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Abstract: We propose robust tests for causal effects on potentially high-dimensional outcomes where there may be selection into treatment. These tests measure conditional independence of outcomes from treatment assignment by asking whether treatment can be predicted from outcomes more accurately than from covariates alone. We design three such tests of conditional independence. First, we propose a conditional randomization test based on an inverse propensity weighted mean squared error test statistic. If the propensity score is known, the test controls size in finite samples. Second, we propose a double-residualization approach, where we partial out covariate effects from both treatment and outcome-based treatment predictions, and test whether the correlation between the residuals is zero. Third, we propose a test based on an augmented inverse propensity weighted estimator built around doubly robust, Neyman-orthogonal population moments that are necessary and sufficient for conditional independence. Simulations show that all three tests improve power over a joint Wald test based on a seemingly unrelated regression across many non-linear settings, particularly under heavy tails, outliers, mixtures, heterogeneous effects, and effects with opposite signs.