Olivier Wintenberger

ASYMPTOTIC NORMALITY OF THE QUASI MAXIMUM LIKELIHOOD ESTIMATOR FOR MULTIDIMENSIONAL CAUSAL PROCESSES

Strong consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) are given for a general class of multidimensional causal processes. For particular cases already studied in the literature (for instance univariate or multivariate GARCH, ARCH, ARMA-GARCH processes) the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes).

Model selection for weakly dependent time series forecasting

Observing a stationary time series, we propose a two-step procedure for the predictionof the next value of the time series. The first step follows machine learning theory paradigmand consists in determining a set of possible predictors as randomized estimators in (possiblynumerous) different predictive models. The second step follows the model selection paradigmand consists in choosing one predictor with good properties among all the predictors of the firststeps. We study our procedure for two different types of observations: causal Bernoulli shifts andbounded weakly dependent processes. In both cases, we give oracle inequalities: the risk of thechosen predictor is close to the best prediction risk in all predictive models that we consider. Weapply our procedure for predictive models such as linear predictors, neural networks predictorsand non-parametric autoregressive predictors.

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.

Multiple breaks detection in general causal time series using penalized quasi-likelihood

This paper is devoted to the off-line multiple breaks detection for a general class of models. The observations are supposed to fit a parametric causal process (such as classical models AR(∞), ARCH(∞) or TARCH(∞)) with distinct parameters on multiple periods. The number and dates of breaks, and the different parameters on each period are estimated using a quasi-likelihood contrast penalized by the number of distinct periods. For a convenient choice of the regularization parameter in the penalty term, the consistency of the estimator is proved when the moment order r of the process satisfies r≥2. If r≥4, the length of each approximative segment tends to infinity at the same rate as the length of the true segment and the parameters estimators on each segment are asymptotically normal. Compared to the existing literature, we added the fact that a dependence is possible over distinct periods. To be robust to this dependence, the chosen regularization parameter in the penalty term is larger than the ones from BIC approach. We detail our results which notably improve the existing ones for the AR(∞), ARCH(∞) and TARCH(∞) models. For the practical applications (when n is not too large) we use a data-driven procedure based on the slope estimation to choose the penalty term. The procedure is implemented using the dynamic programming algorithm. It is an O(n2) complexity algorithm that we apply on AR(1), AR(2), GARCH(1,1) and TARCH(1) processes and on the FTSE index data.

Prediction of time series by statistical learning:general losses and fast rates

Abstract We establish rates of convergences in statistical learning for time series forecasting. Using the PAC-Bayesian approach, slow rates of convergence √ d/n for the Gibbs estimator under the absolute loss were given in a previous work [7], where n is the sample size and d the dimension of the set of predictors. Under the same weak dependence conditions, we extend this result to any convex Lipschitz loss function. We also identify a condition on the parameter space that ensures similar rates for the classical penalized ERM procedure. We apply this method for quantile forecasting of the French GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and for uniformly mixing processes, we prove that the Gibbs estimator actually achieves fast rates of convergence d/n. We discuss the optimality of these different rates pointing out references to lower bounds when they are available. In particular, these results bring a generalization the results of [29] on sparse regression estimation to some autoregression.

Large deviations for bootstrapped empirical measures

We investigate the Large Deviations properties of bootstrapped empirical measure with exchangeable weights. Our main result shows in great generality how the resulting rate function combines the LD properties of both the sample weights and the observations. As an application we recover known conditional and unconditional LDPs and obtain some new ones.