  The study of X-ray time-lag spectra in active galactic nuclei (AGN) is
currently an active research area, since it has the potential to illuminate the
physics and geometry of the innermost region (i.e. close to the putative
super-massive black hole) in these objects. To obtain reliable information from
these studies, the statistical properties of time-lags estimated from data must
be known as accurately as possible.
  Aims: We investigated the statistical properties of Fourier-based time-lag
estimates (i.e. based on the cross-periodogram), using evenly sampled time
series with no missing points. Our aim is to provide practical `guidelines' on
estimating time-lags that are minimally biased (i.e. whose mean is close to
their intrinsic value) and have known errors.}
  Methods: Our investigation is based on both analytical work and extensive
numerical simulations. The latter consisted of generating artificial time
series with various signal-to-noise ratios and sampling patterns/durations
similar to those offered by AGN observations with present and past X-ray
satellites. We also considered a range of different model time-lag spectra
commonly assumed in X-ray analyses of compact accreting systems.
  Results: Discrete sampling, binning and finite light curve duration cause the
mean of the time-lag estimates to have a smaller magnitude than their intrinsic
values. Smoothing (i.e. binning over consecutive frequencies) of the
cross-periodogram can add extra bias at low frequencies. The use of light
curves with low signal-to-noise ratio reduces the intrinsic coherence, and can
introduce a bias to the sample coherence, time-lag estimates, and their
predicted error.
