Estimation in a semiparametric partially linear errors-in-variables model

Abstract

We consider the partially linear model relating a response $Y$ to predictors ($X, T$) with mean function $X^{\top}\beta + g(T)$ when the $X$’s are measured with additive error. The semiparametric likelihood estimate of Severini and Staniswalis leads to biased estimates of both the parameter $\beta$ and the function $g(\cdot)$ when measurement error is ignored. We derive a simple modification of their estimator which is a semiparametric version of the usual parametric correction for attenuation. The resulting estimator of $\beta$ is shown to be consistent and its asymptotic distribution theory is derived. Consistent standard error estimates using sandwich-type ideas are also developed.