Probability inequalities in multivariate distributions

For a bivariate binary response model y, = 1 (x^j β_j+^j > 0), j=1,2, we propose to estimate nonpararnetrically the quadrant correlation E{sgn(u₁) ^*sgn(u₂)} between the two error terms u_l and u₂ without specifjing the error term distribution. The quadrant correlation accounts for the relationship between y_l and y₂ that is not explained by x_l 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.