HANDBOOK OF Volatility Models and Their Applications Edited by Luc BAUWENS CHRISTIAN HAFNER SEBASTIEN LAURENT WILEY A John Wiley & Sons, Inc., Publication
PREFACE CONTRIBUTORS XVII XIX [JQ VOLATILITY MODELS 1 1.1 Introduction, 1 1.2 GARCH, 1 1.2.1 Univariate GARCH, 1 1.2.1.1 Structure of GARCH Models, 3 1.2.1.2 Early GARCH Models, 5 1.2.1.3 Probability Distributions for z t, 7 1.2.1 A New GARCH Models, 9 1.2.1.5 Explanation of Volatility Clustering, 15 1.2.1.6 Literature and Software, 16 1.2.1.7 Applications of Univariate GARCH, 16 1.2.2 Multivariate GARCH, 18 1.2.2.1 Structure of MGARCH Models, 19 1.2.2.2 Conditional Correlations, 19 1.2.2.3 Factor Models, 23 1.3 Stochastic Volatility, 25 1.3.1 Leverage Effect, 26 1.3.2 Estimation, 27 1.3.3 Multivariate SV Models, 28 1.3.4 Model Selection, 30 1.3.5 Empirical Example: S&P 500, 31 1.3.6 Literature, 32 1.4 Realized Volatility, 33 1.4.1 Realised Variance, 33 1.4.1.1 Empirical Application, 40 1.4.2 Realized Covariance, 44
1.4.2.1 Realized Quadratic Covariation, 44 1.4.2.2 Realized Bipower Covariation, 44 Acknowledgments, 45 Autoregressive Conditional Heteroskedasticity and Stochastic Volatility ([jss] NONLINEAR MODELS FOR AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY 49 2.1 Introduction, 49 2.2 The Standard GARCH Model, 50 2.3 Predecessors to Nonlinear GARCH Models, 51 2.4 Nonlinear ARCH and GARCH Models, 52 2.4.1 Engle's Nonlinear GARCH Model, 52 2.4.2 Nonlinear ARCH Model, 53 2.4.3 Asymmetric Power GARCH Model, 53 2.4.4 Smooth Transition GARCH Model, 54 2.4.5 Double Threshold ARCH Model, 56 2.4.6 Neural Network ARCH and GARCH Models, 57 2.4.7 Time-Varying GARCH, 58 2.4.8 Families of GARCH Models and their Probabilistic Properties, 59 2.5 Testing Standard GARCH Against Nonlinear GARCH, 60 2.5.1 Size and Sign Bias Tests, 60 2.5.2 Testing GARCH Against Smooth Transition GARCH, 61 2.5.3 Testing GARCH Against Artificial Neural Network GARCH, 62 2.6 Estimation of Parameters in Nonlinear GARCH Models, 63 2.6.1 Smooth Transition GARCH, 63 2.6.2 Neural Network GARCH, 64 2.7 Forecasting with Nonlinear GARCH Models, 64 2.7.1 Smooth Transition GARCH, 64 2.7.2 Asymmetric Power GARCH, 66 2.8 Models Based on Multiplicative Decomposition of the Variance, 67 2.9 Conclusion, 68 Acknowledgments, 69
vii LID MIXTURE AND REGIME-SWITCHING GARCH MODELS 71 3.1 Introduction, 71 3.2 Regime-Switching GARCH Models for Asset Returns, 73 3.2.1 The Regime-Switching Framework, 73 3.2.2 Modeling the Mixing Weights, 75 3.2.3 Regime-Switching GARCH Specifications, 78 3.3 Stationarity and Moment Structure, 81 3.3.1 Stationarity, 83.._.. 