Impulse Response Weights (IRWs) from strongly persistent time series. A non parametric, time domain estimator based on an autoregressive (AR) approximation is shown to have good theoretical and small sample properties for the estimation of IRWs. An alternative procedure of using a semi-parametric Local Whittle (LW) estimator of the long memory parameter and then obtaining estimates of the short run parameters and IRWs is also considered. The second part of the paper investigates the most appropriate methods for estimating the variability and the construction of condence intervals for the estimated IRWs. Particular attention is given to a generic  semi-parametric sieve bootstrap basedon an autoregressive  approximation of the unknown data generating mechanism.  The validity of bootstrap inference on the IRWs, based on the  autoregressive approximation, is proven under mild  assumptions. The findings in this paper indicate that a good  strategy for analyzing IRWs is to estimate by semi-parametric  AR approximations, and to use the sieve bootstrap for  estimating condence intervals. Simulation evidence indicates  this approach appears to be a very good strategy for processes with either short or long memory. An empirical example  concerning the persistence of real exchange rate series is  included.