  Bayesian inference is used to extract unknown parameters from gravitational
wave signals. Detector noise is typically modelled as stationary, although data
from the LIGO and Virgo detectors is not stationary. We demonstrate that the
posterior of estimated waveform parameters is no longer valid under the
assumption of stationarity. We show that while the posterior is unbiased, the
errors will be under- or overestimated compared to the true posterior. A
formalism was developed to measure the effect of the mismodelling, and found
the effect of any form of non-stationarity has an effect on the results, but
are not significant in certain circumstances. We demonstrate the effect of
short-duration Gaussian noise bursts and persistent oscillatory modulation of
the noise on binary-black-hole-like signals. In the case of short signals,
non-stationarity in the data does not have a large effect on the parameter
estimation, but the errors from non-stationary data containing signals lasting
tens of seconds or longer will be several times worse than if the noise was
stationary. Accounting for this limiting factor in parameter sensitivity could
be very important for achieving accurate astronomical results, including an
estimation of the Hubble parameter. This methodology for handling the
non-stationarity will also be invaluable for analysis of waveforms that last
minutes or longer, such as those we expect to see with the Einstein Telescope.