3.3.2 Moment Structure, 87 3.4 Regime Inference, Likelihood Function, and Volatility Forecasting, 89 3.4.1 Determining the Number of Regimes, 92 3.4.2 Volatility Forecasts, 92 3.4.3 Application of MS-GARCH Models to Stock Return Indices, 93 3.5 Application of Mixture GARCH Models to Density Prediction and Value-at-Risk Estimation, 97 3.5.1 Value-at-Risk, 97 3.5.2 Data and Models, 98 3.5.3 Empirical Results, 99 3.6 Conclusion, 102 Acknowledgments, 102 F4] FORECASTING HIGH DIMENSIONAL COVARIANCE MATRICES 4.1 4.2 4.3 4.4 Introduction, 103 Notation, 104 Rolling Window Forecasts, 104 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 Sample Covariance, 105 Observable Factor Covariance, 105 Statistical Factor Covariance, 106 Equicorrelation, 107 Shrinkage Estimators, 108 Dynamic Models, 109 4.4.1 Covariance Targeting Scalar VEC, 109 4.4.2 Flexible Multivariate GARCH, 110 4.4.3 Conditional Correlation GARCH Models, 111 4.4.4 Orthogonal GARCH, 113 4.4.5 RiskMetrics, 114 4.4.6 Alternative Estimators for Multivariate GARCH Models, 116 1 O3
viii Contents 4.5 High Frequency Based Forecasts, 117 4.5.1 Realized Covariance, 118 4.5.2 Mixed-Frequency Factor Model Covariance, 119 4.5.3 Regularization and Blocking Covariance, 119 4.6 Forecast Evaluation, 123 4.6.1 Portfolio Constraints, 124 4.7 Conclusion, 125 Acknowledgments, 125 [JSJ MEAN, VOLATILITY, ANP SKEWNESS SPILLOVERS IN EQUITY MARKETS 127 5.1 Introduction, 127 5.2 Data and Summary Statistics, 129 5.2.1 Data, 129 5.2.2 Time-Varying Skewness (Univariate Analysis), 132 5.2.3 Spillover Models, 135 5.3 Empirical Results, 138 5.3.1 Parameter Estimates, 138 5.3.2 Spillover Effects in Variance and Skewness, 139 5.3.2.1 Variance Ratios, 139 5.3.2.2 Pattern and Size of Skewness Spillovers, 141 5.4 Conclusion, 144 Acknowledgments, 145 jjj ] RELATING STOCHASTIC VOLATILITY ESTIMATION METHODS. 147 6.1 Introduction, 147 6.2 Theory and Methodology, 149 6.2.1 Quasi-Maximum Likelihood Estimation, 150 6.2.2 Gaussian Mixture Sampling, 151 6.2.3 Simulated Method of Moments, 152 6.2.4 Methods Based on Importance Sampling, 153 6.2.4.1 Approximating in the Basic IS Approach, 154 6.2.4.2 Improving on IS with IIS, 155 6.2.4.3 Alternative Efficiency Gains with EIS, 156
6.2.5 Alternative Sampling Methods: SSS and MMS, 158 6.3 Comparison of Methods, 160 6.3.1 Setup of Data-Generating Process and Estimation Procedures, 160 6.3.2 Parameter Estimates for the Simulation, 161 6.3.3 Precision of IS, 163 6.3.4 Precision of Bayesian Methods, 164 6.4 Estimating Volatility Models in Practice, 165 6.4.1 Describing Return Data*6f Goldman Sachs and IBM Stock, 165 6.4.2 Estimating SV Models, 167 6.4.3 Extracting Underlying Volatility, 168 6.4.4 Relating the Returns in a Bivariate Model, 169 6.5 Conclusion, 172 MULTIVARIATE STOCHASTIC VOLATILITY MODELS 175 7.1 Introduction, 175 7.2 MSV Model, 176 7.2.1 Model, 176 7.2.1.1 Likelihood Function, 177 7.2.1.2 Prior Distribution, 178 7.2.1.3 Posterior Distribution, 179 7.2.2 Bayesian Estimation, 179 7.2.2.1 Generation of a, 179 7.2.2.2 Generation of 0, 181 7.2.2.3 Generation of E, 181 7.2.3 Multivariate-* Errors, 181 7.2.3.1 Generation of v, 182 7.2.3.2 Generation of X, 183 7.3 Factor MSV Model, 183 7.3.1 Model, 183 7.3.1.1 Likelihood Function, 184 7.3.1.2 Prior and Posterior Distributions, 185 7.3.2 Bayesian Estimation, 185 7.3.2.1 Generation of a, 0, and X, 186 7.3.2.2 Generation of/, 187 7.3.2.3 Generation of A., 187 7.3.2.4 Generation of 0, 188
