A common stochastic restriction in econometric models separable in the latent variablesis the assumption of stochastic independence between the unobserved and observed exogenous variables. Both simple and composite tests of this assumption are derived from properties of independence empirical processes and the consistency of these tests is established
Brown, Donald J. and Wegkamp, Marten H., "Tests of Independence in Separable Econometric Models" (2003). Cowles Foundation Discussion Papers. 1660.