Quantile Uncorrelation and Instrumental Regression

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 kdimensionalrandom vector X if and only if the slope from an LAD regression of Yon X is zero. Using this simple definition, we characterize properties of medianuncorrelated random variables, and introduce a notion of multivariate medianuncorrelation. We provide measures of median uncorrelation that are similar to thelinear correlation coefficient and the coefficient of determination. We also extendthis median uncorrelation to other loss functions. As two stage least squaresexploits mean uncorrelation between an instrument vector and the error to deriveconsistent estimators for parameters in linear regressions with endogenousregressors, the main result of this paper shows how a median uncorrelationassumption between an instrument vector and the error can similarly be used toderive consistent estimators in these linear models with endogenous regressors.We also show how median uncorrelation can be used in linear panel models withquantile restrictions and in linear models with measurement errors.