Jean-Michel Zakoïan

GARCH models : structure, statistical inference, and financial applications

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.

Stationarity of multivariate Markov–switching ARMA models

In this article we consider multivariate ARMA models subject to Markov switching. In these models, the parameters are allowed to depend on the state of an unobserved Markov chain. A natural idea when estimating these models is to impose local stationarity conditions, i.e. stationarity within each regime. In this article we show that the local stationarity of the observed process is neither sufficient nor necessary to obtain the global stationarity. We derive stationarity conditions and we compute the autocovariance function of this nonlinear process. Interestingly, it turns out that the autocovariance structure coincides with that of a standard ARMA. Some examples are proposed to illustrate the stationarity conditions. Using Monte Carlo simulations we investigate the consequences of accounting for the stationarity conditions in statistical inference.

Diagnostic Checking in ARMA Models With Uncorrelated Errors

We consider tests for lack of fit in ARMA models with nonindependent innovations. In this framework, the standard Box–Pierce and Ljung–Box portmanteau tests can perform poorly. Specifically, the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration. In this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of ARMA models under weak assumptions on the noise. The asymptotic distribution of the portmanteau statistics follows. A consistent estimator of , and a modification of the portmanteau tests are proposed. This allows us to construct valid asymptotic significance limits for the residual autocorrelations, and (asymptotically) valid goodness-of-fit tests, when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference. A set of Monte Carlo experiments, and an application to the Standard & Poor 500 returns, illustrate the practical relevance of our theoretical results.

Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1, 1) models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1, 1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.

QML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS

We establish the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the parameters of a class of multivariate asymmetric generalized autoregressive conditionally heteroskedastic processes, allowing for cross leverage effects. The conditions required to establish the asymptotic properties of the QMLE are mild and coincide with the minimal ones in the univariate case. In particular, no moment assumption is made on the observed process. Instead, we require strict stationarity, for which a necessary and sufficient condition is established. The asymptotic results are illustrated by Monte Carlo experiments, and an application to a bivariate exchange rates series is proposed.

GARCH models without positivity constraints: Exponential or log GARCH?

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH model, which are already known, with those of an asymmetric version of the log-GARCH. The quasi-maximum likelihood estimation of the log-GARCH parameters is shown to be strongly consistent and asymptotically normal. Similar estimation results are only available for the EGARCH (1,1) model, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.

Bartlett's formula for a general class of nonlinear processes

A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartletts formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. An application to GARCH models is proposed.

Deriving the autocovariances of powers of Markov-switching GARCH models, with applications to statistical inference

A procedure is proposed for computing the autocovariances and the ARMA representations of the squares, and higher-order powers, of Markov-switching GARCH models. It is shown that many interesting subclasses of the general model can be discriminated in view of their autocovariance structures. Explicit derivation of the autocovariances allows for parameter estimation in the general model, via a GMM procedure. It can also be used to determine how many ARMA representations are needed to identify the Markov-switching GARCH parameters. A Monte Carlo study and an application to the Standard & Poor index are presented.

Recent Results for Linear Time Series Models with Non Independent Innovations

In this paper, we provide a review of some recent results for ARMA models with uncorrelated but non independent errors. The standard so-called Box-Jenkins methodology rests on the errors independence. When the errors are suspected to be non independent, which is often the case in real situations, this methodology needs to be adapted. We study in detail the main steps of this methodology in the above-mentioned framework.

ESTIMATION-ADJUSTED VAR

Standard risk measures, such as the Value-at-Risk (VaR), or the Expected Shortfall, have to be estimated and their estimated counterparts are subject to estimation uncertainty. Replacing, in the theoretical formulas, the true parameter value by an estimator based on n observations of the Profit and Loss variable, induces an asymptotic bias of order 1/n in the coverage probabilities. This paper shows how to correct for this bias by introducing a new estimator of the VaR, called Estimation adjusted VaR (EVaR). This adjustment allows for a joint treatment of theoretical and estimation risks, taking into account for their possible dependence. The estimator is derived for a general parametric dynamic model and is particularized to stochastic drift and volatility models. The finite sample properties of the EVaR estimator are studied by simulation and an empirical study of the S&P Index is proposed