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by Jean-Michel Zakoian, Christian Francq
GARCH Models, 2nd Edition
Cover
Preface to the Second Edition
Preface to the First Edition
Notation
1 Classical Time Series Models and Financial Series
1.1 Stationary Processes
1.2 ARMA and ARIMA Models
1.3 Financial Series
1.4 Random Variance Models
1.5 Bibliographical Notes
1.6 Exercises
Part I: Univariate GARCH Models
2 GARCH(p, q) Processes
2.1 Definitions and Representations
2.2 Stationarity Study
2.3 ARCH(∞)Representation
2.4 Properties of the Marginal Distribution
2.5 Autocovariances of the Squares of a GARCH
2.6 Theoretical Predictions
2.7 Bibliographical Notes
2.8 Exercises
3 Mixing*
3.1 Markov Chains with Continuous State Space
3.2 Mixing Properties of GARCH Processes
3.3 Bibliographical Notes
3.4 Exercises
4 Alternative Models for the Conditional Variance
4.1 Stochastic Recurrence Equation (SRE)
4.2 Exponential GARCH Model
4.3 Log‐GARCH Model
4.4 Threshold GARCH Model
4.5 Asymmetric Power GARCH Model
4.6 Other Asymmetric GARCH Models
4.7 A GARCH Model with Contemporaneous Conditional Asymmetry
4.8 Empirical Comparisons of Asymmetric GARCH Formulations
4.9 Models Incorporating External Information
4.10 Models Based on the Score: GAS and Beta‐t‐(E)GARCH
4.11 GARCH‐type Models for Observations Other Than Returns
4.12 Complementary Bibliographical Notes
4.13 Exercises
Part II: Statistical Inference
5 Identification
5.1 Autocorrelation Check for White Noise
5.2 Identifying the ARMA Orders of an ARMA‐GARCH
5.3 Identifying the GARCH Orders of an ARMA‐GARCH Model
5.4 Lagrange Multiplier Test for Conditional Homoscedasticity
5.5 Application to Real Series
5.6 Bibliographical Notes
5.7 Exercises
6 Estimating ARCH Models by Least Squares
6.1 Estimation of ARCH( q ) models by Ordinary Least Squares
6.2 Estimation of ARCH( q ) Models by Feasible Generalised Least Squares
6.3 Estimation by Constrained Ordinary Least Squares
6.4 Bibliographical Notes
6.5 Exercises
7 Estimating GARCH Models by Quasi‐Maximum Likelihood
7.1 Conditional Quasi‐Likelihood
7.2 Estimation of ARMA–GARCH Models by Quasi‐Maximum Likelihood
7.3 Application to Real Data
7.4 Proofs of the Asymptotic Results*
7.5 Bibliographical Notes
7.6 Exercises
8 Tests Based on the Likelihood
8.1 Test of the Second‐Order Stationarity Assumption
8.2 Asymptotic Distribution of the QML When θ 0 is at the Boundary
8.3 Significance of the GARCH Coefficients
8.4 Diagnostic Checking with Portmanteau Tests
8.5 Application: Is the GARCH(1,1) Model Overrepresented?
8.6 Proofs of the Main Results
8.7 Bibliographical Notes
8.8 Exercises
9 Optimal Inference and Alternatives to the QMLE*
9.1 Maximum Likelihood Estimator
9.2 Maximum Likelihood Estimator with Mis‐specified Density
9.3 Alternative Estimation Methods
9.4 Bibliographical Notes
9.5 Exercises
Part III: Extensions and Applications
10 Multivariate GARCH Processes
10.1 Multivariate Stationary Processes
10.2 Multivariate GARCH Models
10.3 Stationarity
10.4 QML Estimation of General MGARCH
10.5 Estimation of the CCC Model
10.6 Looking for Numerically Feasible Estimation Methods
10.7 Proofs of the Asymptotic Results
10.8 Bibliographical Notes
10.9 Exercises
11 Financial Applications
11.1 Relation Between GARCH and Continuous‐Time Models
11.2 Option Pricing
11.3 Value at Risk and Other Risk Measures
11.4 Bibliographical Notes
11.5 Exercises
12 Parameter‐Driven Volatility Models
12.1 Stochastic Volatility Models
12.2 Markov Switching Volatility Models
12.3 Bibliographical Notes
12.4 Exercises
Appendix B: Ergodicity, Martingales, Mixing
A.1. Ergodicity
A.2. Martingale Increments
A.3 Mixing
Appendix B: Autocorrelation and Partial Autocorrelation
B.1. Partial Autocorrelation
B.2. Generalised Bartlett Formula for Non‐linear Processes
Appendix C: Markov Chains on Countable State Spaces
C.1. Definition of a Markov Chain
C.2. Transition Probabilities
C.3. Classification of States
C.4. Invariant Probability and Stationarity
C.5. Ergodic Results
C.6. Limit Distributions
C.7. Examples
Appendix D: The Kalman Filter
D.1. General Form of the Kalman Filter
D.2. Prediction and Smoothing with the Kalman Filter
D.3. Kalman Filter in the Stationary Case
D.4. Statistical Inference with the Kalman Filter
Appendix E: Solutions to the Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
