The extremogram: a correlogram for extreme events

By Richard A. Davis, Thomas Mikosch. 2009

223 citations recorded by the source corpus.

We consider a strictly stationary sequence of random vectors whose finite-dimensional distributions are jointly regularly varying with some positive index. This class of processes includes among others ARMA processes with regularly varying noise, GARCH processes with normally or student distributed noise, and stochastic volatility models with regularly varying multiplicative noise. We define an analog of the autocorrelation function, the extremogram, which only depends on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under �-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram. 1. Measures of extremal dependence in a strictly stationary sequence The motivation for this research comes from the problem of choosing between two popular and commonly used families of models, the generalized autoregressive conditional heteroscedastic (GARCH) process and the heavy-tailed stochastic volatility (SV) process, for modeling a particular financial time series. Both GARCH and SV models possess the stylized features exhibited by log- returns of financial assets. Specifically, these time series have heavy-tailed marginal distributions, are dependent but uncorrelated, and display stochastic volatility. The latter property is manifested via the often slow decay of the sample autocorrelation function (ACF) of the absolute values and squares of the time series. Since both GARCH and SV models can be chosen to have virtually identical behavior in the tails of the marginal distribution and in the ACF of the squares of the process, it is difficult for a given time series of returns to decide between the two models on the basis of routine time series diagnostic tools. The problem of finding probabilistically reasonable and statistically estimable measures of ex- tremal dependence in a strictly stationary sequence is to some extent an open one. In classical time series analysis, which mostly deals with second order structure of stationary sequences, the ACF is a well accepted object for describing meaningful information about serial dependence. The ACF is sometimes over-valued as a tool for measuring dependence especially if one is only interested in extremes. It does of course determine the distribution of a stationary Gaussian sequence, but for non-Gaussian and non-linear time series the ACF often provides little insight into the dependence structure of the process. This is particularly the case when one considers heavy-tailed non-linear time series such as the GARCH model. In this case, the estimation of the ACF via the sample ACF is also rather imprecise and even misleading since the asymptotic confidence bands are typ- ically larger than the estimated autocorrelations, see for example the results in Basrak et al. (1) for bilinear processes; Davis and Mikosch (12), Mikosch and Stuaricua (26) and Basrak et al. (2) for ARCH and GARCH processes, Resnick (32) for teletraffic models. 1.1. The extremal index. The asymptotic behavior of the extremes leads to one clear difference between GARCH and SV processes. It was shown in Davis and Mikosch (12), Basrak et al. (2), Davis and Mikosch (13) (see also Breidt and Davis (7) for the light-tailed SV case) that GARCH processes exhibit extremal clustering (i.e., clustering of extremes), while SV processes lack this

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/f3448a897de9286ac5915b8ba27751fcbf4b09ca

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Citations in the corpus

Functional peaks‐over‐threshold analysis (2020, background): “Equation (20) summarises the pairwise extremal dependence between X(s) and X(s′); it extends the extremogram (Davis & Mikosch, 2009) to r-exceedances and generalises the extremal dependence coefficient 𝜒 (Ledford & Tawn, 1996) to processes.”
Towards estimating extremal serial dependence via the bootstrapped extremogram (2012, methodology): “1) exists, it was shown in Davis and Mikosch [3] that ρ̂A,B(h) is asymptotically normal; i.”
Geostatistics of Dependent and Asymptotically Independent Extremes (2013, background): “Similar dependence measures for extremes of time series are described by Ledford and Tawn (2003) and Davis and Mikosch (2010).”
Quantile Spectral Processes: Asymptotic Analysis and Inference (2014, unknown): “reme events, which is related in spirit but quite diﬀerent in many other respects, was considered by Davis, Mikoschand Zhao [11]. Finally, in the time domain, Linton and Whang [41], Davis and Mikosch [10] and Han et al. [24] Quantile spectral processes 3 introduced the related concepts of quantilogr”
The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series (2014, unknown): “Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles, and Davis et al. (2012) has provided the inference methods based on bootstrap and permutation for the extremogram.”
Threshold modeling of extreme spatial rainfall (2013, background): “Statistical models that are capable of consistent extrapolation beyond available data are needed to estimate probabilities for higher rainfall levels, and these are provided by extreme-value theory.”
Modelling Time Series Extremes (2012, methodology): “If ηh = 1 and Lh(u) → ρh > 0 as u → ∞, then X0 and Xh are asymptotically dependent, so the first of these plots, called an extremogram by Davis & Mikosch (2009b), provides an estimate of the extremal dependence at lag h.”
Max-stable processes for modelling extremes observed in space and time (2011, methodology): “Univariate extreme value theory is well developed and standard introductions can be found in many books, including for example Coles [7], Embrechts, Klüppelberg and Mikosch [19], and Leadbetter, Lindgren and Rootzén [29].”
Measures of serial extremal dependence and their estimation (2013, methodology): “This fact was proved in Mikosch and Samorodnitsky [47] for the distribution of X .”
Statistical inference for max‐stable processes in space and time (2012, methodology): “The situation here is similar to the estimation of the extremogram, where bootstrap methods have been suggested to construct 806 R. A. Davis, C. Klüppelberg and C. Steinkohl asymptotically correct confidence bands (see Davis and Mikosch (2009) and Davis et al. (2012)).”
Predictive quantile regression with persistent covariates: IVX-QR approach (2015, background): “If we consider a quantile-quantile predictability (Han et al., 2014) or a extreme quantile version of it (Davis and Mikosch, 2009), L ( ) or L will play the contributing role for the distortion.”
A functional limit theorem for dependent sequences with infinite variance stable limits (2010, methodology): “y an application of the multivariate version of Breiman’s lemma, the sequence (Xn)n is regularly varying with index αtoo; see Basrak et al. [4], Proposition 5.1. From the results in Davis and Mikosch [14], it follows that n Xrn i=1 (4.1) P(|Xi|>tan,|X0|>tan)→0 as n→∞ for any t>0, where (rn)”
Local Gaussian correlation: A new measure of dependence (2013, background): “Another form of correlation based on extreme values is given by Davis and Mikosch (2009).”
Quantile coherency: A general measure for dependence between cyclical economic variables (2015, unknown): “Linton and Whang (2007), and Davis et al. (2009) introduced the (univariate) concepts of quantilograms and extremograms, respectively.”