Bounding the effects of proxy variables on instrumental-variables coefficients

Abstract We consider instrumental-variables models in which one of the observed variables is a proxy for some unobserved ‘true’ variable. We seek the smallest correlation between the proxy and the unobserved true variable sufficient to guarantee that, regardless of any other correlations involving unobserved variables or disturbances, a coefficient's sign has not been reversed by substituting the proxy. We derive exact solutions when the proxy is the dependent variable, the endogenous explanatory variable whose coefficient is of concern, some other endogenous explanatory variable, or a solely instrumental variable. For an exogenous explanatory proxy, we derive either exact solutions or useful lower bounds.