Bayesian Estimation -- 7.2.2.1 Generation of a -- 7.2.2.2 Generation of o -- 7.2.2.3 Generation of s -- 7.2.3 Multivariate-t Errors -- 7.2.3.1 Generation of v -- 7.2.3.2 Generation of l -- 7.3 Factor MSV Model -- 7.3.1 Model -- 7.3.1.1 Likelihood Function -- 7.3.1.2 Prior and Posterior Distributions -- 7.3.2 Bayesian Estimation.

7.3.2.1 Generation of a, o, and s -- 7.3.2.2 Generation of f -- 7.3.2.3 Generation of l -- 7.3.2.4 Generation of b -- 7.3.2.5 Generation of v -- 7.4 Applications to Stock Indices Returns -- 7.4.1 S&P 500 Sector Indices -- 7.4.2 MSV Model with Multivariate t Errors -- 7.4.2.1 Prior Distributions -- 7.4.2.2 Estimation Results -- 7.4.3 Factor MSV Model -- 7.4.3.1 Prior Distributions -- 7.4.3.2 Estimation Results -- 7.5 Conclusion -- 7.6 Appendix: Sampling a in the MSV Model -- 7.6.1 Single-Move Sampler -- 7.6.2 Multi-move Sampler -- 8 Model Selection and Testing of Conditional and Stochastic Volatility Models -- 8.1 Introduction -- 8.1.1 Model Specifications -- 8.2 Model Selection and Testing -- 8.2.1 In-Sample Comparisons -- 8.2.2 Out-of-Sample Comparisons -- 8.2.2.1 Direct Model Evaluation -- 8.2.2.2 Indirect Model Evaluation -- 8.3 Empirical Example -- 8.4 Conclusion -- part two Other Models and Methods -- 9 Multiplicative Error Models -- 9.1 Introduction -- 9.2 Theory and Methodology -- 9.2.1 Model Formulation -- 9.2.1.1 Specifications for u -- 9.2.1.2 Specifications for et -- 9.2.2 Inference -- 9.2.2.1 Maximum Likelihood Inference -- 9.2.2.2 Generalized Method of Moments Inference -- 9.3 MEMs for Realized Volatility -- 9.4 MEM Extensions -- 9.4.1 Component Multiplicative Error Model -- 9.4.2 Vector Multiplicative Error Model -- 9.5 Conclusion -- 10 Locally Stationary Volatility Modeling -- 10.1 Introduction -- 10.2 Empirical Evidences -- 10.2.1 Structural Breaks, Nonstationarity, and Persistence -- 10.2.2 Testing Stationarity -- 10.3 Locally Stationary Processes and their Time-Varying Autocovariance Function -- 10.4 Locally Stationary Volatility Models -- 10.4.1 Multiplicative Models -- 10.4.2 Time-Varying ARCH Processes -- 10.4.3 Adaptive Approaches -- 10.5 Multivariate Models for Locally Stationary Volatility -- 10.5.1 Multiplicative Models.

10.5.2 Adaptive Approaches -- 10.6 Conclusions -- Acknowledgments -- 11 Nonparametric and Semiparametric Volatility Models: Specification, Estimation, and Testing -- 11.1 Introduction -- 11.2 Nonparametric and Semiparametric Univariate Volatility Models -- 11.2.1 Stationary Volatility Models -- 11.2.1.1 The Simplest Nonparametric Volatility Model -- 11.2.1.2 Additive Nonparametric Volatility Model -- 11.2.1.3 Functional-Coefficient Volatility Model -- 11.2.1.4 Single-Index Volatility Model -- 11.2.1.5 Stationary Semiparametric ARCH (D) Models -- 11.2.1.6 Semiparametric Combined Estimator of Volatility -- 11.2.1.7 Semiparametric Inference in GARCH-in-Mean Models -- 11.2.2 Nonstationary Univariate Volatility Models -- 11.2.3 Specification of the Error Density -- 11.2.4 Nonparametric Volatility Density Estimation -- 11.3 Nonparametric and Semiparametric Multivariate Volatility Models -- 11.3.1 Modeling the Conditional Covariance Matrix under Stationarity -- 11.3.1.1 Hafner, van Dijk, and Franses' Semiparametric Estimator -- 11.3.1.2 Long, Su, and Ullah's Semiparametric Estimator -- 11.3.1.3 Test for the Correct Specification of Parametric Conditional Covariance Models -- 11.3.2 Specification of the Error Density -- 11.4 Empirical Analysis -- 11.5 Conclusion -- Acknowledgments -- 12 Copula-Based Volatility Models -- 12.1 Introduction -- 12.2 Definition and Properties of Copulas -- 12.2.1 Sklar's Theorem -- 12.2.2 Conditional Copula -- 12.2.3 Some Commonly Used Bivariate Copulas -- 12.2.4 Copula-Based Dependence Measures -- 12.3 Estimation -- 12.3.1 Exact Maximum Likelihood -- 12.3.2 IFM -- 12.3.3 Bivariate Static Copula Models -- 12.4 Dynamic Copulas -- 12.4.1 Early Approaches -- 12.4.2 Dynamics Based on the DCC Model -- 12.4.3 Alternative Methods -- 12.5 Value-at-Risk -- 12.6 Multivariate Static Copulas -- 12.6.1 Multivariate Archimedean Copulas.

12.6.2 Vines.