Volatility Models and Their Applications

Transcription

1 HANDBOOK OF Volatility Models and Their Applications Edited by Luc BAUWENS CHRISTIAN HAFNER SEBASTIEN LAURENT WILEY A John Wiley & Sons, Inc., Publication

2 PREFACE CONTRIBUTORS XVII XIX [JQ VOLATILITY MODELS Introduction, GARCH, Univariate GARCH, Structure of GARCH Models, Early GARCH Models, Probability Distributions for z t, A New GARCH Models, Explanation of Volatility Clustering, Literature and Software, Applications of Univariate GARCH, Multivariate GARCH, Structure of MGARCH Models, Conditional Correlations, Factor Models, Stochastic Volatility, Leverage Effect, Estimation, Multivariate SV Models, Model Selection, Empirical Example: S&P 500, Literature, Realized Volatility, Realised Variance, Empirical Application, Realized Covariance, 44

3 Realized Quadratic Covariation, Realized Bipower Covariation, 44 Acknowledgments, 45 Autoregressive Conditional Heteroskedasticity and Stochastic Volatility ([jss] NONLINEAR MODELS FOR AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY Introduction, The Standard GARCH Model, Predecessors to Nonlinear GARCH Models, Nonlinear ARCH and GARCH Models, Engle's Nonlinear GARCH Model, Nonlinear ARCH Model, Asymmetric Power GARCH Model, Smooth Transition GARCH Model, Double Threshold ARCH Model, Neural Network ARCH and GARCH Models, Time-Varying GARCH, Families of GARCH Models and their Probabilistic Properties, Testing Standard GARCH Against Nonlinear GARCH, Size and Sign Bias Tests, Testing GARCH Against Smooth Transition GARCH, Testing GARCH Against Artificial Neural Network GARCH, Estimation of Parameters in Nonlinear GARCH Models, Smooth Transition GARCH, Neural Network GARCH, Forecasting with Nonlinear GARCH Models, Smooth Transition GARCH, Asymmetric Power GARCH, Models Based on Multiplicative Decomposition of the Variance, Conclusion, 68 Acknowledgments, 69

4 vii LID MIXTURE AND REGIME-SWITCHING GARCH MODELS Introduction, Regime-Switching GARCH Models for Asset Returns, The Regime-Switching Framework, Modeling the Mixing Weights, Regime-Switching GARCH Specifications, Stationarity and Moment Structure, Stationarity, 83.._ Moment Structure, Regime Inference, Likelihood Function, and Volatility Forecasting, Determining the Number of Regimes, Volatility Forecasts, Application of MS-GARCH Models to Stock Return Indices, Application of Mixture GARCH Models to Density Prediction and Value-at-Risk Estimation, Value-at-Risk, Data and Models, Empirical Results, Conclusion, 102 Acknowledgments, 102 F4] FORECASTING HIGH DIMENSIONAL COVARIANCE MATRICES Introduction, 103 Notation, 104 Rolling Window Forecasts, Sample Covariance, 105 Observable Factor Covariance, 105 Statistical Factor Covariance, 106 Equicorrelation, 107 Shrinkage Estimators, 108 Dynamic Models, Covariance Targeting Scalar VEC, Flexible Multivariate GARCH, Conditional Correlation GARCH Models, Orthogonal GARCH, RiskMetrics, Alternative Estimators for Multivariate GARCH Models, O3

5 viii Contents 4.5 High Frequency Based Forecasts, Realized Covariance, Mixed-Frequency Factor Model Covariance, Regularization and Blocking Covariance, Forecast Evaluation, Portfolio Constraints, Conclusion, 125 Acknowledgments, 125 [JSJ MEAN, VOLATILITY, ANP SKEWNESS SPILLOVERS IN EQUITY MARKETS Introduction, Data and Summary Statistics, Data, Time-Varying Skewness (Univariate Analysis), Spillover Models, Empirical Results, Parameter Estimates, Spillover Effects in Variance and Skewness, Variance Ratios, Pattern and Size of Skewness Spillovers, Conclusion, 144 Acknowledgments, 145 jjj ] RELATING STOCHASTIC VOLATILITY ESTIMATION METHODS Introduction, Theory and Methodology, Quasi-Maximum Likelihood Estimation, Gaussian Mixture Sampling, Simulated Method of Moments, Methods Based on Importance Sampling, Approximating in the Basic IS Approach, Improving on IS with IIS, Alternative Efficiency Gains with EIS, 156

