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Notation
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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Preface to the First Edition
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1 Classical Time Series Models and Financial Series
Notation
General notation
‘is defined as’
(or
in Chapter 10)
Sets and spaces
positive integers, integers, rational
numbers, real numbers
positive real line
‐dimensional Euclidean space
complement of the set
half‐closed interval
Matrices
‐dimensional identity matrix
the set of
real matrices
Processes
iid
independent and identically distributed
iid (0,1)
iid centered with unit variance
or
discrete‐time process
GARCH process
conditional variance or volatility
strong white noise with unit variance
kurtosis coefficient of
or
lag operator
or
sigma‐field generated by the past of
Functions
1 if
, 0 otherwise
integer part of
autocovariance and autocorrelation functions of
sample autocovariance and autocorrelation
Probability
Gaussian law with mean
and covariance matrix
chi‐square distribution with
degrees of freedom
quantile of order
of the
distribution
convergence in distribution
a.s.
almost surely
in probability
equals
up to the stochastic order
Estimation
Fisher information matrix
asymptotic variance of the QML
true parameter value
parameter set
element of the parameter set
estimators of
volatility built with the value
as
but with initial values
approximation of
, built with initial values
asymptotic variance and covariance
Some abbreviations
ES
expected shortfall
FGLS
feasible generalized least squares
OLS
ordinary least squares
QML
quasi‐maximum likelihood
RMSE
root mean square error
SACR
sample autocorrelation
SACV
sample autocovariance
SPAC
sample partial autocorrelation
VaR
value at risk
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