GARCH Models

By Christian Francq, Jean-Michel Zakoian. 2010

823 citations recorded by the source corpus.

Preface. 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 Temporal Aggregation and Weak GARCH Models. 4.1 Temporal Aggregation of GARCH Processes. 4.2 Weak GARCH. 4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class. 4.4 Bibliographical Notes. 4.5 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 Generalized 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 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 Misspecified Density. 9.3 Alternative Estimation Methods. 9.4 Bibliographical Notes. 9.5 Exercises. Part III Extensions and Applications. 10 Asymmetries. 10.1 Exponential GARCH Model. 10.2 Threshold GARCH Model. 10.3 Asymmetric Power GARCH Model. 10.4 Other Asymmetric GARCH Models. 10.5 A GARCH Model with Contemporaneous Conditional Asymmetry. 10.6 Empirical Comparisons of Asymmetric GARCH Formulations. 10.7 Bibliographical Notes. 10.8 Exercises. 11 Multivariate GARCH Processes. 11.1 Multivariate Stationary Processes. 11.2 Multivariate GARCH Models. 11.3 Stationarity. 11.4 Estimation of the CCC Model. 11.5 Bibliographical Notes. 11.6 Exercises. 12 Financial Applications. 12.1 Relation between GARCH and Continuous-Time Models. 12.2 Option Pricing. 12.3 Value at Risk and Other Risk Measures. 12.4 Bibliographical Notes. 12.5 Exercises. Part IV Appendices. A Ergodicity, Martingales, Mixing. A.1 Ergodicity. A.2 Martingale Increments. A.3 Mixing. B Autocorrelation and Partial Autocorrelation. B.1 Partial Autocorrelation. B.2 Generalized Bartlett Formula for Nonlinear Processes. C Solutions to the Exercises. D Problems. References. Index.

Subfields: Economics and Econometrics

View Paper: https://www.semanticscholar.org/paper/b9a3d23682a6059f617905fadd8d7973d2b04b30

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Citations in the corpus

Bootstrap prediction intervals for linear, nonlinear, and nonparametric autoregressions (2016, methodology): “ARCH models are typically estimated using quasi-maximum likelihood estimation (QMLE); see e.g. Weiss(1986)[41], Bollerslev and Wooldridge (1992)[5], and Francq and Zakoian (2010) [17].”
Poisson QMLE of Count Time Series Models (2016, methodology): “Noting that equation (9) is similar to that satisfied by the volatility in a GARCH.p; q/ model, it is easy to show (by, for instance, the arguments used to show (7.30) in Francq and Zakoian (2010), J. Time.”
ECONOMETRIC ANALYSIS OF VOLATILITY COMPONENT MODELS (2014, methodology): “The proof uses an argument similar to Francq and Zakoian (2010), p. 159 so the proof is omitted. n Proof of Lemma 4.5.”
Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models (2011, background): “When wt ≡ 1, the θ̂sn is the global QMELE and it needs the moment condition E|εt|(3) <∞ for its asymptotic normality, which is weaker than the moment condition Eε(4)t <∞ as for the QMLE of θ0 in Francq and Zaköıan (2004). It is well known that the higher is the moment condition of εt, the smaller is”
Dynamic Conditional Correlation: On Properties and Estimation (2013, neutral): “As Gaussian score losses (Amisano and Giacomini 2007) we set dt ≡ EWSCOREt ≡ log(w Ĥt|t−1w) + (wŷt|t−1)(2)/(w Ĥt|t−1w) (20) for the DCC forecasts of the EW conditional variance, and dt ≡ CORRSCOREt ≡ log |R̂t|t−1|+ ε̂t|t−1{R̂t|t−1}ε̂t|t−1 (21) for the DCC correlation forecasts, where ε̂t|t−1 ≡ [ε̂1,”
Multivariate Variance Targeting in the BEKK-GARCH Model (2014, neutral): “We prove this lemma arguing in line with the proof of Theorem 3.2 in Francq and Zakoïan (2010), see also p.77-78 in Comte and Lieberman (2003).”
GARCH Modelling of Cryptocurrencies (2017, neutral): “…1) model; Nelson (1990) (Theorem 6, p. 326, 1990) for the SGARCH (1, 1) and IGARCH (1, 1) models; Zakoian (1994) (Proposition 7, 1994) for the TGARCH (1, 1) model; Karanasos and Kim (2003) (Theorem 1, pp. 149–150, 2003) and Francq and Zakoian (2010) (Theorem 10.2, 2010) for the EGARCH (1, 1) model.”