local parameter $d$. In other words the estimation of $d(u)$ can be obtained from an analysis over a fixed set of frequencies taken away from zero. This would however induce a high sensitivity to model misspecification. In contrast, as usual in semi-parametric vs parametric estimation, we want our approach to be robust against model misspecification. To impose this robustness in the semi-parametric model, the memory parameter $d$ is disconnected from the spectral content out of the origin so that short range correlations does not carry any information about this parameter. This fully justifies the semi-parametric context even though it is more involved as it necessitates a *low frequency* analysis (where the long memory behavior occurs) which, at first sight, seems contradictory to the *local* stationarity framework. In fact, this contradiction is inherent to any locally stationary model which rely on a compromise between stationarity, which appears at small scales, and analyzing bandwidth, which requires a large scale to decrease the randomness in the data. As a consequence practical applications of these models require very long data sets. Our results will prove that this apparent contradiction can still be overcome for locally stationary long memory models, but with some price to 