Detecting non-causal artifacts in multivariate linear regression models

We consider linear models where $d$ potential causes $X_1,...,X_d$ are correlated with one target quantity $Y$ and propose a method to infer whether the association is causal or whether it is an artifact caused by overfitting or hidden common causes. We employ the idea that in the former case the vector of regression coefficients has 'generic' orientation relative to the covariance matrix $\Sigma_{XX}$ of $X$. Using an ICA based model for confounding, we show that both confounding and overfitting yield regression vectors that concentrate mainly in the space of low eigenvalues of $\Sigma_{XX}$.