Decomposition of Variables and Correlated Measurement Errors

Abstract

This paper examines the bias in the OLS estimators when the regressors have measurement errors correlated in a particular manner. When a variable is decomposed into two components but only one of them is observed with error, the induced measurement error in the other component is identical but has the opposite sign. This specific correlation pattern enables us to assess the direction of the bias in the OLS estimators from observed data. In the standard EIV case this would require knowledge of the relative variances of the measurement errors. Examples of this type of decomposition in applied work are presented. Regression models that allow for measurement errors in the explanatory variables usually assume that these errors are uncorrelated among them.2 This is, perhaps, the natural assumption when nothing is known about the data process generating the regressors. In this paper it is argued that in empirical work there are many cases where, due to a trivial manipulation of the data, this assumption cannot be invoked. The bias in the OLS estimators when the explanatory variables are measured with correlated errors is examined. The results both complement and differ from the standard case of uncorrelated measurement errors in interesting ways. Specifically, the case that motivates the paper is one in which an original explanatory variable, X, is decomposed into two (or more) components because one is interested in testing whether these components have different effects on the dependent variable. This case occurs frequently in applied work. In addition, it is common to have data on the total variable X and on only one of its components; the other is computed as a residual. Provided that X is measured without error, any measurement error in one of its components is transmitted to the second compo