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Building regression models to test the impact of X on Y, but sometimes there is reserve causality between X on Y, which is Y may also have an impact on X. How can we test this hypothesis?

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  • $\begingroup$ maybe there's something I'm missing but it sounds to me like you could do a granger causality test in the opposite direction. $\endgroup$
    – mlofton
    Dec 21 '19 at 22:57
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True underlying causality is very difficult to test, this being said two of the most used tests for causality are:

  1. Granger causality test, as mlofton pointed out. A Granger causality test is based on auxiliary (vector) autoregression of following form (here x is the variable for which you test causality):

$$y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \cdots + a_my_{t-m} + b_px_{t-p} + \cdots + b_qx_{t-q} + u_t$$

The null hypothesis here is that $x$ does not Granger-cause $y$, which is tested based on the joint significance of the lagged coefficients of $x$.

The test basically tries to see if past values of $x$ have any explanatory power on $y$ and to check for a causality that goes other way you can just exchange the role of $x$ and $y$.

The downsides of this test are that it tests for Granger-causality which is weaker concept than the "true" causality. It requires series to be stationary, and also for the test to have good power its preferable to have data with decent number of time periods which is not always easy to get.

  1. Durbin-Wu-Hausman endogeneity test. This test is based on comparison of the coefficients from two models, most commonly OLS and IV. If we denote the coefficient from OLS as $\beta_{OLS}$ and IV as $\beta_{IV}$ then after estimating auxiliary regressions to get the coefficients the test statistics would be:

$$H=(\beta_{OLS}-\beta_{IV})'\big(\operatorname{Var}(\beta_{IV})-\operatorname{Var}(\beta_{OLS})\big)^{-1}(\beta_{OLS}-\beta_{IV})$$

which has $\chi^2$ distribution where degrees of freedom are given by the rank of the matrix in the middle of the expression. Here the null hypothesis is that both coefficients are consistent but OLS is more efficient. And alternative hypothesis is that OLS is inconsistent and IV consistent.

However, a downside of this test is that to even perform it you first have to find some good instruments as if you dont have valid instruments IV will be inconsistent due to that reason, not due to the presence of endogeneity. Also endogeneity could be caused not just by reverse causality but also by measurement error or omitted variables. So unless you are confident that you have good instruments and that the models are not misspecified this test wont tell you much.

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    $\begingroup$ Thank you very much for your answers. I greatly appreciate it. $\endgroup$
    – Amy
    Dec 23 '19 at 8:16

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