X-rays illuminating the accretion disc in active galactic nuclei give rise to an iron K line and its associated reflection spectrum which are lagged behind the continuum variability by the light-travel time from the source to the disc. The measured lag timescales in the iron band can be as short as $sim R_g/c$, where $R_g$ is the gravitational radius, which is often interpreted as evidence for a very small continuum source close to the event horizon of a rapidly spinning black hole. However, the short lags can also be produced by reflection from more distant material, because the primary photons with no time-delay dilute the time-lags caused by the reprocessed photons. We perform a Monte-Carlo simulation to calculate the dilution effect in the X-ray reverberation lags from a half-shell of neutral material placed at $100,R_g$ from the central source. This gives lags of $sim2,R_g/c$, but the iron line is a distinctly narrow feature in the lag-energy plot, whereas the data often show a broader line. We show that both the short lag and the line broadening can be reproduced if the scattering material is outflowing at $sim0.1c$. The velocity structure in the wind can also give shifts in the line profile in the lag-energy plot calculated at different frequencies. Hence we propose that the observed broad iron reverberation lags and shifts in profile as a function of frequency of variability can arise from a disc wind at fairly large distances from the X-ray source.
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