Long-run restrictions are a very popular method for identifying structural vector autoregressions, but they suffer from weak identification when the data is very persistent, i.e., when the highest autoregressive roots are near unity. Near unit roots introduce additional nuisance parameters and make standard weak-instrument robust methods of inference inapplicable. We develop a method of inference that is robust to both weak identification and strong persistence. The method is based on a combination of the Anderson-Rubin test with instruments derived by filtering potentially non-stationary variables to make them near stationary using the IVX instrumentation method of Magdalinos and Phillips (2009). We apply our method to obtain robust confidence bands on impulse responses in two leading applications in the literature.