Testing for changes in (extreme) VaR

Abstract

Summary In this paper, we develop tests for a change in an unconditional small quantile (Value‐at‐Risk, VaR, in financial time series analysis) based on an estimator motivated by extreme value theory. This so‐called Weissman estimator allows tests to be applied for extreme VaR, where extant tests mostly fail. In view of applications, we allow for weakly dependent observations. Our test statistics rely on self‐normalization, which obviates the need to estimate the complicated asymptotic variance. Consistency is shown under local alternatives, where multiple breaks can occur. A simulation study shows that in finite samples our tests compare favourably in the tail region with extant tests based on order statistic estimators and also with tail index break tests. Two empirical examples serve to illustrate the practical use of our tests.

Sequential monitoring of the tail behavior of dependent data

“…As for the mixing assumption (A2) we remark that GARCH-models, which are widely used to model financial return data like the one considered here, are geometrically strong mixing, i.e., β(k) = η k , η ∈ (0, 1). Further, applying the retrospective tests of Hoga (2014) and Hoga (2015) we find no evidence of extreme quantile or tail index breaks during that period which would violate the stationarity assumption. Hence, we proceed with our monitoring procedure.…”

“…Tests based on other estimators than the Hill (1975) estimator were first proposed by Einmahl, de Haan and Zhou (2015+) for independent and Hoga (2014) for dependent data. To the best of our knowledge the only paper dealing with changes in extreme quantiles is Hoga (2015). All these tests are of a retrospective nature.…”

“…This assumption can of course be tested by any of the retrospective change point tests proposed in, e.g., Hoga (2014Hoga ( , 2015.…”

“…For a discussion of these conditions, as well as some easier-to-verify sufficient conditions for (A3) and (A4) see Hoga (2014), where also some examples of time series satisfying these conditions are presented. Condi- Hoga (2015). (A6) provides a range for p: lim n→∞ np k = 0 provides an upper bound for the decay of p (indicating the limitations of the extreme value theory approach towards the center of the distribution) while lim n→∞ k −1/2 log (np/k) = lim n→∞ k −1/2 log (np) = 0 provides a lower bound, beyond which estimation is no longer feasible.…”

“…For the verification of the conditions (A2)-(A7) we refer to Hoga (2015) and the references cited therein.…”