  Context: Theoretical modelling of time-lags between variations in the Fe
K$\alpha$ emission and the X-ray continuum might shed light on the physics and
geometry of the X-ray emitting region in active galaxies (AGN) and X-ray
binaries. We here present the results from a systematic analysis of time-lags
between variations in two energy bands ($5-7$ vs $2-4\,\mathrm{keV}$) for seven
X-ray bright and variable AGN.
  Aims: We estimate time-lags as accurately as possible and fit them with
theoretical models in the context of the lamp-post geometry. We also constrain
the geometry of the X-ray emitting region in AGN.
  Methods: We used all available archival \textit{XMM-Newton} data for the
sources in our sample and extracted light curves in the $5-7$ and
$2-4\,\mathrm{keV}$ energy bands. We used these light curves and applied a
thoroughly tested (through extensive numerical simulations) recipe to estimate
time-lags that have minimal bias, approximately follow a Gaussian distribution,
and have known errors. Using traditional $\chi^2$ minimisation techniques, we
then fitted the observed time-lags with two different models: a
phenomenological model where the time-lags have a power-law dependence on
frequency, and a physical model, using the reverberation time-lags expected in
the lamp-post geometry. The latter were computed assuming a point-like primary
X-ray source above a black hole surrounded by a neutral and prograde accretion
disc with solar iron abundance. We took all relativistic effects into account
for various X-ray source heights, inclination angles, and black hole spin
values.
