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Published online by Cambridge University Press:  12 April 2022

Sven Otto
Institute of Finance and Statistics, University of Bonn
Jörg Breitung*
Institute of Econometrics and Statistics, University of Cologne
Address correspondence to Jörg Breitung, Institute of Econometrics and Statistics, University of Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany; e-mail:


It is well known that the conventional cumulative sum (CUSUM) test suffers from low power and large detection delay. In order to improve the power of the test, we propose two alternative statistics. The backward CUSUM detector considers the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector sequentially cumulates a triangular array of backwardly cumulated residuals. A multivariate invariance principle for partial sums of recursive residuals is given, and the limiting distributions of the test statistics are derived under local alternatives. In the retrospective context, the local power of the tests is shown to be substantially higher than that of the conventional CUSUM test if a break occurs in the middle or at the end of the sample. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is found to be much shorter than that of the conventional monitoring CUSUM procedure. Furthermore, we propose an estimator of the break date based on the backward CUSUM detector and show that in monitoring exercises this estimator tends to outperform the usual maximum likelihood estimator. Finally, an application of the methodology to COVID-19 data is presented.

© The Author(s), 2022. Published by Cambridge University Press

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We are thankful to Holger Dette, Josua Gösmann, Alexander Mayer, Dominik Wied, and three referees for their very helpful comments and suggestions which helped to improve the paper a lot. Furthermore, the usage of the CHEOPS HPC cluster for parallel computing is gratefully acknowledged.



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