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This paper discusses conditions for identification of structural shocks with external instruments in VARs, under partial invertibility. This is a very general condition, often of empirical relevance, and less stringent than the standard full invertibility, sometimes required for SVARs and LPs with controls. We show that in this case dynamic responses can be recovered provided that a condition of limited lead-lag exogeneity of the instrument holds. This condition allows the instrument to be contaminated by leads and lags other invertible shocks. We evaluate our results in a simulated environment and provide an empirical application to the case of monetary policy shocks.