Les modèles vectoriels et multiplicatifs avec erreurs non-négatives de séries chronologiques

The objective of this master thesis is to present models for multivariate time series involving random vectors where each component is non-negative. We consider vMEM models (vector multiplicative error models) presented by Cipollini, Engle and Gallo (2006) and Cipollini and Gallo (2010). These models represent a generalization to the multivariate case of MEM models introduced by Engle (2002). These models are applied especially with financial time series. The vMEM models can be used to model time series involving asset volumes, durations, conditional variances, among these applications. It is also possible to model the variables jointly and to study the dynamics between these time series forming the system under study.  To model multivariate time series with non-negative components, several specifications for the term vector error have been proposed in the literature. One approach is to consider the use of random vectors where the distribution of the error term is such that each component is non-negative. However, finding a sufficiently flexible multivariate distribution defined on the positive support is rather difficult, at least with the applications mentioned above. As indicated by Cipollini, Engle and Gallo (2006), a possible candidate is the multivariate gamma distribution, which  however imposes severe restrictions on the contemporaneous correlations between variables. Since the possibilities are limited, one possible approach is to use the theory of copulas. Thus, according to this approach, margins can be specified, such that the distributions in question have non-negative supports, and a copula function takes into account the dependency between components. A possible estimation technique is the method of maximum likelihood. An alternative approach is the generalized method of moments (GMM). This latter method has the advantage of being semi-parametric in the sense that unlike the approach imposing a multivariate distribution, it is not necessary to specify a multivariate distribution for the error term.  Generally, estimating vMEM models is complicated. Existing algorithms must take into account the large number of parameters and the elaborate nature of the likelihood function. In the case of the GMM estimation method, solving the system requires the use of solvers for non linear systems. In this project, considerable energy has been devoted to the development of computer code (in the R language) to estimate the different parameters of the model.   In the first chapter, we define the stationary processes, autoregressive processes, the autoregressive conditional heteroskedasticity processes (ARCH) and the generalized ARCH processes (GARCH). We also present duration models ACD and MEM models. In the second chapter, we present the theory of copulas needed for our work, under the vector multiplicative error models vMEM. We also discuss the possible estimation methods. In the third chapter, we discuss the simulation results for several estimation methods. In the last chapter, applications to financial time series are presented. The R code is provided in an Appendix. A conclusion completes this thesis.

Keyphrases: vMEM, GMM, Copula, vMEM, GMM, Multivariate time series, Multivariate Ljung-Box statistics