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Perron and Zhu (2005) established the consistency, convergence rate and limiting distributions of parameter estimates in time trends with a change in slope with or without a concurrent level change for the cases with I(1) or I(0) errors. We extend their analysis to the general case of fractionally integrated errors with memory parameter d^∗. Our results uncover interesting features; e.g., with a level shift allowed, the convergence rate for the break date estimate is the same for all d^∗∈(−0.5,0.5). In other cases, it is decreasing as d^∗ increases. We also provide results about the so‐called spurious break issue.