This paper considers estimation and variable selection for multiplicative linear regression models when neither the response variable nor the covariates can be directly observed, but are distorted by unknown functions of a commonly observable confounding variable. After taking logarithmic transformation on the response variable, we propose two estimation methods for the parameter. One is the least squares estimator, the second one is the moment-based estimator linked with varying coefficient models. The third one is the least product relative error estimator without logarithmic transformation. For the hypothesis testing of parametric components, restricted estimators under the null hypothesis and test statistics are proposed. The asymptotic properties for the estimators and test statistics are established. A bootstrap procedure is proposed to calculate critical values. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.
- Distortion measurement errors
- Least product relative error estimator
- Local linear smoothing
- Variable selection
- Varying coefficient models