Estimation of Partial Linear Error-in-Variables Models under Martingale Difference Sequence

Abstract:

      Consider the partly linear model $Y=x\beta+g(t)+e$ where the explanatory $x$ is erroneously measured, and both $t$ and the response $Y$ are measured exactly, the random error $e$ is a martingale difference sequence. Let $\widetilde{x}$ be a surrogate variable observed instead of the true $x$ in the primary survey data. Assume that in addition to the primary data set containing $N$ observations of $\{(Y_{j},\widetilde{x}_{j},t_{j})_{j=n+1}^{n+N}\}$, the independent validation data containing $n$ observations of $\{(\widetilde{x}_{j},x_{j},t_{j})_{j=1}^{n}\}$ is available. In this paper, a semiparametric method with the primary data is employed to obtain the estimator of $\beta$ and $g(\cdot)$ based on the least squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. Finite sample behavior of the estimators is investigated via simulations too.