6 6.2.5 Alternative Sampling Methods: SSS and MMS, Comparison of Methods, Setup of Data-Generating Process and Estimation Procedures, Parameter Estimates for the Simulation, Precision of IS, Precision of Bayesian Methods, Estimating Volatility Models in Practice, Describing Return Data*6f Goldman Sachs and IBM Stock, Estimating SV Models, Extracting Underlying Volatility, Relating the Returns in a Bivariate Model, Conclusion, 172 MULTIVARIATE STOCHASTIC VOLATILITY MODELS Introduction, MSV Model, Model, Likelihood Function, Prior Distribution, Posterior Distribution, Bayesian Estimation, Generation of a, Generation of 0, Generation of E, Multivariate-* Errors, Generation of v, Generation of X, Factor MSV Model, Model, Likelihood Function, Prior and Posterior Distributions, Bayesian Estimation, Generation of a, 0, and X, Generation of/, Generation of A., Generation of 0, 188

7 Generation of v, Applications to Stock Indices Returns, S&P 500 Sector Indices, MSV Model with Multivariate t Errors, Prior Distributions, Estimation Results, Factor MSV Model, Prior Distributions, Estimation Results, Conclusion, Appendix: Sampling a ih the MSV Model, Single-Move Sampler, Multi-move Sampler, 196 MODEL SELECTION AND TESTING OF CONDITIONAL AND STOCHASTIC VOLATILITY MODELS Introduction, Model Specifications, Model Selection and Testing, In-Sample Comparisons, Out-of-Sample Comparisons, Direct Model Evaluation, Indirect Model Evaluation, Empirical Example, Conclusion, 221 irart TWO Other Models and Methods 01 MULTIPLICATIVE ERROR MODELS Introduction, Theory and Methodology, Model Formulation, Specifications for fi t, Specifications for e t, Inference, Maximum Likelihood Inference, Generalized Method of Moments Inference, MEMs for Realized Volatility, MEM Extensions, 242

8 xi Component Multiplicative Error Model, Vector Multiplicative Error Model, Conclusion, 247 [To) LOCALLY STATIONARY VOLATILITY MODELING Introduction, Empirical Evidences, Structural Breaks, Nonstationarity, and Persistence, Testing Stationarity, Locally Stationary Processes and their Time-Varying Autocovariance Function, Locally Stationary Volatility Models, Multiplicative Models, Time-Varying ARCH Processes, Adaptive Approaches, Multivariate Models for Locally Stationary Volatility, Multiplicative Models, Adaptive Approaches, Conclusions, 267 Acknowledgments, 268 [IF] NONPARAMETRIC AND SEMIPARAMETRIC VOLATILITY MODELS: SPECIFICATION, ESTIMATION, AND TESTING Introduction, Nonparametric and Semiparametric Univariate Volatility Models, Stationary Volatility Models, The Simplest Nonparametric Volatility Model, Additive Nonparametric Volatility Model, Functional-Coefficient Volatility Model, Single-Index Volatility Model, Stationary Semiparametric ARCH (oo) Models, Semiparametric Combined Estimator ofvolatility, 279

9 xii Contents Semiparametric Inference in GARCH-in-Mean Models, Nonstationary Univariate Volatility Models, Specification of the Error Density, Nonparametric Volatility Density Estimation, Nonparametric and Semiparametric Multivariate Volatility Models, Modeling the Conditional Covariance Matrix under Stationarity, Hafner, van Dijk, and Franses' Semiparametric Estimator, Long, Su, and Ullah's Semiparametric Estimator, Test for the Correct Specification of Parametric Conditional Covariance Models, Specification of the Error Density, Empirical Analysis, Conclusion, 291 Acknowledgments, 291 COPULA-BASED VOLATILITY MODELS Introduction, Definition and Properties of Copulas, Sklar's Theorem, Conditional Copula, Some Commonly Used Bivariate Copulas, Copula-Based Dependence Measures, Estimation, Exact Maximum Likelihood, IFM, Bivariate Static Copula Models, Dynamic Copulas, Early Approaches, Dynamics Based on the DCC Model, Alternative Methods, Value-at-Risk, Multivariate Static Copulas, Multivariate Archimedean Copulas, Vines, Conclusion, 315

