  We explore the systematic uncertainties of using Type Ia supernovae (SNe Ia)
as cosmological probes, using the Supernova Legacy Survey Three Year data
(SNLS3). We focus on studying the possible evolution of the stretch-luminosity
parameter $\alpha$ and the color-luminosity parameter $\beta$, by allowing
$\alpha$ and $\beta$ to be function of redshift, $z$. We find no evidence for
the redshift evolution of $\alpha$. We find that without flux-averaging SNe,
$\beta$ is consistent with being a constant when only statistical uncertainties
are included, but it increases significantly with $z$ when systematic
uncertainties are also included. The evolution of $\beta$ becomes marginal when
all the SNe are flux-averaged, and $\beta$ is consistent with being a constant
when only SNe at $z\ge 0.04$ are flux-averaged. Our results are insensitive to
the lightcurve fitter used to derive the SNLS3 sample, or the functional form
of $\alpha(z)$ and $\beta(z)$ assumed. It is likely that the apparent evolution
of $\beta$ with $z$ for SNe without flux-averaging is a consequence of unknown
systematic effects; flux-averaging reduces the impact of these effects by
averaging them within each redshift bin. Assuming constant $\alpha$ and
$\beta$, we find that the flux-averaging of SNe has a significant impact on the
distance-redshift relation.
