Detecting Tail Risk Differences in Multivariate Time Series

Abstract

We derive functional central limit theory for tail index estimates in multivariate time series under mild conditions on the extremal dependence between the components. We use this result to also derive convergence results for extreme value‐at‐risk and extreme expected shortfall estimates. This allows us to construct tests for equality of ‘tail risk’ in multivariate data, which can be useful in a number of empirical contexts. In constructing test statistics, we avoid estimating long‐run variances by using self‐normalization. Size and power of the tests for equal ‘tail risk’ are assessed in simulations. An empirical application to exchange returns illustrates the practical usefulness of the tests.

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

“…Theorem 4 also contains Proposition 1 in Jiang et al (2017), which is limited to the case d = 2 and X (2) = −X (1) . Another result related to Theorem 4 is Proposition 3 in Hoga (2018), although the present result and that of Hoga (2018) are more difficult to compare since the latter is stated within the particular context of time series analysis. Our proof of Theorem 4 is also conceptually less involved than that of Dematteo and Clémençon (2016), which rests upon a multivariate functional central limit theorem (see Theorem 7.1 and Corollary 7.2 therein).…”

“…The benefit of writing this is that while showing directly the joint convergence of the random pair on the left-hand side is difficult and appears to require advanced theoretical arguments (see Dematteo and Clémençon (2016) and Hoga (2018)), the convergence of the right-hand side is much easier to obtain since it is nothing but a pair of (ratios of) sums of independent and identically distributed random variables. We will return to this in Section 3 to show how this observation leads to conceptually simple proofs of the joint asymptotic normality of several Hill estimators.…”

“…This condition appears, among others, in Cai et al (2015) as well as, in a slightly different form, in Hoga (2018). It is a convenient way to describe the asymptotic dependence structure of the bivariate random vector (X (j ) , X (l) ), while being weaker than a pairwise bivariate regular variation assumption (in the sense of e.g.…”

“…Such joint convergence results have only been proven very recently by Dematteo and Clémençon (2016), Kinsvater et al (2016) and Hoga (2018). The methods of proof therein rely on advanced theoretical methodologies, namely multivariate vague convergence in Dematteo and Clémençon (2016) and Kinsvater et al (2016) and multivariate empirical process theory in Hoga (2018), as well as on various ad hoc technical conditions.…”

“…Such joint convergence results have only been proven very recently by Dematteo and Clémençon (2016), Kinsvater et al (2016) and Hoga (2018). The methods of proof therein rely on advanced theoretical methodologies, namely multivariate vague convergence in Dematteo and Clémençon (2016) and Kinsvater et al (2016) and multivariate empirical process theory in Hoga (2018), as well as on various ad hoc technical conditions. These joint asymptotic results have been found useful for testing tail homogeneity across marginals of a multivariate heavy-tailed distribution in environmental or financial contexts (see Kinsvater et al (2016) and Hoga (2018)) and constructing improved tail index estimators by pooling (see Section 3.2 in Dematteo and Clémençon (2016)).…”