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Consistency of modified BIC under autocorrelation and time-varying second moments

Prove that the modified BIC with the penalties of Kurozumi and Tuvaandorj (2011) for OLS and Hall, Osborn, and Sakkas (2017) for 2SLS consistently estimates the number of structural breaks in linear time series regressions when errors are autocorrelated and second moments change over time.

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Background

Existing simulation evidence suggests modified information criteria can outperform sequential tests with autocorrelated errors, but formal consistency results under autocorrelation and time-varying second moments have not been established. The authors conjecture that consistency holds with specific penalties for OLS and 2SLS.

References

However, we conjecture that the modified BIC is consistent both for OLS and 2SLS using the penalties in \citeasnoun{Kurozumi/Tuvaandorj:2011} and \citeasnoun{Hall/Osborn/Sakkas:2017} respectively, even in the presence of autocorrelation or changes in the second moment of the data.

— Testing for multiple change-points in macroeconometrics: an empirical guide and recent developments (2507.22204 - Boldea et al., 29 Jul 2025) in Section 2, Other approaches to break estimation (Information criteria)