  The iron K$\alpha$ line commonly observed in the X-ray spectrum of both
stellar-mass and supermassive black hole candidates is produced by the
illumination of a cold accretion disk by a hot corona. In this framework, the
activation of a new flaring region in the hot corona imprints a time variation
on the iron line spectrum. Future X-ray facilities with high time resolution
and large effective areas may be able to measure the so-called 2-dimensional
transfer function; that is, the iron line profile detected by a distant
observer as a function of time in response to an instantaneous flare from the
X-ray primary source. This work is a preliminary study to determine if and how
such a technique can provide more information about the spacetime geometry
around the compact object than the already possible measurements of the
time-integrated iron line profile. Within our simplified model, we find that a
measurement of iron line reverberation can improve constraints appreciably
given a sufficiently strong signal, though that most of the information is
present in the time-integrated spectrum. Our aim is to test the Kerr metric. We
find that current X-ray facilities and data are unable to provide strong tests
of the Kerr nature of supermassive black hole candidates. We consider an
optimistic case of $10^5$ iron line photons from a next-generation data set.
With such data, the reverberation model improves upon the spectral constraint
by an order of magnitude.