7.3.2.5 Generation of v, 188 7.4 Applications to Stock Indices Returns, 188 7.4.1 S&P 500 Sector Indices, 188 7.4.2 MSV Model with Multivariate t Errors, 189 7.4.2.1 Prior Distributions, 189 7.4.2.2 Estimation Results, 189 7.4.3 Factor MSV Model, 192 7.4.3.1 Prior Distributions, 192 7.4.3.2 Estimation Results, 192 7.5 Conclusion, 195 7.6 Appendix: Sampling a ih the MSV Model, 195 7.6.1 Single-Move Sampler, 195 7.6.2 Multi-move Sampler, 196 MODEL SELECTION AND TESTING OF CONDITIONAL AND STOCHASTIC VOLATILITY MODELS 199 8.1 Introduction, 199 8.1.1 Model Specifications, 200 8.2 Model Selection and Testing, 202 8.2.1 In-Sample Comparisons, 202 8.2.2 Out-of-Sample Comparisons, 206 8.2.2.1 Direct Model Evaluation, 206 8.2.2.2 Indirect Model Evaluation, 209 8.3 Empirical Example, 211 8.4 Conclusion, 221 irart TWO Other Models and Methods 01 MULTIPLICATIVE ERROR MODELS 225 9.1 Introduction, 225 9.2 Theory and Methodology, 226 9.2.1 Model Formulation, 226 9.2.1.1 Specifications for fi t, 227 9.2.1.2 Specifications for e t, 230 9.2.2 Inference, 230 9.2.2.1 Maximum Likelihood Inference, 230 9.2.2.2 Generalized Method of Moments Inference, 233 9.3 MEMs for Realized Volatility, 235 9.4 MEM Extensions, 242
xi 9.4.1 Component Multiplicative Error Model, 242 9.4.2 Vector Multiplicative Error Model, 243 9.5 Conclusion, 247 [To) LOCALLY STATIONARY VOLATILITY MODELING 249 10.1 Introduction, 249 10.2 Empirical Evidences, 251 10.2.1 Structural Breaks, Nonstationarity, and Persistence, 251 10.2.2 Testing Stationarity, 253 10.3 Locally Stationary Processes and their Time-Varying Autocovariance Function, 256 10.4 Locally Stationary Volatility Models, 260 10.4.1 Multiplicative Models, 260 10.4.2 Time-Varying ARCH Processes, 261 10.4.3 Adaptive Approaches, 264 10.5 Multivariate Models for Locally Stationary Volatility, 266 10.5.1 Multiplicative Models, 266 10.5.2 Adaptive Approaches, 267 10.6 Conclusions, 267 Acknowledgments, 268 [IF] NONPARAMETRIC AND SEMIPARAMETRIC VOLATILITY MODELS: SPECIFICATION, ESTIMATION, AND TESTING 269 11.1 Introduction, 269 11.2 Nonparametric and Semiparametric Univariate Volatility Models, 271 11.2.1 Stationary Volatility Models, 271 11.2.1.1 The Simplest Nonparametric Volatility Model, 271 11.2.1.2 Additive Nonparametric Volatility Model, 273 11.2.1.3 Functional-Coefficient Volatility Model, 276 11.2.1.4 Single-Index Volatility Model, 277 11.2.1.5 Stationary Semiparametric ARCH (oo) Models, 278 11.2.1.6 Semiparametric Combined Estimator ofvolatility, 279
