ZAKOIAN Jean Michel

This paper addresses the problem of deriving the asymptotic distribution of the empirical distribution function F n of the residuals in a general class of time series models, including conditional mean and conditional heteroscedaticity, whose independent and identically distributed errors have unknown distribution F. We show that, for a large class of time series models (including the standard ARMA-GARCH), the asymptotic distribution of √ n{ F n (·) − F (·)} is impacted by the estimation but does not depend on the model parameters. It is thus neither asymptotically estimation free, as is the case for purely linear models, nor asymptotically model dependent, as is the case for some nonlinear models. The asymptotic stochastic equicontinuity is also established. We consider an application to the estimation of the conditional Value-at-Risk.

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This paper deals with the high dimensionality problem in multivariate GARCH processes. The author proposes a new vine-GARCH dynamics for correlation processes parameterized by an undirected graph called "vine". This approach directly generates definite-positive matrices and encourages parsimony. After establishing existence and uniqueness results for stationary solutions of the vine-GARCH model, the author analyzes the asymptotic properties of the model. He then proposes a general framework of penalized M-estimators for dependent processes and focuses on the asymptotic properties of the adaptive Sparse Group Lasso estimator. The high dimension is treated by considering the case where the number of parameters diverges with the sample size. The asymptotic results are illustrated by simulated experiments. Finally in this framework the author proposes to generate the sparsity for dynamics of variance-covariance matrices. To do so, the class of multivariate ARCH models is used and the corresponding processes are estimated by penalized ordinary least squares.

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Maximum likelihood estimation is a widely used method for identifying a parameterized time series model from a sample of observations. For well-specified models, it is essential to obtain the consistency of the estimator, i.e. its convergence to the true parameter when the size of the sample of observations tends to infinity. For many time series models, for example hidden Markov models (HMM), the property of "strong" consistency can however be difficult to establish. We can then focus on the consistency of the maximum likelihood estimator (MLE) in a weak sense, i.e. when the sample size tends to infinity, the MLE converges to a set of parameters that are all associated with the same probability distribution of the observations as the true parameter. Consistency in this sense, which remains a preferred property in many time series applications, is referred to as equivalence class consistency. Obtaining equivalence class consistency generally requires two important steps: 1) showing that the MLE converges to the set that maximizes the asymptotic normalized log-likelihood . and 2) showing that each parameter in this set produces the same distribution of the observation process as the true parameter. The main purpose of this thesis is to establish the equivalence class consistency of partially observed Markov models (PMMs), such as HMMs and observation-driven models (ODMs).

This paper studies goodness of fit tests and specification tests for an extension of the log-GARCH model which is stable by scaling. A Lagrange-Multiplier test is derived for testing the null assumption of extended log-GARCH against more general formulations including the Exponential GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended log-GARCH. Simulations illustrating the theoretical results and an application to real financial data are proposed.

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This doctoral thesis, composed of three chapters, contributes to the development of the problematic on the selection of GARCH-type volatility models. The first chapter proposes a simulation study on model selection in the specific framework of regime-switching models. Simulation experiments are proposed to highlight the inefficiency of the usual selection criteria in particular cases, which can lead to misspecification during model selection. The second chapter proposes a test of the Lagrange multiplier of misspecification in univariate GARCH models. The null hypothesis assumes that the data generating process is a linear GARCH model while under the alternative hypothesis it corresponds to an unknown functional form that is linearized using a Taylor expansion. The test is illustrated in an empirical application on exchange rates. The last chapter studies the impact of oil prices on the sovereign credit default swap spreads of two oil exporting countries: Venezuela and Russia. Using recent data, we find that oil price returns impact Venezuela's sovereign CDS spreads directly while it goes through the exchange rate channel for Russia. This chapter employs advanced statistical methods, including the use of Markov regime-switching models. Finally, the appendix provides a manual for the MSGtool (Matlab) toolbox which provides a collection of functions for studying Markovian regime-switching models. The toolbox is very user-friendly.

Until recently the liquidity of financial assets has typically beenviewed as a second-order consideration. Liquidity was frequently associatedwith simple transaction costs that impose - temporary if any- effect on assetprices, and whose shocks could be easily diversified away. Yet the evidenceespeciallythe recent liquidity crisis- suggests that liquidity is now a primaryconcern. This paper aims at disentangling market risk and liquidity riskin the context of conditional volatility models. Our approach allows theisolation of the intrisic liquidity of any asset, and thus makes it possible todeduce a liquidity risk even when volumes are not observed.

