Structural changes and outliers often coexist, complicating statistical inference. This paper addresses the problem of testing for parameter changes in conditionally heteroscedastic time series models, particularly in the presence of outliers. To mitigate the impact of outliers, we introduce a two-step procedure comprising robust estimation and residual truncation. Based on this procedure, we propose a residual-based robust CUSUM test and its self-normalized counterpart. We derive the limiting null distributions of the proposed robust tests and establish their consistency. We also investigate the breakdown of the naive tests under additive and innovation outlier contamination and provide theoretical support for the robustness of the proposed tests. Simulation results demonstrate the strong robustness of our tests against outliers. Finally, we analyze Bitcoin data to illustrate the practical application.

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