The measurement errors problem is endemic in many econometric studies, and one of the oldest known statistical problems. Instrumental variable (IV) method is one of the popular solutions adopted to deal with the mismeasured variables in statistical and econometric analyses. This paper proposes an efficient IV estimator to the parameters of the simple regression model where both variables are subject to measurement errors. The proposed IV is defined using simple mathematical transformation of the manifest independent variable (mismeasured variable). The proposed method is straightforward, and easy to implement. The theoretical superiority of the proposed estimator over the existing IV based estimators due to Wald (1940), Bartlett (1949), and Durbin (1954) is established by analytical comparison and geometric expositions. Simulation based numerical comparisons of the proposed estimator with four different existing estimators are also included.

This paper proposes an estimation method based on the reflection of the (manifest) explanatory variable to estimate the parameters of a simple linear regression model when both response and explanatory variables are subject to measurement error (ME). The reflection method (RM) uses all observed data points, and does not exclude or ignore part of the data or replace them by their ranks. The RM is straightforward, and easy to implement. We show that the RM is equivalent or asymptotically equivalent to the orthogonal regression (OR) method. Simulation studies show that the RM produces estimators that are nearly asymptotically unbiased and efficient under the assumption that the ratio of the error variances equals one. Moreover, it allows to define the sum of squares of errors uniquely, the same way as in the case of no measurement error. Simulation based numerical comparisons of the RM with the ordinary least square (OLS) and OR methods are also included.