Regularly varying multivariate time series

By Bojan Basrak, Johan Segers. 2007

233 citations recorded by the source corpus.

Extreme values of a stationary, multivariate time series may exhibit dependence across coordinates and over time. The aim of this paper is to offer a new and potentially useful tool called tail process to describe and model such extremes. The key property is the following fact: existence of the tail process is equivalent to multivariate regular variation of finite cuts of the original process. Certain remarkable properties of the tail process are exploited to shed new light on known results on certain point processes of extremes. The theory is shown to be applicable with great ease to stationary solutions of stochastic autoregressive processes with random coefficient matrices, an interesting special case being a recently proposed factor GARCH model. In this class of models, the distribution of the tail process is calculated by a combination of analytical methods and a novel sampling algorithm.

Subfields: Economics and Econometrics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3c724e4add9d6e526072c6566b02534b81878ed4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

In Finance

In Statistics and Probability

Citations in the corpus

A functional limit theorem for dependent sequences with infinite variance stable limits (2010, background): “In particular, it allows us to build on the theory developed in [5, 12] to determine the asymptotic behavior of partial sums over shorter blocks of indices.”
Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model (2013, methodology): “By Corollary 3.2 in Basrak and Segers (2009) we have the following equivalent characterization of the jointly regular variation property: Theorem 3.5 Let {Xk : k ∈ Z} be a stationary sequence in Rd and assume x 7→ P(‖X0‖ > x) is regularly varying with index −α for some α > 0.”
Towards estimating extremal serial dependence via the bootstrapped extremogram (2012, background): “We refer to Basrak and Segers [1] who give an enlightening interpretation of the structure of a regularly varying sequence.”
Measures of serial extremal dependence and their estimation (2013, methodology): “Basrak and Segers [2] also gave an expression for the extremal index in terms of the spectral tail process: θ = E [ sup t≥0 |Θt|α − sup t≥1 |Θt|α ] .”
Heavy tailed time series with extremal independence (2013, background): “ments. This convergence can be applied to study risk measures such as the conditional tail expectation; this will be discussed in Section 2.3. In Section 2.4 we extend the tail process, introduced by [BS09] in the extremally dependent case, to the extremally independent case and we give a representa”
The extremogram: a correlogram for extreme events (2009, background): “Basrak and Segers [3] provide a useful representation for regularly varying strictly stationary sequences.”
MAX-STABLE MODELS FOR MULTIVARIATE EXTREMES (2012, background): “The device is not new as it appears already for instance in the theory of regularly varying multivariate time series in Basrak and Segers (2009) or in the concept of a D-norm in Falk et al. (2010).”
Stable limits for sums of dependent infinite variance random variables (2009, background): “It follows from recent work by Basrak and Segers [6] that the index set {i : d ≤ |i | ≤ mn} can be replaced by {i : d ≤ i ≤ mn}, reducing the efforts for verifying (2.”