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This paper introduces the concept of risk parameter in conditional volatility models of the form $\epsilon_t=\sigma_t(\theta_0)\eta_t$ and develops statistical procedures to estimate this parameter. For a given risk measure $r$, the risk parameter is expressed as a function of the volatility coefficients $\theta_0$ and the risk, $r(\eta_t)$, of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate these findings.

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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 particular EGARCH models, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.

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In this thesis, we study the probabilistic properties and the statistical inference of parametric models of conditional volatility, whose coefficients are functions of an observed exogenous process. A first part of the thesis is devoted to the study of the stability properties of a GARCH (1,1) model belonging to this class. Necessary and sufficient conditions for the existence of a solution, generally non-stationary, are established, as well as conditions for the existence of moments for these solutions. These conditions concern the coefficients of the GARCH model in the various regimes of the exogenous process and the stationary probabilities of these regimes. In a second part, the asymptotic properties of the quasi-maximum likelihood estimator are studied. The convergence and the asymptotic normality of this estimator are demonstrated under regularity assumptions implying the stability of the solution and the strict positivity of the parameters but not requiring the existence of moments of the observed process. The study of the asymptotic behavior of the estimator when some coefficients of the model are zero is the subject of a last part. In this case, the asymptotic distribution of the estimator is non-standard and corresponds to the projection of a Gaussian distribution on a convex cone. We also obtain the asymptotic distributions of nullity tests of some coefficients of the model as well as their local asymptotic power. The main asymptotic results are illustrated by stimulus experiments. The model appears to be particularly suitable for energy price dynamics. For gas prices, we highlight the existence of GARCH volatilities depending on several temperature-related regimes.

In this thesis, we study the estimation and hypothesis testing problems of two large classes of nonlinear GARCH models. First, we consider several methods for estimating a class of GARCH models with a power threshold. Under very weak conditions, we study the asymptotic properties of these estimators in the following two situations. First, we assume the power is known. We establish the properties of the quasi-maximum likelihood estimator (QMV). We also consider two sequences of ordinary least squares estimators, in the pure ARCH case of the model and we show that, for some values of the power, these estimators can be more efficient than the QMV estimator. In a second step we consider the case where the power is unknown, and is jointly estimated with the other parameters. The asymptotic properties of the QMV are established under the assumption that the noise has a density. Moreover, we study a class of non-Gaussian quasi-Maximum Likelihood estimators in the concrete situation where the error density is misspecified. We show that this class of estimators can provide efficient alternatives to the standard QMV estimator, in particular, when the errors have thick distribution tails. Skewness tests are proposed. In the second part of this thesis, we introduce a general class of weak GARCH processes containing a large family of models with conditional heteroscedasticity. We propose a representation consisting of two ARMA equations: the first one deals with the observed process, and the second one with some function of the linear innovation of the observed process. Under ergodicity and mixing assumptions, and certain moment conditions on the observed process, we establish the convergence and asymptotic normality of the two-stage least squares estimator. We also consider the estimation of the asymptotic covariance matrix of this estimator. Most of these asymptotic results are illustrated by simulation experiments and are applied to financial series.

In this thesis we extend the scope of vector ARMA (AutoRegressive Moving-Average) models by considering error terms that are uncorrelated but may contain nonlinear dependencies. These models are called vector weak ARMAs and allow to deal with processes that may have very general nonlinear dynamics. In contrast, we call strong ARMAs the models typically used in the literature in which the error term is assumed to be iid noise. Since weak ARMA models are in particular dense in the set of regular stationary processes, they are much more general than strong ARMA models. The problem we will be concerned with is the statistical analysis of vector weak ARMA models. More precisely, we study the estimation and validation problems. First, we study the asymptotic properties of the near-maximum likelihood estimator and the least squares estimator. The asymptotic variance matrix of these estimators is of the "sandwich" form, and can be very different from the asymptotic variance obtained in the strong case. Then, we pay particular attention to the validation problems. First, we propose modified versions of the Wald, Lagrange multiplier and likelihood ratio tests to test linear restrictions on the parameters of weak vector ARMA models. Second, we are interested in residual-based tests, which aim at verifying that the residuals of the estimated models are indeed white noise estimates. In particular, we are interested in portmanteau tests, also called autocorrelation tests. We show that the asymptotic distribution of the residual autocorrelations is normally distributed with a covariance matrix different from the strong case (i.e. under idd assumptions on the noise). We derive the asymptotic behavior of the port-mantle statistics. In the standard strong ARMA framework, it is known that the asymptotic distribution of portmanteau tests is correctly approximated by a chi-square. In the general case, we show that this asymptotic distribution is that of a weighted sum of chi-squares. This distribution can be very different from the usual chi-deux approximation of the strong case. We therefore propose modified portmanteau tests to test the adequacy of weak vector ARMA models. Finally, we are interested in the choice of weak vector ARMA models based on the minimization of an information criterion, namely the one introduced by Akaike (AIC). With this criterion, we try to approximate the distance (often called Kullback-Leibler information) between the true law of the observations (unknown) and the law of the estimated model. We will see that the corrected criterion (AICc) in the context of weak vector ARMA models can, again, be very different from the strong case.

