IT IS WELL KNOWN that least-squares estimates of the coefficients of a regression equation are inconsistent if any of the regressors are measured with error. The nature of these inconsistencies has been examined by Aigner [1], Blomqvist [2], Chow [3], Levi [5], McCallum [6], and Wickens [10] for the case in which a single regressor is subject to measurement error. The purpose of this study is to examine the nature of these inconsistencies when more than one variable is measured with error. We begin by reviewing the case of one variable measured with error, developing a unified treatment of issues which previously have been discussed separately. Concentrating on the case in which two regressors are measured with error, we then examine how the predictions of the one erroneously measured regressor model must be qualified when more than one regressor is subject to measurement error.

[We consider the nature of the inferences that can be made when all variables in a linear regression are measured with error. Assuming that the measurement errors are orthogonal to each other and the unobserved correctly measured regressors, we demonstrate that the true regression coefficient vector can be restricted to the convex hull of all possible regressions iff all these regressions yield coefficient vectors lying in the same orthant. Otherwise, the set of feasible coefficient vectors is unbounded. For the unbounded case, we demonstrate that prior information concerning the "seriousness" of the measurement errors in the variables can bound the feasible region. Two diagnostics are proposed to indicate the sensitivity of conventional inferences to measurement error in the regressors, and an illustrative example is presented.]