Bootstrap validity for multivariate sequential procedures without pre-testing

Prove that a bootstrap analogue of the Qu and Perron (2007) sequential procedure for detecting multiple structural breaks in multivariate linear time series models is valid without any pre-testing for variance changes, under appropriate regularity conditions.

Background

Sequential testing in multivariate models often involves assumptions about second-moment stability or pre-tests to segment the sample. The authors conjecture that a direct bootstrap approach could validate sequential tests for multiple breaks without pre-testing, expanding practical applicability in empirical macroeconometrics.

References

However, given extant results for univariate models and for multivariate models we conjecture that a bootstrap equivalent of the \citeasnoun{Qu/Perron:2007} sequential procedure, without further pre-testing, is valid under appropriate regularity conditions.

— Testing for multiple change-points in macroeconometrics: an empirical guide and recent developments (2507.22204 - Boldea et al., 29 Jul 2025) in Section 3, Linear multivariate time series models (Multivariate models)