["Oxford Bulletin of Economics and Statistics, Volume 88, Issue 3, Page 545-552, June 2026. ", "\nABSTRACT\nOne of the attractions of panel data is the ability to pool information regarding parameters that are common across the cross‐section. In fact, pooling is standard even though there are often reasons to believe that parameters are not common. One reason for this practice is that there is quite some evidence to suggest that estimators of common parameters are robust to random heterogeneity, so much so that this type of robustness is sometimes taken for granted. In the present paper, we consider the problem of least squares estimation of the breakpoint in a panel data model with a breaking mean. This part of the literature standardly assumes that there is a common breakpoint, and all results available are based on this assumption. Yet, it is not difficult to find studies suggesting that the least squares breakpoint estimator should be robust to random heterogeneity. We show that it is not, except in special circumstances.\n"]