This paper investigates the problem of testing conditional independence of Y and Z given λθ(X) for some unknown θ ∈ Θ ⊂ Rd, for a parametric function λè(·). For instance, such a problem is relevant in recent literatures of heterogeneous treatment effects and contract theory. First, this paper finds that using Rosenblatt transforms in a certain way, we can construct a class of tests that are asymptotically pivotal and asymptotically unbiased against √n-converging Pitman local alternatives. The asymptotic pivotalness is convenient especially because the asymptotic critical values remain invariant over different estimators of the unknown parameter θ. Even when tests are asymptotically pivotal, however, it is often the case that simulation methods to obtain asymptotic critical values are yet unavailable or complicated, and hence this paper suggests a simple wild bootstrap procedure. A special case of the proposed testing framework is to test the presence of quantile treatment effects in a program evaluation data set. Using the JTPA training data set, we investigate the validity of nonexperimental procedures for inferences about quantile treatment effects of the job training program.