Sequential monitoring of the tail behavior of dependent data

Journal Time Series Analysis

2018

Self Cite

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.

T_{n} = \frac{{[{\hat{γ}}_{H, X} (1) - {\hat{γ}}_{H, Y} (1)]}^{2}}{\int_{t_{0}}^{1} t^{2} {\{[{\hat{γ}}_{H, X} (t) - {\hat{γ}}_{H, Y} (t)] - [{\hat{γ}}_{H, X} (1) - {\hat{γ}}_{H, Y} (1)]\}}^{2} d t}, t_{0} \in (0, 1),

and reject

{scriptH}_{0}^{γ}

if T n is too large.…”

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 97%

See 1 more Smart Citation

Detecting Tail Risk Differences in Multivariate Time Series

Journal Time Series Analysis

2018

Self Cite

“…Remark Hoga (, Theorem 1) and Hoga and Wied (, Proof of Theorem 1) show that under Assumptions – we have under a Skorohod construction that for some small

\tilde{δ} > 0

and every

ν \in [0, 1 / 2)

\underset{\frac{0pt}{t \in false[0, 1 false]}}{falseprefixsup} y^{ν / γ} | \sqrt{k} (\frac{1}{k} \sum_{i = 1}^{⌊ n t ⌋} I_{{X_{i} > y U (n / k)}} - y^{- 1 / γ} t) - W false(t, y^{- 1 / γ} false) | \overset{a . s .}{\underset{false(n \to \infty false)}{\to}} 0

holds, where

W (\cdot, \cdot)

is a continuous, zero‐mean Gaussian process with covariance function

Cov false(W (t_{1}, y_{1}), W (t_{2}, y_{2}) false) = min false(t_{1}, t_{2} false) r false(y_{1}, y_{2} false),

and

r (\cdot, \cdot)

defined in Assumption . This result will be crucial in the proof of Theorem .…”

Section: Resultsmentioning

confidence: 99%

“…Remark Assumptions – have been discussed for ARMA, ARCH and SV (stochastic volatility) models in Drees (, Section ) and Hoga and Wied (, Section 2.3).…”

Section: Resultsmentioning

confidence: 99%

Testing for changes in (extreme) VaR

The Econometrics Journal

2017

Self Cite

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.

Quantifying the data-dredging bias in structural break tests

2021

Stat Papers

Self Cite

Structural break tests are often applied as a pre-step to ensure the validity of subsequent statistical analyses. Without any a priori knowledge of the type of breaks to expect, eye-balling the data can indicate changes in some parameter, e.g., the mean. This, however, can distort the result of a structural break test for that parameter, because the data themselves suggested the hypothesis. In this paper, we formalize the eye-balling procedure and theoretically derive the implied size distortion of the structural break test. We also show that eye-balling a stretch of historical data for possible changes in a parameter does not invalidate the subsequent procedure that monitors for structural change in new incoming observations. An empirical application to Bitcoin returns shows that taking into account the data-dredging bias, which is incurred by looking at the data, can lead to different test decisions.