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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.

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  • $\begingroup$ Thanks a lot for all the answers! I will be sure to mention that establishing a causal relationship is not my aim with this regression. Just wanted to make sure I didn't overlook some method to rule endogeneity out. $\endgroup$
    – rpguerrera
    Jan 31 '14 at 14:27
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Unfortunately there is no actual test for reverse causality. Your correlation test between $X$ and the residuals does not work because they are uncorrelated by construction. This does not mean that $X$ is uncorrelated with the error (note the difference between residuals and error term) because the true error is unobserved to you.

If you cannot find an instrument for $X$ you may try to find a proxy variable which does not suffer from the same reverse causality problem. Or you may try to find a variable which, once conditioned on in your regression, removes the correlation between $X$ and $u$ via the conditional independence assumption. However, it is usually equally difficult to find such variables as is finding good instruments. Researchers spend a great deal of effort to cope with these type of problems but not always is it possible and even the most clever identification strategies rest on additional assumptions.

The book by Angrist and Pischke (2009) "Mostly Harmless Econometrics" discusses many ways for causal inference which seeks to get around endogeneity issues, so you might give it a look. Depending on your particular research question and the data at hand there might be alternative solutions.

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You cannot rule out (or determine) double causation purely on the basis of regression.

See Causality for an extensive treatment of this.

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In short: There is no statistical a-priori method to identify causality in non-time-series data, and there cannot be.

Regression analysis always only measures some sort of coincidence, that is a relationship between functions of variables, whose coefficients’s interpretation is conditional on the model being true, which includes assumptions about causality.

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    $\begingroup$ What do you mean with "statistical a-priori method"? Because there are certainly ways to identify causality from purely observational data even under the absence of time. See the book linked by Peter Flom. $\endgroup$
    – ziggystar
    Jan 31 '14 at 11:48

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