  Misspecified models often provide useful information about the true data
generating distribution. For example, if $y$ is a non-linear function of $x$
the least squares estimator $\hat{\beta}$ is an estimate of $\beta$, the slope
of the best linear approximation to the non-linear function. Motivated by
problems in astronomy, we study how to incorporate observation measurement
error variances into fitting parameters of misspecified models. Our asymptotic
theory focuses on the particular case of linear regression where often weighted
least squares procedures are used to account for heteroskedasticity. We find
that when the response is a non-linear function of the independent variable,
the standard procedure of weighting by the inverse of the observation variances
can be counter-productive. In particular, ordinary least squares may have lower
asymptotic variance. We construct an adaptive estimator which has lower
asymptotic variance than either OLS or standard WLS. We demonstrate our theory
in a small simulation and apply these ideas to the problem of estimating the
period of a periodic function using a sinusoidal model.