xii Contents 11.2.1.7 Semiparametric Inference in GARCH-in-Mean Models, 280 11.2.2 Nonstationary Univariate Volatility Models, 281 11.2.3 Specification of the Error Density, 282 11.2.4 Nonparametric Volatility Density Estimation, 283 11.3 Nonparametric and Semiparametric Multivariate Volatility Models, 284 11.3.1 Modeling the Conditional Covariance Matrix under Stationarity, 285 11.3.1.1 Hafner, van Dijk, and Franses' Semiparametric Estimator, 285 11.3.1.2 Long, Su, and Ullah's Semiparametric Estimator, 286 11.3.1.3 Test for the Correct Specification of Parametric Conditional Covariance Models, 286 11.3.2 Specification of the Error Density, 287 11.4 Empirical Analysis, 288 11.5 Conclusion, 291 Acknowledgments, 291 COPULA-BASED VOLATILITY MODELS 293 12.1 Introduction, 293 12.2 Definition and Properties of Copulas, 294 12.2.1 Sklar's Theorem, 295 12.2.2 Conditional Copula, 296 12.2.3 Some Commonly Used Bivariate Copulas, 296 12.2.4 Copula-Based Dependence Measures, 298 12.3 Estimation, 300 12.3.1 Exact Maximum Likelihood, 300 12.3.2 IFM, 301 12.3.3 Bivariate Static Copula Models, 301 12.4 Dynamic Copulas, 304 12.4.1 Early Approaches, 305 12.4.2 Dynamics Based on the DCC Model, 305 12.4.3 Alternative Methods, 307 12.5 Value-at-Risk, 308 12.6 Multivariate Static Copulas, 310 12.6.1 Multivariate Archimedean Copulas, 310 12.6.2 Vines, 313 12.7 Conclusion, 315
xiii PART THREE Realized Volatility REALIZED VOLATILITY: THEORY AND APPLICATIONS 319 13.1 Introduction, 319 13.2 Modeling Framework, 320 13.2.1 Efficient Price, 320 13.2.2 Measurement Error, 322 13.3 Issues in Handling Intraday Transaction Databases, 323 13.3.1 Which Price to Use?, 324 13.3.2 High Frequency Data Preprocessing, 326 13.3.3 How to and How Often to Sample?, 326 13.4 Realized Variance and Covariance, 329 13.4.1 Univariate Volatility Estimators, 329 13.4.1.1 Measurement Error, 330 13.4.2 Multivariate Volatility Estimators, 333 13.4.2.1 Measurement Error, 336 13.5 Modeling and Forecasting, 337 13.5.1 Time Series Models of (co) Volatility, 337 13.5.2 Forecast Comparison, 339 13.6 Asset Pricing, 340 13.6.1 Distribution of Returns Conditional on the Volatility Measure, 340 13.6.2 Application to Factor Pricing Model, 341 13.6.3 Effects of Algorithmic Trading, 342 13.6.4 Application to Option Pricing, 342 13.7 Estimating Continuous Time Models, 344 14J LIKELIHOOD-BASED VOLATILITY ESTIMATORS IN THE PRESENCE OF MARKET MLCROSTRUCTURE NOISE 347 14.1 Introduction, 347 14.2 Volatility Estimation, 349 14.2.1 Constant Volatility and Gaussian Noise Case: MLE, 349 14.2.2 Robustness to Non-Gaussian Noise, 351 14.2.3 Implementing Maximum Likelihood, 351 14.2.4 Robustness to Stochastic Volatility: QMLE, 352 14.2.5 Comparison with Other Estimators, 355 14.2.6 Random Sampling and Non-i.i.d. Noise, 356 14.3 Covariance Estimation, 356
xiv Contents 14.4 Empirical Application: Correlation between Stock and Commodity Futures, 359 14.5 Conclusion, 360 Acknowledgments, 361 FJ[T HAR MODELING FOR REALIZED VOLATILITY FORECASTING 363 15.1 Introduction, 363 15.2 Stylized Facts on Realized Volatility, 365 15.3 Heterogeneity and Volatility Persistence, 366 15.3.1 Genuine Long Memory or Superposition of Factors?, 369 15.4 HAR Extensions, 370 15.4.1 Jump Measures and Their Volatility Impact, 370 15.4.2 Leverage Effects, 372 15A.3 General Nonlinear Effects in Volatility, 373 15.5 