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

A Note on Parameter Estimation for Misspecified Regression Models with Heteroskedastic Errors