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The effective lower bound on a short term interest rate may not constrain a central bank's capacity to achieve its objectives if unconventional monetary policy (UMP) is powerful enough. We formalize this 'irrelevance hypothesis' using a dynamic stochastic general equilibrium model with UMP and test it empirically for the United States and Japan using a structural vector autoregressive model that includes variables subject to occasionally binding constraints. The hypothesis is strongly rejected for both countries. However, a comparison of the impulse responses to a monetary policy shock across regimes shows that UMP has had strong delayed effects in each country.