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Optimality of sup-F and sup-Wald tests for multiple breaks (k>1)

Determine the optimality properties of the sup-F and sup-Wald statistics for testing zero versus k>1 structural breaks in linear regression models estimated by OLS under covariance stationarity assumptions, extending the analysis beyond the k=1 case studied by Kim and Perron (2009).

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Background

The chapter reviews sup-F and sup-Wald tests for detecting multiple structural breaks in linear OLS models and notes that optimality properties have been studied for a single break under covariance stationarity. For more than one break, optimality results have not been established, creating a gap in theoretical understanding of test performance when k>1.

References

The optimality properties of the $sup\mathcal{F}_T(k)$ and $sup\mathcal{W}_T(k)$ tests are discussed in \citeasnoun{Kim/Perron:2009} for $k=1$ under covariance stationarity assumptions, but, to the best of our knowledge, are not known for $k>1$.

— Testing for multiple change-points in macroeconometrics: an empirical guide and recent developments (2507.22204 - Boldea et al., 29 Jul 2025) in Section 2, Univariate linear models (OLS tests)