References
Index
End User License Agreement
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Prev
Previous Chapter
Cover
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Title Page
Table of Contents
Cover
Preface to the Second Edition
Preface to the First Edition
Notation
1 Classical Time Series Models and Financial Series
1.1 Stationary Processes
1.2 ARMA and ARIMA Models
1.3 Financial Series
1.4 Random Variance Models
1.5 Bibliographical Notes
1.6 Exercises
Part I: Univariate GARCH Models
2 GARCH(
p
,
q
) Processes
2.1 Definitions and Representations
2.2 Stationarity Study
2.3 ARCH(∞)Representation
2.4 Properties of the Marginal Distribution
2.5 Autocovariances of the Squares of a GARCH
2.6 Theoretical Predictions
2.7 Bibliographical Notes
2.8 Exercises
3 Mixing*
3.1 Markov Chains with Continuous State Space
3.2 Mixing Properties of GARCH Processes
3.3 Bibliographical Notes
3.4 Exercises
4 Alternative Models for the Conditional Variance
4.1 Stochastic Recurrence Equation (SRE)
4.2 Exponential GARCH Model
4.3 Log‐GARCH Model
4.4 Threshold GARCH Model
4.5 Asymmetric Power GARCH Model
4.6 Other Asymmetric GARCH Models
4.7 A GARCH Model with Contemporaneous Conditional Asymmetry
4.8 Empirical Comparisons of Asymmetric GARCH Formulations
4.9 Models Incorporating External Information
4.10 Models Based on the Score: GAS and Beta‐t‐(E)GARCH
4.11 GARCH‐type Models for Observations Other Than Returns
4.12 Complementary Bibliographical Notes
4.13 Exercises
Part II: Statistical Inference
5 Identification
5.1 Autocorrelation Check for White Noise
5.2 Identifying the ARMA Orders of an ARMA‐GARCH
5.3 Identifying the GARCH Orders of an ARMA‐GARCH Model
5.4 Lagrange Multiplier Test for Conditional Homoscedasticity
5.5 Application to Real Series
5.6 Bibliographical Notes
5.7 Exercises
6 Estimating ARCH Models by Least Squares
6.1 Estimation of ARCH( q ) models by Ordinary Least Squares
6.2 Estimation of ARCH( q ) Models by Feasible Generalised Least Squares
6.3 Estimation by Constrained Ordinary Least Squares
6.4 Bibliographical Notes
6.5 Exercises
7 Estimating GARCH Models by Quasi‐Maximum Likelihood
7.1 Conditional Quasi‐Likelihood
7.2 Estimation of ARMA–GARCH Models by Quasi‐Maximum Likelihood
7.3 Application to Real Data
7.4 Proofs of the Asymptotic Results*
7.5 Bibliographical Notes
7.6 Exercises
8 Tests Based on the Likelihood
8.1 Test of the Second‐Order Stationarity Assumption
8.2 Asymptotic Distribution of the QML When θ 0 is at the Boundary
8.3 Significance of the GARCH Coefficients
8.4 Diagnostic Checking with Portmanteau Tests
8.5 Application: Is the GARCH(1,1) Model Overrepresented?
8.6 Proofs of the Main Results
8.7 Bibliographical Notes
8.8 Exercises
9 Optimal Inference and Alternatives to the QMLE*
9.1 Maximum Likelihood Estimator
9.2 Maximum Likelihood Estimator with Mis‐specified Density
9.3 Alternative Estimation Methods
9.4 Bibliographical Notes
9.5 Exercises
Part III: Extensions and Applications
10 Multivariate GARCH Processes
10.1 Multivariate Stationary Processes
10.2 Multivariate GARCH Models
10.3 Stationarity
10.4 QML Estimation of General MGARCH
10.5 Estimation of the CCC Model
10.6 Looking for Numerically Feasible Estimation Methods
10.7 Proofs of the Asymptotic Results
10.8 Bibliographical Notes
10.9 Exercises
11 Financial Applications
11.1 Relation Between GARCH and Continuous‐Time Models
11.2 Option Pricing
11.3 Value at Risk and Other Risk Measures
11.4 Bibliographical Notes
11.5 Exercises
12 Parameter‐Driven Volatility Models
12.1 Stochastic Volatility Models
12.2 Markov Switching Volatility Models
12.3 Bibliographical Notes
12.4 Exercises
Appendix B: Ergodicity, Martingales, Mixing
A.1. Ergodicity
A.2. Martingale Increments
A.3 Mixing
Appendix B: Autocorrelation and Partial Autocorrelation
B.1. Partial Autocorrelation
B.2. Generalised Bartlett Formula for Non‐linear Processes
