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GARCH Models

Author Jean-Michel Zakoian , Christian Francq
Release Date: 2010/08/01
ISBN: 9780470683910
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Book Description

This book provides a comprehensive and systematic approach to understanding GARCH time series models and their applications whilst presenting the most advanced results concerning the theory and practical aspects of GARCH. The probability structure of standard GARCH models is studied in detail as well as statistical inference such as identification, estimation and tests. The book also provides coverage of several extensions such as asymmetric and multivariate models and looks at financial applications.

Key features:

  • Provides up-to-date coverage of the current research in the probability, statistics and econometric theory of GARCH models.
  • Numerous illustrations and applications to real financial series are provided.
  • Supporting website featuring R codes, Fortran programs and data sets.
  • Presents a large collection of problems and exercises.

This authoritative, state-of-the-art reference is ideal for graduate students, researchers and practitioners in business and finance seeking to broaden their skills of understanding of econometric time series models.

Table of Contents

  1. Cover
  2. Title page
  3. Copyright page
  4. Preface
  5. Notation
  6. 1: Classical Time Series Models and Financial Series
    1. 1.1 Stationary Processes
    2. 1.2 ARMA and ARIMA Models
    3. 1.3 Financial Series
    4. 1.4 Random Variance Models
    5. 1.5 Bibliographical Notes
    6. 1.6 Exercises
  7. Part I: Univariate GARCH Models
    1. 2: GARCH(p, q) Processes
      1. 2.1 Definitions and Representations
      2. 2.2 Stationarity Study
        1. 2.2.1 The GARCH (1, 1) Case
        2. 2.2.2 The General Case
      3. 2.3 ARCH (∞) Representation*
        1. 2.3.1 Existence Conditions
        2. 2.3.2 ARCH (∞) Representation of a GARCH
        3. 2.3.3 Long-Memory ARCH
      4. 2.4 Properties of the Marginal Distribution
        1. 2.4.1 Even-Order Moments
        2. 2.4.2 Kurtosis
      5. 2.5 Autocovariances of the Squares of a GARCH
        1. 2.5.1 Positivity of the Autocovariances
        2. 2.5.2 The Autocovariances Do Not Always Decrease
        3. 2.5.3 Explicit Computation of the Autocovariances of the Squares
      6. 2.6 Theoretical Predictions
      7. 2.7 Bibliographical Notes
      8. 2.8 Exercises
    2. 3: Mixing*
      1. 3.1 Markov Chains with Continuous State Space
      2. 3.2 Mixing Properties of GARCH Processes
      3. 3.3 Bibliographical Notes
      4. 3.4 Exercises
    3. 4: Temporal Aggregation and Weak GARCH Models
      1. 4.1 Temporal Aggregation of GARCH Processes
        1. 4.1.1 Nontemporal Aggregation of Strong Models
        2. 4.1.2 Nonaggregation in the Class of Semi-Strong GARCH Processes
      2. 4.2 Weak GARCH
      3. 4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class
      4. 4.4 Bibliographical Notes
      5. 4.5 Exercises
  8. Part II: Statistical Inference
    1. 5: Identification
      1. 5.1 Autocorrelation Check for White Noise
        1. 5.1.1 Behavior of the Sample Autocorrelations of a GARCH Process
        2. 5.1.2 Portmanteau Tests
        3. 5.1.3 Sample Partial Autocorrelations of a GARCH
        4. 5.1.4 Numerical Illustrations
      2. 5.2 Identifying the ARMA Orders of an ARMA-GARCH
        1. 5.2.1 Sample Autocorrelations of an ARMA-GARCH
        2. 5.2.2 Sample Autocorrelations of an ARMA-GARCH Process When the Noise is Not Symmetrically Distributed
        3. 5.2.3 Identifying the Orders (P,Q) 106
      3. 5.3 Identifying the GARCH Orders of an ARMA-GARCH Model
        1. 5.3.1 Corner Method in the GARCH Case
        2. 5.3.2 Applications
      4. 5.4 Lagrange Multiplier Test for Conditional Homoscedasticity
        1. 5.4.1 General Form of the LM Test
        2. 5.4.2 LM Test for Conditional Homoscedasticity
      5. 5.5 Application to Real Series
      6. 5.6 Bibliographical Notes
      7. 5.7 Exercises
    2. 6: Estimating ARCH Models by Least Squares
      1. 6.1 Estimation of ARCH(q) models by Ordinary Least Squares
      2. 6.2 Estimation of ARCH(q) Models by Feasible Generalized Least Squares
      3. 6.3 Estimation by Constrained Ordinary Least Squares
        1. 6.3.1 Properties of the Constrained OLS Estimator
        2. 6.3.2 Computation of the Constrained OLS Estimator
      4. 6.4 Bibliographical Notes
      5. 6.5 Exercises
    3. 7: Estimating GARCH Models by Quasi-Maximum Likelihood
