The extremal index for GARCH(1, 1) processes

Laurini, Fabrizio; Tawn, Jonathan A.

doi:10.1007/s10687-012-0148-z

Cited by 13 publications

(14 citation statements)

References 20 publications

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“…For the GARCH(1, 1) process, Laurini and Tawn (2012) provided an expression for the spectral measure, for a different description of the angular variable to that used here. For the choice of the angular variable (2.8), for all t, their result translates to…”

Section: Basrakmentioning

confidence: 99%

“…Existing theoretical and computational methods for deriving extremal properties are well established for special cases of the GARCH(p, q) process, namely: for symmetric Z t with p = 0, q = 1, corresponding to the ARCH(1) process (de Haan et al, 1989) and for p = q = 1, corresponding to a GARCH(1,1) (Laurini and Tawn, 2012); and for asymmetric Z t with p = q = 1 (Ehlert et al, 2015). Additional results of other tails probabilities are derived in the two-dimensional case by Collamore et al (2014) but are effective only for ARCH(1) as further complications arise for the GARCH(1, 1).…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of extremal properties of GARCH(p,q) processes

Laurini¹,

Fearnhead²,

Tawn³

2019

Preprint

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Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling features commonly found in observed financial returns. The extremal properties of these processes are of considerable interest for market risk management. For the simplest GARCH(p, q) process, with max(p, q) = 1, all extremal features have been fully characterised. Although the marginal features of extreme values of the process have been theoretically characterised when max(p, q) ≥ 2, much remains to be found about both marginal and dependence structure during extreme excursions. Specifically, a reliable method is required for evaluating the tail index, which regulates the marginal tail behaviour and there is a need for methods and algorithms for determining clustering. In particular, for the latter, the mean number of extreme values in a short-term cluster, i.e., the reciprocal of the extremal index, has only been characterised in special cases which exclude all GARCH(p, q) processes that are used in practice. Although recent research has identified the multivariate regular variation property of stationary GARCH(p, q) processes, currently there are no reliable methods for numerically evaluating key components of these characterisations. We overcome these issues and are able to generate the forward tail chain of the process to derive the extremal index and a range of other cluster functionals for all GARCH(p, q) processes including integrated GARCH processes and processes with unbounded and asymmetric innovations. The new theory and methods we present extend to assessing the strict stationarity and extremal properties for a much broader class of stochastic recurrence equations.

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Section: Basrakmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Evaluation of extremal properties of GARCH(p,q) processes

Laurini¹,

Fearnhead²,

Tawn³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Figure 2 we find the proportions of anti-D (3) (u n ) to anti-D (5) (u n ) of a GARCH(1,1) process with Gaussian innovations, autoregressive parameter λ = 0.25 and variance parameter β = 0.7 (Laurini and Tawn, [16] 2012). More precisely, in the first two panels are plotted the proportions of anti-D (3) (u n ) by choosing k n = [(log n) 3 ] and k n = [(log n) 3.3 ], respectively, and the last two plots correspond to the proportions of anti-D (4) (u n ) and anti-D (5) (u n ), with k n = [(log n) 3.3 ].…”

Section: Estimationmentioning

confidence: 99%

Estimating the extremal index through local dependence

Ferreira¹,

Феррейра²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. Our new estimation approach for this parameter is based on the extremal behavior under the local dependence condition D (k) (un). We compare a process satisfying one of this hierarchy of increasingly weaker local mixing conditions with a process of cycles satisfying the D (2) (un) condition. We also analyze local dependence within moving maxima processes and derive a necessary and sufficient condition for D (k) (un). In order to evaluate the performance of the proposed estimators, we apply an empirical diagnostic for local dependence conditions, we conduct a simulation study and compare with existing methods. An application to a financial time series is also presented.

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“…This property is illustrated in Sections 4 and 5. A nice illustration of this feature arises in time series: a Gaussian autoregressive process has temporal dependence as measured by the autocorrelation function but no tail dependence as measured by χ (Sibuya, 1960); whereas the reverse holds for the ARCH and GARCH processes (de Haan and Resnick (1989), Laurini and Tawn (2012)).…”

Section: Different Dependence Measures For Typical and Extreme Value mentioning

confidence: 99%

Assessing extremal dependence of North Sea storm severity

2016

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Extreme value theory provides asymptotically motivated methods for modelling the occurrences of extreme values of oceanographic variables, such as significant wave height at a single location. Problems are often multivariate or spatial in nature, where interest lies in the risk associated with joint occurrences of rare events, e.g. the risk to multiple offshore structures from a storm event. There are two different classes of joint tail behaviour that have very different implications: asymptotic independence suggesting that extreme events are unlikely to occur together, and asymptotic dependence implying that extreme events can occur simultaneously. It is vital to have good diagnostics to identify the appropriate dependence class. If variables are asymptotically independent, incorrectly assuming an asymptotically dependent model can lead to overestimation of the joint risk of extreme events, and hence to higher than necessary design costs of offshore structures. We develop improved diagnostics for differentiating between these two classes, which leads to increased confidence in model selection. Application to samples of North Sea sea-state and storm-peak significant wave height suggests that tail dependence changes with wave direction and distance between spatial locations, and that in most cases asymptotic independence seems to be the more appropriate assumption.

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The extremal index for GARCH(1, 1) processes

Cited by 13 publications

References 20 publications

Evaluation of extremal properties of GARCH(p,q) processes

Evaluation of extremal properties of GARCH(p,q) processes

Estimating the extremal index through local dependence

Assessing extremal dependence of North Sea storm severity

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