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We consider multivariate stationary processes (Xt) satisfying a stochastic recurrence equation of the form Xt=𝕄tXt−1+Qt, where (Qt) are i.i.d. random vectors and 𝕄t=Diag(b1+c1Mt,…,bd+cdMt) are i.i.d. diagonal matrices and (Mt) are i.i.d. random variables. We obtain a full characterization of the vector scaling regular variation properties of (Xt), proving that some coordinates Xt, i and Xt, j are asymptotically independent even though all coordinates rely on the same random input (Mt). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK‐ARCH and CCC‐GARCH) processes and to log‐returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

International audience; We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form$$ \boldsymbol{X}_t= \mathbb{ M}_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t,$$where $(\boldsymbol{Q}_t)$ are iid random vectors and $$\mathbb{M}_t=\mathrm{Diag}(b_1+c_1 M_t, \dots, b_d+c_d M_t)$$ are iid diagonal matrices and $(M_t)$ are iid random variables. % It is known that under suitable assumptions the marginals $X_{t,i}$ of $\boldsymbol{X}_t$ are regularly varying. We obtain a full characterization of the Vector Scaling Regular Variation properties of $(\boldsymbol{X}_t)$, proving that some coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent even though all coordinates rely on the same random input $(M_t)$. We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK-ARCH and CCC-GARCH) processes and to log-returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form$$ \boldsymbol{X}_t= \boldsymbol{m}M_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t,$$where $(M_t)$ and $(\boldsymbol{Q}_t)$ are iid random variables and random vectors, respectively, and $\boldsymbol{m}=\mathrm{diag}(m_1, \dots, m_d)$ is a deterministic diagonal matrix. We obtain a full characterization of the multivariate regular variation properties of $(\boldsymbol{X}_t)$, proving that coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent if and only if $m_i \neq m_j$; even though all coordinates rely on the same random input $(M_t)$. We describe extremal properties of $(\boldsymbol{X}_t)$ in the framework of vector scaling regular variation. Our results are applied to multivariate autoregressive conditional heteroskedasticity (ARCH) processes.

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation, in the spirit of Perfekt (1997, Advances in Applied Probability, 29, pp. 138-164). The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices., Extremes, Springer Verlag (Germany), A Para\^itre

In this thesis, we propose a systematic approach called the doubly adaptive LASSO tailored to time series analysis, which includes four specific methods for four time series models, respectively: The PAC-weighted adaptive LASSO for univariate autoregressive (AR) models. Although the LASSO methodology has been applied to AR models, the existing methods in the literature ignore the temporal dependence information embedded in AR time series data. Consequently, the methods may not reflect the characteristics of underlying AR processes, especially, the lag order of AR models. The PAC-weighted adaptive LASSO incorporates the partial autocorrelation (PAC) into the adaptive LASSO weights. The PAC-weighted adaptive LASSO estimator has asymptotic oracle properties and a Monte Carlo study shows promising results. The PAC-weighted adaptive positive LASSO for autoregressive conditional heteroscedastic (ARCH) models. We have not found any results in the literature that apply the LASSO methodology to ARCH models. The PAC-weighted adaptive positive LASSO incorporates the PAC information embedded in squared ARCH process into adaptive LASSO weights. The word positive reflects the fact that the parameters in ARCH models are non-negative. We introduce a new concept named the surrogate of the second-order approximate likelihood, and propose a modified shooting algorithm to implement the PAC-weighted adaptive positive LASSO computationally. The PAC-weighted adaptive positive LASSO estimator has asymptotic oracle properties and a Monte Carlo study shows promising results. The PLAC-weighted adaptive LASSO for vector autoregressive (VAR) models. Although the LASSO methodology has been applied to building VAR time series models, the existing methods in the literature ignore the temporal dependence information embedded in VAR time series data. Consequently, the methods may not reflect the characteristics of VAR time series data, especially, the lag order of VAR models. The PLAC-weighted adaptive LASSO incorporates the partial lag autocorrelation (PLAC) into the adaptive LASSO weights. The PLAC-weighted adaptive LASSO estimator has oracle properties and Monte Carlo studies show promising results. The PLAC-weighted adaptive LASSO for BEKK vector ARCH (VARCH) models. We have not found any results in the literature that apply the LASSO methodology to VARCH processes. We focus on the BEKK VARCH models. The PLAC-weighted adaptive LASSO incorporates the PLAC information embedded in the squared BEKK VARCH process into the adaptive LASSO weights. We extend the concept of the surrogate of the second-order approximate likelihood, and propose a modified shooting algorithm to implement the PLAC-weighted adaptive LASSO computationally. We conduct a Monte Carlo study and have preliminary results from the study. Keywords: Time series, financial time series, data mining, oracle property, LASSO, adaptive LASSO, doubly adaptive LASSO, positive LASSO, PAC-weighted adaptive LASSO, PAC-weighted adaptive positive LASSO, PLAC-weighted adaptive LASSO, autoregressive, AR(P), autoregressive conditional heteroscedastic, ARCH(q), vector autoregressive, multivariate autoregressive, VAR(p), vector ARCH, multivariate ARCH, VARCH(q), analytical score, analytical Hessian, quadratic approximation, surrogate to approximate likelihood, S\&P 500, Nikkei.