      1. 7.1 Conditional Quasi-Likelihood
        1. 7.1.1 Asymptotic Properties of the QMLE
        2. 7.1.2 The ARCH (1) Case: Numerical Evaluation of the Asymptotic Variance
        3. 7.1.3 The Nonstationary ARCH (1)
      2. 7.2 Estimation of ARMA-GARCH Models by Quasi-Maximum Likelihood
      3. 7.3 Application to Real Data
      4. 7.4 Proofs of the Asymptotic Results*
      5. 7.5 Bibliographical Notes
      6. 7.6 Exercises
    4. 8: Tests Based on the Likelihood
      1. 8.1 Test of the Second-Order Stationarity Assumption
      2. 8.2 Asymptotic Distribution of the QML When θ0 is at the Boundary
        1. 8.2.1 Computation of the Asymptotic Distribution
      3. 8.3 Significance of the GARCH Coefficients
        1. 8.3.1 Tests and Rejection Regions
        2. 8.3.2 Modification of the Standard Tests
        3. 8.3.3 Test for the Nullity of One Coefficient
        4. 8.3.4 Conditional Homoscedasticity Tests with ARCH Models
        5. 8.3.5 Asymptotic Comparison of the Tests
      4. 8.4 Diagnostic Checking with Portmanteau Tests
      5. 8.5 Application: Is the GARCH(1,1) Model Overrepresented?
      6. 8.6 Proofs of the Main Results*
      7. 8.7 Bibliographical Notes
      8. 8.8 Exercises
    5. 9: Optimal Inference and Alternatives to the QMLE*
      1. 9.1 Maximum Likelihood Estimator
        1. 9.1.1 Asymptotic Behavior
        2. 9.1.2 One-Step Efficient Estimator
        3. 9.1.3 Semiparametric Models and Adaptive Estimators
        4. 9.1.4 Local Asymptotic Normality
      2. 9.2 Maximum Likelihood Estimator with Misspecified Density
        1. 9.2.1 Condition for the Convergence of θ̂ n,h to θ0
        2. 9.2.2 Reparameterization Implying the Convergence of θ ̂n,h to θ0
        3. 9.2.3 Choice of Instrumental Density h 233
        4. 9.2.4 Asymptotic Distribution of θ ̂n,h234
      3. 9.3 Alternative Estimation Methods
        1. 9.3.1 Weighted LSE for the ARMA Parameters
        2. 9.3.2 Self-Weighted QMLE
        3. 9.3.3 Lp Estimators
        4. 9.3.4 Least Absolute Value Estimation
        5. 9.3.5 Whittle Estimator
      4. 9.4 Bibliographical Notes
      5. 9.5 Exercises
  9. Part III: Extensions and Applications
    1. 10: Asymmetries
      1. 10.1 Exponential GARCH Model
      2. 10.2 Threshold GARCH Model
      3. 10.3 Asymmetric Power GARCH Model
      4. 10.4 Other Asymmetric GARCH Models
      5. 10.5 A GARCH Model with Contemporaneous Conditional Asymmetry
      6. 10.6 Empirical Comparisons of Asymmetric GARCH Formulations
      7. 10.7 Bibliographical Notes
      8. 10.8 Exercises
    2. 11: Multivariate GARCH Processes
      1. 11.1 Multivariate Stationary Processes
      2. 11.2 Multivariate GARCH Models
        1. 11.2.1 Diagonal Model
        2. 11.2.2 Vector GARCH Model
        3. 11.2.3 Constant Conditional Correlations Models
        4. 11.2.4 Dynamic Conditional Correlations Models
        5. 11.2.5 BEKK-GARCH Model
        6. 11.2.6 Factor GARCH Models
      3. 11.3 Stationarity
        1. 11.3.1 Stationarity of VEC and BEKK Models
        2. 11.3.2 Stationarity of the CCC Model
      4. 11.4 Estimation of the CCC Model
        1. 11.4.1 Identifiability Conditions
        2. 11.4.2 Asymptotic Properties of the QMLE of the CCC-GARCH model
        3. 11.4.3 Proof of the Consistency and the Asymptotic Normality of the QML
      5. 11.5 Bibliographical Notes
      6. 11.6 Exercises
    3. 12: Financial Applications
      1. 12.1 Relation between GARCH and Continuous-Time Models
        1. 12.1.1 Some Properties of Stochastic Differential Equations
        2. 12.1.2 Convergence of Markov Chains to Diffusions
      2. 12.2 Option Pricing
        1. 12.2.1 Derivatives and Options
        2. 12.2.2 The Black–Scholes Approach
        3. 12.2.3 Historic Volatility and Implied Volatilities
        4. 12.2.4 Option Pricing when the Underlying Process is a GARCH
      3. 12.3 Value at Risk and Other Risk Measures
        1. 12.3.1 Value at Risk
        2. 12.3.2 Other Risk Measures
        3. 12.3.3 Estimation Methods
      4. 12.4 Bibliographical Notes
      5. 12.5 Exercises
  10. Part IV: Appendices
    1. A: Ergodicity, Martingales, Mixing
      1. A.1 Ergodicity
      2. A.2 Martingale Increments
      3. A.3 Mixing
        1. A.3.1 α-Mixing and β-Mixing Coefficients
        2. A.3.2 Covariance Inequality
        3. A.3.3 Central Limit Theorem
    2. B: Autocorrelation and Partial Autocorrelation
      1. B.1 Partial Autocorrelation
      2. B.2 Generalized Bartlett Formula for Nonlinear Processes
    3. C: Solutions to the Exercises
    4. D: Problems
  11. References
  12. Index
3.20.224.107