We provide a comprehensive set of new results on the impact of mis-specifying the short run dynamics in fractionally integrated processes. We show that four alternative parametric estimators - frequency domain maximum likelihood, Whittle, time domain maximum likelihood and conditional sum of squares - converge to the same pseudo-true value under common mis-specification, and that they possess a common asymptotic distribution. The results are derived assuming a completely general parametric specification for the short run dynamics of the estimated (mis-specified) fractional model, and with long memory, short memory and antipersistence in both the model and the true data generating process accommodated. As well as providing new theoretical insights, we undertake an extensive set of numerical explorations, beginning with the numerical evaluation, and implementation, of the (common) asymptotic distribution that holds under the most extreme form of mis-specification. Simulation experiments are then conducted to assess the relative finite sample performance of all four mis-specified estimators, initially under the assumption of a known mean, as accords with the theoretical derivations. The importance of the known mean assumption is illustrated via the production of an alternative set of bias and mean squared error results, in which the estimators are applied to demeaned data. The paper concludes with a discussion of open problems.