Appendix C: Markov Chains on Countable State Spaces
C.1. Definition of a Markov Chain
C.2. Transition Probabilities
C.3. Classification of States
C.4. Invariant Probability and Stationarity
C.5. Ergodic Results
C.6. Limit Distributions
C.7. Examples
Appendix D: The Kalman Filter
D.1. General Form of the Kalman Filter
D.2. Prediction and Smoothing with the Kalman Filter
D.3. Kalman Filter in the Stationary Case
D.4. Statistical Inference with the Kalman Filter
Appendix E: Solutions to the Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
References
Index
End User License Agreement
List of Tables
Chapter 1
Sample autocorrelations of returns
ε
t
(CAC 40 index, 2 Janua...
Chapter 2
Estimations of
γ
obtained from 1000 simulations of size 1000 ...
Chapter 4
Table 4.1 Empirical autocorrelations (CAC 40 series, period 1988–1998)....
Table 4.2 Empirical autocorrelations (CAC 40, for the period 1988–1998)...
Table 4.3 Portmanteau test of the white noise hypothesis for the CAC 40...
Table 4.4 Likelihoods of the different models for the CAC 40 series.
Table 4.5 Means of the squared differences between the estimated volati...
Table 4.6 Variance (
×10
4
) and kurtosis of the CAC 40 index and of...
Table 4.7 Number of CAC returns outside the limits
(THEO being the ...
Table 4.8 Means of the squares of the differences between the estimated...
Table 4.9 SAS program for the fitting of a TGARCH
(1, 1)
model with inte...
Chapter 5
Portmanteau tests on a simulation of size
n
= 5000
of...
As Table 5.1, for tests based on partial autocorrelations instead o...
White noise portmanteau tests on a simulation of size
n = 100
...
Portmanteau tests on the squared CAC 40 returns (2 March 1990 to 29 ...
LM tests for conditional homoscedasticity of the CAC 40 and FTSE 100...
Portmanteau tests on the CAC 40 (2 March 1990 to 29 December 2006). ...
Portmanteau tests on the FTSE 100 (3 April 1984 to 3 April 2007).
Studentised statistics for the corner method for the CAC 40 series a...
Studentised statistics for the corner method for the FTSE 100 series...
Studentised statistics for the corner method for the squared CAC 40 ...
Studentised statistics for the corner method for the squared FTSE 10...
Chapter 6
Table 6.1 Strict stationarity and moment conditions for the ARCH
(1)
mod...
Table 6.2 Asymptotic variance of the OLS estimator of an ARCH
(1)
model ...
Chapter 7
Asymptotic variance for the QMLE of an ARCH
(1)
process with
η t ∼풩
...
Comparison of the empirical and theoretical asymptotic variances, fo...
Matrices
Σ
of asymptotic variance of the estimator of
(a 0, α 0
...
GARCH
(1, 1)
models estimated by QML for 11 indices.
Chapter 8
Test of the infinite variance assumption for 11 stock market returns...
Asymptotic critical value
c
q
,
α
, at level
α
, of the...
Exact asymptotic level (%) of erroneous Wald tests, of rejection reg...
Portmanteau test
p
‐values for adequacy of the ARCH(5) and GARCH(1,1)...
p
‐values for tests of the null of a GARCH(1,1) model against the GA...
Chapter 9
Table 9.1 Asymptotic relative efficiency (ARE) of the MLE with respect ...
Table 9.2 QMLE and efficient estimator
, on
N
= 1000
...
Table 9.3 Identifiability constraint under which
is consistent.
Table 9.4 Choice of
h
as function of the prediction problem.
Table 9.5 Asymptotic relative efficiency of the MLE with respect to the...
Chapter 10
Table 10.1 Seconds of CPU time for computing the VTE and QMLE (average ...
Table 10.2 Computation time (CPU time in seconds) and relative efficien...
Chapter 11
Table 11.1 Comparison of the four VaR estimation methods for the CAC 40...
5
Number of parameters as a function of
m
.
List of Illustrations
Chapter 1
Figure 1.1 CAC 40 index for the period from 1 March 1990 to 15 October ...
Figure 1.2 CAC 40 returns (2 March 1990 to 15 October 2008). 19 August ...
Figure 1.3 Returns of the CAC 40 (2 January 2008 to 15 October 2008).
