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

The Identification Power of Equilibrium in Simple Games

We examine the identification power that (Nash) equilibrium assumptions play in conducting inference about parameters in some simple games. We focus on three static games in which we drop the Nash equilibrium assumption and instead use rationalizability as the basis for strategic play. The first example examines a bivariate discrete game with complete information of the kind studied in entry models. The second example considers the incomplete-information version of the discrete bivariate game. Finally, the third example considers a first-price auction with independent private values. In each example, we study the inferential question of what can be learned about the parameter of interest using a random sample of observations, under level-k rationality, where k is an integer ≥ 1. As k increases, our identified set shrinks, limiting to the identified set under full rationality or rationalizability (as k → ∞). This is related to the concepts of iterated dominance and higher-order beliefs, which are incorporated into the econometric analysis in our framework. We are then able to categorize what can be learned about the parameters in a model under various maintained levels of rationality, highlighting the roles of different assumptions. We provide constructive identification results that lead naturally to consistent estimators.