Taylor rules and their implications for monetary policy analysis can be misleading if the inflation target is held fixed while being in fact time-varying. We offer a theoretical analysis showing why assuming a fixed inflation target in place of a time-varying target can lead to a downward bias in the estimated policy rate response to the inflation gap and wrong statistical inference about indeterminacy. Our analysis suggests the bias is stronger in periods where inflation target movements are large. This is confirmed by simulation evidence about the magnitude of the bias obtained from a New Keynesian model featuring positive trend inflation. We further estimate medium-scale NK models with positive trend inflation and a time-varying inflation target using a novel population-based MCMC routine known as parallel tempering. The estimation results confirm our theoretical analysis while favouring a determinacy outcome for both pre and post-Volcker periods and shedding new light about the type of rule the Fed likely followed.