Taking as data-generation process a standard DSGE model, we show via Monte Carlo that reliably detecting hysteresis, defined as the presence of aggregate demand shocks with a permanent impact on output, is a significant challenge, as model-consistent identification schemes (i) spuriously detect it with non-negligible probability when in fact the data-generation process features none, and (ii) have a low power to discriminate between alternative extents of hysteresis. We propose a simple approach to test for the presence of hysteresis, and to estimate its extent, based on the notion of simulating specific statistics (e.g., the fraction of frequency-zero variance of GDP due to hysteresis shocks) conditional on alternative values of hysteresis we impose upon the VAR, and then comparing the resulting Monte Carlo distributions to the corresponding distributions computed based on the actual data via the Kullback-Leibler divergence. Based on two alternative identification schemes, evidence suggests that post-WWII U.S. data are compatible with the notion of no hysteresis, although the most plausible estimate points towards a modest extent, equal to 7 per cent of the frequency-zero variance of GDP.