Quantile Uncorrelation and Instrumental Regressions

We introduce a notion of median uncorrelation that is a natural extension of mean (linear) uncorrelation. A scalar random variable Y is median uncorrelated with a k-dimensional random vector X if and only if the slope from an LAD regression of Y on X is zero. Using this simple definition, we characterize properties of median uncorrelated random variables, and introduce a notion of multivariate median uncorrelation. We provide measures of median uncorrelation that are similar to the linear correlation coefficient and the coefficient of determination. We also extend this median uncorrelation to other loss functions. As two stage least squares exploits mean uncorrelation between an instrument vector and the error to derive consistent estimators for parameters in linear regressions with endogenous regressors, the main result of this paper shows how a median uncorrelation assumption between an instrument vector and the error can similarly be used to derive consistent estimators in these linear models with endogenous regressors. We also show how median uncorrelation can be used in linear panel models with quantile restrictions and in linear models with measurement errors. Date: September 9, 2010. 1 London School of Economics and Political Science, Department of Economics, Houghton Street, London WC2A 2AE, England. Tel.: +44-(0)20-7852-3707; fax: + 44-(0)20-7955-6592. E-mail: [email protected]. 2 Northwestern University, Department of Statistics. Support from the National Science Foundation is gratefully acknowledged. 3 Northwestern University, Department of Economics. Partial support from the National Science Foundation is gratefully acknowledged. ∗ We thank the seminar participants at the London School of Economics, the University College London and the University of Toronto for their comments. We also appreciate feedback from the participants of the Canadian Econometric Study Group and the participants of the all UC Econometrics Conference. 1