Multivariate Models, 375 15.6 Applications, 378 15.7 Conclusion, 381 FORECASTING VOLATILITY WITH MIDAS 383 16.1 Introduction, 383 16.2 MIDAS Regression Models and Volatility Forecasting, 384 16.2.1 MIDAS Regressions, 384 16.2.2 Direct Versus Iterated Volatility Forecasting, 386 16.2.3 Variations on the Theme of MIDAS Regressions, 389 16.2.4 Microstructure Noise and MIDAS Regressions, 390 16.3 Likelihood-Based Methods, 391 16.3.1 Risk-Return Trade-Off, 391 16.3.2 HYBRID GARCH Models, 393 16.3.3 GARCH-MIDAS Models, 398 16.4 Multivariate Models, 399 16.5 Conclusion, 401 JUMPS 17.1 Introduction, 403 17.1.1 Some Models Used in Finance and Our Framework, 403 17.1.2 Simulated Models Used in This Chapter, 407 17A3 Realized Variance and Quadratic Variation, 409 4O3
xv j 17.1.4 Importance of Disentangling, 410 17.1.5 Further Notation, 411 17.2 How to Disentangle: Estimators of Integrated Variance and Integrated Covariance, 411 17.2.1 Bipower Variation, 413 17.2.2 Threshold Estimator, 416 17.2.3 Threshold Bipower Variation, 419 17.2.4 Other Methods, 421 17.2.4.1 Realized Quantile, 421 17.2.4.2 MinRVandMedRV, 422 17.2.4.3 Realised Outlyingness Weighted Variation, 422 17.2.4.4 Range Bipower Variation, 423 17..2.4.5 Generalization of the Realized Range, 424 17.2.4.6 Duration-Based Variation, 425 17.2.4.7 Irregularly Spaced Observations, 425 17.2.5 Comparative Implementation on Simulated Data, 426 17.2.6 Noisy Data, 427 17.2.7 Multivariate Assets, 432 17.3 Testing for the Presence of Jumps, 433 17.3.1 Confidence Intervals, 434 17.3.2 Tests Based on IV B - RV B or on 1 - IV B /RV B, 434 17.3.3 Tests Based on Normalized Returns, 436 17.3.4 PV-Based Tests, 439 17.3.4.1 Remarks, 440 17.3.5 Tests Based on Signature Plots, 441 17.3.6 Tests Based on Observation of Optioii Prices, 442 17.3.6.1 Remarks, 442 17.3.7 Indirect Test for the Presence of Jumps, 443 17.3.7.1 In the Presence of Noise, 443 17.3.8 Comparisons, 443 17.4 Conclusions, 444 Acknowledgments, 445 18] NONPARAMETRIC TESTS FOR INTRADAY JUMPS: IMPACT OF PERIODICITY AND MICROSTRUCTURE NOISE 447 18.1 Introduction, 447 18.2 Model, 449 18.3 Price Jump Detection Method, 450
xvi Contents 18.3.1 Estimation of the Noise Variance, 451 18.3.2 Robust Estimators of the Integrated Variance, 451 18.3.3 Periodicity Estimation, 452 18.3.4 Jump Test Statistics, 454 18.3.5 Critical Value, 454 18.4 Simulation Study, 455 18.4.1 Intraday Differences in the Value of the Test Statistics, 455 18.4.2 Comparison of Size and Power, 457 18.4.3 Simulation Setup, 457 18.4.4 Results, 458! 18.5 Comparison on NYSE Stock Prices, 460 18.6 Conclusion, 462 [jj9j VOLATILITY FORECASTS EVALUATION AND COMPARISON 465 19.1 Introduction, 465 19.2 Notation, 467 19.3 Single Forecast Evaluation, 468 19.4 Loss Functions and the Latent Variable Problem, 471 19.5 Pairwise Comparison, 474 19.6 Multiple Comparison, 477 19.7 Consistency of the Ordering and Inference on Forecast Performances, 481' 19.8 Conclusion, 485 BIBLIOGRAPHY 487 INDEX - 537