  This paper shows that testability of reverse causality is possible even in
the absence of exogenous variation, such as in the form of instrumental
variables. Instead of relying on exogenous variation, we achieve testability by
imposing relatively weak model restrictions and exploiting that a dependence of
residual and purported cause is informative about the causal direction. Our
main assumption is that the true functional relationship is nonlinear and that
error terms are additively separable. We extend previous results by
incorporating control variables and allowing heteroskedastic errors. We build
on reproducing kernel Hilbert space (RKHS) embeddings of probability
distributions to test conditional independence and demonstrate the efficacy in
detecting the causal direction in both Monte Carlo simulations and an
application to German survey data.
