GARCH Models: Structure, Statistical Inference and Financial Applications

✍ Scribed by Christian Francq, Jean-Michel Zakoian

• Publisher: Wiley
• Year: 2010
• Tongue: English
• Leaves: 505
• Category: Library

✦ Synopsis

This book provides a complete coverage to GARCH modeling, including probability properties, identifying an appropriate model, estimation and testing, multivariate extensions including EGARCH, TGARCH and APGARCH, volatility features such as asymmetries and financial applications.

Many sections are based on up to date research, featured in econometric and statistic journals. GARCH models is accessible to a wide audience who have worked in time series analysis and wish to become familiar with the use and modeling techniques specially devoted to financial time series.

✦ Table of Contents

GARCH Models......Page 5
Contents......Page 7
Preface......Page 13
Notation......Page 15
1.1 Stationary Processes......Page 17
1.2 ARMA and ARIMA Models......Page 19
1.3 Financial Series......Page 23
1.4 Random Variance Models......Page 26
1.6 Exercises......Page 28
Part I Univariate GARCH Models......Page 33
2.1 Definitions and Representations......Page 35
2.2.1 The GARCH(1, 1) Case......Page 40
2.2.2 The General Case......Page 44
2.3.1 Existence Conditions......Page 55
2.3.2 ARCH (∞) Representation of a GARCH......Page 58
2.3.3 Long-Memory ARCH......Page 59
2.4.1 Even-Order Moments......Page 61
2.4.2 Kurtosis......Page 64
2.5.1 Positivity of the Autocovariances......Page 66
2.5.2 The Autocovariances Do Not Always Decrease......Page 67
2.5.3 Explicit Computation of the Autocovariances of the Squares......Page 68
2.6 Theoretical Predictions......Page 69
2.7 Bibliographical Notes......Page 73
2.8 Exercises......Page 74
3.1 Markov Chains with Continuous State Space......Page 79
3.2 Mixing Properties of GARCH Processes......Page 84
3.4 Exercises......Page 92
4.1 Temporal Aggregation of GARCH Processes......Page 95
4.1.1 Nontemporal Aggregation of Strong Models......Page 96
4.1.2 Nonaggregation in the Class of Semi-Strong GARCH Processes......Page 97
4.2 Weak GARCH......Page 98
4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class......Page 101
4.4 Bibliographical Notes......Page 104
4.5 Exercises......Page 105
Part II Statistical Inference......Page 107
5.1 Autocorrelation Check for White Noise......Page 109
5.1.1 Behavior of the Sample Autocorrelations of a GARCH Process......Page 110
5.1.3 Sample Partial Autocorrelations of a GARCH......Page 113
5.1.4 Numerical Illustrations......Page 114
5.2 Identifying the ARMA Orders of an ARMA-GARCH......Page 116
5.2.1 Sample Autocorrelations of an ARMA-GARCH......Page 117
5.2.2 Sample Autocorrelations of an ARMA-GARCH Process When the Noise is Not Symmetrically Distributed......Page 120
5.2.3 Identifying the Orders (P,Q)......Page 122
5.3 Identifying the GARCH Orders of an ARMA-GARCH Model......Page 124
5.3.2 Applications......Page 125
5.4.1 General Form of the LM Test......Page 127
5.4.2 LM Test for Conditional Homoscedasticity......Page 131
5.5 Application to Real Series......Page 133
5.6 Bibliographical Notes......Page 136
5.7 Exercises......Page 138
6.1 Estimation of ARCH(q) models by Ordinary Least Squares......Page 143
6.2 Estimation of ARCH(q) Models by Feasible Generalized Least Squares......Page 148
6.3.1 Properties of the Constrained OLS Estimator......Page 151
6.3.2 Computation of the Constrained OLS Estimator......Page 153
6.5 Exercises......Page 154
7.1 Conditional Quasi-Likelihood......Page 157
7.1.1 Asymptotic Properties of the QMLE......Page 159
7.1.2 The ARCH(1) Case: Numerical Evaluation of the Asymptotic Variance......Page 163
7.1.3 The Nonstationary ARCH(1)......Page 164