10 xiii PART THREE Realized Volatility REALIZED VOLATILITY: THEORY AND APPLICATIONS Introduction, Modeling Framework, Efficient Price, Measurement Error, Issues in Handling Intraday Transaction Databases, Which Price to Use?, High Frequency Data Preprocessing, How to and How Often to Sample?, Realized Variance and Covariance, Univariate Volatility Estimators, Measurement Error, Multivariate Volatility Estimators, Measurement Error, Modeling and Forecasting, Time Series Models of (co) Volatility, Forecast Comparison, Asset Pricing, Distribution of Returns Conditional on the Volatility Measure, Application to Factor Pricing Model, Effects of Algorithmic Trading, Application to Option Pricing, Estimating Continuous Time Models, J LIKELIHOOD-BASED VOLATILITY ESTIMATORS IN THE PRESENCE OF MARKET MLCROSTRUCTURE NOISE Introduction, Volatility Estimation, Constant Volatility and Gaussian Noise Case: MLE, Robustness to Non-Gaussian Noise, Implementing Maximum Likelihood, Robustness to Stochastic Volatility: QMLE, Comparison with Other Estimators, Random Sampling and Non-i.i.d. Noise, Covariance Estimation, 356

11 xiv Contents 14.4 Empirical Application: Correlation between Stock and Commodity Futures, Conclusion, 360 Acknowledgments, 361 FJ[T HAR MODELING FOR REALIZED VOLATILITY FORECASTING Introduction, Stylized Facts on Realized Volatility, Heterogeneity and Volatility Persistence, Genuine Long Memory or Superposition of Factors?, HAR Extensions, Jump Measures and Their Volatility Impact, Leverage Effects, A.3 General Nonlinear Effects in Volatility, Multivariate Models, Applications, Conclusion, 381 FORECASTING VOLATILITY WITH MIDAS Introduction, MIDAS Regression Models and Volatility Forecasting, MIDAS Regressions, Direct Versus Iterated Volatility Forecasting, Variations on the Theme of MIDAS Regressions, Microstructure Noise and MIDAS Regressions, Likelihood-Based Methods, Risk-Return Trade-Off, HYBRID GARCH Models, GARCH-MIDAS Models, Multivariate Models, Conclusion, 401 JUMPS 17.1 Introduction, Some Models Used in Finance and Our Framework, Simulated Models Used in This Chapter, A3 Realized Variance and Quadratic Variation, 409 4O3

12 xv j Importance of Disentangling, Further Notation, How to Disentangle: Estimators of Integrated Variance and Integrated Covariance, Bipower Variation, Threshold Estimator, Threshold Bipower Variation, Other Methods, Realized Quantile, MinRVandMedRV, Realised Outlyingness Weighted Variation, Range Bipower Variation, Generalization of the Realized Range, Duration-Based Variation, Irregularly Spaced Observations, Comparative Implementation on Simulated Data, Noisy Data, Multivariate Assets, Testing for the Presence of Jumps, Confidence Intervals, Tests Based on IV B - RV B or on 1 - IV B /RV B, Tests Based on Normalized Returns, PV-Based Tests, Remarks, Tests Based on Signature Plots, Tests Based on Observation of Optioii Prices, Remarks, Indirect Test for the Presence of Jumps, In the Presence of Noise, Comparisons, Conclusions, 444 Acknowledgments, ] NONPARAMETRIC TESTS FOR INTRADAY JUMPS: IMPACT OF PERIODICITY AND MICROSTRUCTURE NOISE Introduction, Model, Price Jump Detection Method, 450

13 xvi Contents Estimation of the Noise Variance, Robust Estimators of the Integrated Variance, Periodicity Estimation, Jump Test Statistics, Critical Value, Simulation Study, Intraday Differences in the Value of the Test Statistics, Comparison of Size and Power, Simulation Setup, Results, 458! 18.5 Comparison on NYSE Stock Prices, Conclusion, 462 [jj9j VOLATILITY FORECASTS EVALUATION AND COMPARISON Introduction, Notation, Single Forecast Evaluation, Loss Functions and the Latent Variable Problem, Pairwise Comparison, Multiple Comparison, Consistency of the Ordering and Inference on Forecast Performances, 481' 19.8 Conclusion, 485 BIBLIOGRAPHY 487 INDEX - 537