Figure 1.4 Sample autocorrelations of (a) returns and (b) squared retur...
Figure 1.5 Kernel estimator of the CAC 40 returns density (solid line) ...
Figure 1.6 Sample autocorrelations
(
h
= 1, …, 36
Chapter 2
Figure 2.1 Simulation of size 500 of the ARCH
(1)
process with
ω = 1
...
Figure 2.2Figure 2.2 Simulation of size 500 of the ARCH
(1)
process with ...
Figure 2.3 Simulation of size 500 of the ARCH
(1)
process with
ω = 1
...
Figure 2.4 Simulation of size 200 of the ARCH
(1)
process with
ω = 1
...
Figure 2.5 Observations 100–140 of Figure 2.4.
Figure 2.6 Simulation of size 500 of the GARCH
(1, 1)
process with
ω =
...
Figure 2.7 Simulation of size 500 of the GARCH
(1, 1)
process with
ω =
...
Figure 2.8 Stationarity regions for the GARCH
(1, 1)
model when
: 1, ...
Figure 2.9 Stationarity regions for the ARCH(2) model: 1, second‐order ...
Figure 2.10 Regions of moments existence for the GARCH
(1, 1)
model: 1, ...
Figure 2.11 Autocorrelation function (a) and partial autocorrelation fu...
Figure 2.2 Autocorrelation function (a) and partial autocorrelation fun...
Figure 2.3 Prediction intervals at horizon 1, at 95
%
, for the strong
풩
...
Figure 2.4 Prediction intervals at horizon 1, at 95%, for the GARCH
(1,
...
Figure 2.15Figure 2.15 Prediction intervals at horizon 1, at 95%, for the...
Figure 2.6 Prediction intervals at horizon 1, at 95%, for the GARCH
(1,
...
Chapter 4
Figure 4.1 Volatility
(in full line) and volatility estimates
(...
Figure 4.2 Theoretical autocorrelation function of the squares of a Log...
Figure 4.3 The set of the Extended Log‐GARCH volatilities contains the ...
Figure 4.4 News impact curves for the ARCH
(1)
model,
(dashed line),...
Figure 4.5 Stationarity regions for the TARCH
(1)
model with
: 1, sec...
Figure 4.6 Stationarity regions for the APARCH(1,0) model with
: 1, ...
Figure 4.7 The first 500 values of the CAC 40 index (a) and of the squa...
Figure 4.8 Correlograms of the CAC 40 index (a) and the squared index (...
Figure 4.9 Correlogram
of the absolute CAC 40 returns (a) and cross...
Figure 4.10 From left to right and top to bottom, graph of the first 50...
Figure 4.11 Returns
r
t
of the CAC 40 index (solid lines) and confi...
Figure 4.12 Comparison of the estimated volatilities of the EGARCH and ...
Figure 4.13 Correlogram
h ↦ ρ(∣r t ∣ , ∣ r t − h ∣)
...
Chapter 5
Figure 5.1 SACR of exchange rates against the euro, standard significan...
Figure 5.2 SACRs of a simulation of a strong white noise (a) and of the...
Figure 5.3 Sample autocorrelations of a simulation of size
n = 5000
...
Figure 5.4 Sample partial autocorrelations of a simulation of size
n =
...
Figure 5.5 Autocorrelations (a) and partial autocorrelations (b) for mo...
Figure 5.6 SACRs (a) and SPACs (b) of a simulation of size
n = 1000
...
Figure 5.7 Correlograms of returns and squared returns of the CAC 40 in...
Chapter 7
Figure 7.1 GARCH
(1, 1)
: zones of strict and second‐order stationa...
Figure 7.2 Box‐plots of the QML estimation errors for the parameters
ω
...
Chapter 8
Figure 8.1 ARCH
(1)
model with
θ
0
= (
ω
0
, 0)
a...
Figure 8.2 Concentrated log‐likelihood (solid line)
for an ARCH
(1)
...
Figure 8.3 Comparison between a kernel density estimator of the Wald st...
Figure 8.4 Comparison of the observed powers of the Wald test (thick li...
Figure 8.5 Local asymptotic power of the Wald test (solid line) and of ...
Chapter 9
Figure 9.1 Density ( 9.9) for different values of
a > 0
...
Figure 9.2 Local asymptotic power of the optimal Wald test
(solid l...
Chapter 11
Figure 11.1 (a) VaR is the
(1 −
α
)
‐quantile of t...
Figure 11.2 Effective losses of the CAC 40 (solid lines) and estimated ...
Chapter 12
Figure 12.1 Simulation of length 100 of a three‐regime HMM. The full li...
Figure 12.2 CAC 40 and SP 500 from March 1, 1990 to December 29, 2006, ...
Guide
Cover
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