7.2 Estimation of ARMA-GARCH Models by Quasi-Maximum Likelihood......Page 166
7.3 Application to Real Data......Page 171
7.4 Proofs of the Asymptotic Results*......Page 172
7.6 Exercises......Page 196
8 Tests Based on the Likelihood......Page 201
8.1 Test of the Second-Order Stationarity Assumption......Page 202
8.2 Asymptotic Distribution of the QML When ¥è0 is at the Boundary......Page 203
8.2.1 Computation of the Asymptotic Distribution......Page 207
8.3.1 Tests and Rejection Regions......Page 210
8.3.2 Modification of the Standard Tests......Page 212
8.3.3 Test for the Nullity of One Coefficient......Page 213
8.3.4 Conditional Homoscedasticity Tests with ARCH Models......Page 215
8.3.5 Asymptotic Comparison of the Tests......Page 217
8.5 Application: Is the GARCH(1,1) Model Overrepresented?......Page 220
8.6 Proofs of the Main Results......Page 223
8.8 Exercises......Page 231
9.1 Maximum Likelihood Estimator......Page 235
9.1.1 Asymptotic Behavior......Page 236
9.1.2 One-Step Efficient Estimator......Page 238
9.1.3 Semiparametric Models and Adaptive Estimators......Page 239
9.1.4 Local Asymptotic Normality......Page 242
9.2.1 Condition for the Convergence of [omitted]......Page 247
9.2.2 Reparameterization Implying the Convergence of [omitted]......Page 248
9.2.3 Choice of Instrumental Density h......Page 249
9.2.4 Asymptotic Distribution of [omitted]......Page 250
9.3.1 Weighted LSE for the ARMA Parameters......Page 252
9.3.3 Lp Estimators......Page 253
9.3.5 Whittle Estimator......Page 254
9.5 Exercises......Page 255
Part III Extensions and Applications......Page 259
10 Asymmetries......Page 261
10.1 Exponential GARCH Model......Page 262
10.2 Threshold GARCH Model......Page 266
10.3 Asymmetric Power GARCH Model......Page 272
10.4 Other Asymmetric GARCH Models......Page 274
10.5 A GARCH Model with Contemporaneous Conditional Asymmetry......Page 275
10.6 Empirical Comparisons of Asymmetric GARCH Formulations......Page 277
10.7 Bibliographical Notes......Page 285
10.8 Exercises......Page 286
11.1 Multivariate Stationary Processes......Page 289
11.2 Multivariate GARCH Models......Page 291
11.2.1 Diagonal Model......Page 292
11.2.2 Vector GARCH Model......Page 293
11.2.3 Constant Conditional Correlations Models......Page 295
11.2.5 BEKK-GARCH Model......Page 297
11.2.6 Factor GARCH Models......Page 300
11.3.1 Stationarity of VEC and BEKK Models......Page 302
11.3.2 Stationarity of the CCC Model......Page 305
11.4 Estimation of the CCC Model......Page 307
11.4.1 Identifiability Conditions......Page 308
11.4.2 Asymptotic Properties of the QMLE of the CCC-GARCH model......Page 310
11.4.3 Proof of the Consistency and the Asymptotic Normality of the QML......Page 312
11.5 Bibliographical Notes......Page 323
11.6 Exercises......Page 324
12.1.1 Some Properties of Stochastic Differential Equations......Page 327
12.1.2 Convergence of Markov Chains to Diffusions......Page 329
12.2.2 The Black–Scholes Approach......Page 335
12.2.4 Option Pricing when the Underlying Process is a GARCH......Page 337
12.3.1 Value at Risk......Page 343
12.3.2 Other Risk Measures......Page 347
12.3.3 Estimation Methods......Page 350
12.4 Bibliographical Notes......Page 353
12.5 Exercises......Page 354
Part IV Appendices......Page 357
A.1 Ergodicity......Page 359
A.2 Martingale Increments......Page 360
A.3 Mixing......Page 363
A.3.1 α-Mixing and β-Mixing Coefficients......Page 364
A.3.2 Covariance Inequality......Page 365
A.3.3 Central Limit Theorem......Page 368
B.1 Partial Autocorrelation......Page 369
B.2 Generalized Bartlett Formula for Nonlinear Processes......Page 375
C Solutions to the Exercises......Page 381
D Problems......Page 455
References......Page 489
Index......Page 503