I have a model where I suspect (purely on theoretical grounds) that double causation might be an issue. How can I test this hypothesis?
I.e. I have something like
$Y_i = \beta_0 + \beta_1 X_i + \beta_2 W_i + u_i$.
And I suspect that $X_i$ may also be causing $Y_i$.
What I did is simply calculate the correlation between the covariate in question ($X_i$) and the error term in the regression ($u_i$). It is basically zero (-3.946634e-18). Can I then conclude that there is no double causation? Or is there a more formal way of doing this? Note: I don't think I can get an IV for $X_i$, so any test involving an IV seems infeasible.