Bounding the Effects of Proxy Variables on Regression Coefficients

We consider a regression in which one of the observed variables is a proxy for some unobserved "true" variable. Given a lower bound for the correlation between the proxy and the unobserved true variable for which it substitutes, we derive intervals in which the coefficients of the unobserved true regression must lie, regardless of any other correlations involving unobserved variables or disturbances. We present a simple solution for the important special case in which only the signs of the coefficients are of concern and one seeks the smallest correlation between the proxy and the true variable that guarantees the correctness of the signs of the coefficients in the proxy regression. We also present an algorithm for extending these results to the multiple-proxy problem.