Probability inequalities in multivariate distributions

For a bivariate binary response model y, = 1 (xj βj+j > 0), j=1,2, we propose to estimate nonpararnetrically the quadrant correlation E{sgn(u1) *sgn(u2)} between the two error terms ul and u2 without specifjing the error term distribution. The quadrant correlation accounts for the relationship between yl and y2 that is not explained by xl and x2, and can be used in testing for the specification of endogenous dummy variable models. The quadrant correlation is further generalized into orthant dependence allowing unknown regression functions, unknown error term distribution and arbitrary forms of heteroskedasticity. A simulation study is provided, followed by a brief application to a real data set.