We study equilibria in second price auctions when bidders are independently and privately informed about both their values and participation costs and their joint distributions across bidders are not necessarily identical. We show that there always exists an equilibrium in this general setting...

This paper develops a general framework for conducting inference on the rank of an unknown matrix Π0. A defining feature of our setup is the null hypothesis of the form . The problem is of first-order importance because the previous literature focuses on by implicitly assuming away , which...