Partially linear models have been given increasing attention in the last two decades since Engle et al. (1986) used them to study the relationship between temperature and electricity usage in United States cities. We develop methodology for the estimation of regression parameters in partially linear models when the covariates are measured with errors or may be missing. We are particularly concerned with two cases where we observe a surrogate of the covariate. The second case focuses on the linear covariate being incompletely observable. We give the corresponding solutions for the above problems. The first solution employs the technique of correcting for attenuation. The second is proposed using inverse weight probability. The estimators given are proven to be asymptotically normal. The model is used to analyze a data set from the Framingham Heart Study for the purpose of illustrating the methods. We also investigate the semiparametric partially linear single index errors-in-variables models, for which two classes of estimators are proposed, and the corresponding theoretical properties are derived and compared.
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