This doctoral thesis is divided into two parts dealing with identification and selection problems in econometrics. We study the following topics: (1) the problem of identifying time series models using autocorrelation, partial autocorrelation, inverse autocorrelation and inverse partial autocorrelation functions. (2) the estimation of the inverse autocorrelation function in the framework of nonlinear time series. In a first part, we consider the problem of identifying time series models using the above mentioned autocorrelation functions. We construct statistical tests based on empirical estimators of these functions and study their asymptotic distribution. Using the Bahadur and Pitman approach, we compare the performance of these autocorrelation functions in detecting the order of a moving average and an autoregressive model. Then, we identify the inverse process of an ARMA model and study its probabilistic properties. Finally, we characterize the temporal reversibility using the dual and inverse processes. The second part is devoted to the estimation of the inverse autocorrelation function in the framework of nonlinear processes. Under certain regularity conditions, we study the asymptotic properties of the empirical inverse autocorrelations for a stationary and strongly mixing process. We obtain the convergence and the asymptotic normality of the estimators. Next, we consider the case of a linear process generated by a white noise of GARCH type. We obtain an explicit formula for the asymptotic autocovariance matrix. Using examples, we show that the standard formula for this matrix is not valid when the process generating the data is nonlinear. Finally, we apply the previous results to show the asymptotic normality of estimators of the parameters of a weak moving average. Our results are illustrated by experiments.

In this thesis we extend the scope of vector autoregressive models (VAR) by considering uncorrelated but dependent errors. More precisely, by studying estimation problems and the behavior of statistical tools, we are interested in the validity in our framework of results valid under the assumption of iid Gaussian iinovations. We show that the asymptotic behavior of the estimators of short term parameters and residual autocorrelations is different from the standard case. Thus, modified portmanteau tests whose asymptotic distribution is a weighted sum of chi-squares are proposed. We present an algorithm to implement these tests. The behavior of the estimators of the long-run parameters and of the likelihood ratio test for the cointegration rank is studied. It is shown that the standard results for long-run relationships extend to our framework. We also show that the asymptotic behavior of the estimators of the adjustment parameters is different from the Gaussian iid case. Theoretical examples that justify our approach are shown. The finite distance behavior of different tests is studied by Monte Carlo experiments.

In this thesis, we study the probabilistic and/or statistical properties of linear or non-linear time series models with time-dependent coefficients. The first part of the thesis is devoted to the statistics of ARMA models whose coefficients vary according to recurrent, but non-periodic events. The asymptotic properties (strong convergence and normality) of the least squares estimators are established. The special case of ARMA models with Markovian regime switching is then considered. The second part of the thesis studies the asymptotic influence of the time series mean correction on the least squares estimation of periodic ARMA models. In the last part of the thesis, we extend our research to bilinear models with periodic coefficients. The results obtained are regularly illustrated at finite distance using Monte Carlo experiments.

We study a class of nonlinear conditionally heteroskedastic models (ARCH). The volatility of the variable at date t depends on the relative position of the past variables. We show that a family of such models admits a Markovian representation allowing to study its stability. Using Lyapunov criteria, we establish conditions for the existence of moments. The asymptotic properties (strong and law convergence) of three types of parameter estimators are established. These results are illustrated at finite distance using simulated methods and on real series.

This work is based on the introduction of thresholds in time series models. We start by presenting the theory of homogeneous Markov chains, necessary for the study of nonlinear processes. Autoregressive models of order one with a threshold are the subject of the second part. Distributional properties are obtained in the neighborhood of the linear model, which allows to obtain approximate formulas for various quantities: mean, variance, moments... . Finally, two methods for testing the linearity hypothesis are proposed. The next part proposes a new class of ARCH (autogressive conditionally heteroskedastic) models. The introduction of thresholds in the specification of the conditional variance allows to take into account specific effects on the volatility (persistence, asymmetry according to the sign of the prior errors...). A complete study is proposed: weak stationarity, strict stationarity, computation of moments, analysis of the leptokurtic effect, comparison with ARCH models, estimation of various parameters, test of the homoscedasticity hypothesis. Finally, the last part of the thesis deals with the transition to continuous time of heteroscedastic threshold models.