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?
Solved – How to test reverse causality
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@John is correct, but, in addition you cannot prove causation with any experimental design: You can only have weaker or stronger evidence of causality.
In any study, but especially in an observational study, evidence for causality is increased by including relevant covariates, giving a scientifically plausible causal path, replicating results and so on.
However, even in the best experimental design, you don't prove causality.
As for t-tests vs. regression - your friend does not know what he/she is talking about. T-tests results can be duplicated exactly with regression procedures: Just use a single independent variable that is dichotomous.
This is quite a well known issue in economics and I would say that you would most likely have to estimate your equation by IV rather than OLS since your error term will most likely be correlated with your regressors, i.e. $E\left(x_{i}error_{i}\right)\neq0 $. In other words you'll have an endogeneity problem due to simultaneity bias ( http://en.wikipedia.org/wiki/Endogeneity_(applied_statistics) ). Often lagged values of the regressors are useable as instruments. There are of course other options in order to estimate the relationship at hand! Note that if your coefficient is 0.60 or not does not play any role since your estimates will be biased and inconsistent in case you have a simultaneity problem which you do.
Answering your questions step-by-step:
Does this model suffer from reverse causality? Yes it does as explained above.
In other words, is it the case that the relationship is because higher consumption is driving down income? Income affects consumption and consumption affects income as is known from economic theory.
Can I use this as a rule-of-thumb to rule out the reverse causality in this case? No. This is because your estimates are inconsistent and biased.
Is this generalizable to other cases with two variables? No it is not as it really depends on each case. There is no rule of thumb in this case other than using the Hausman test to test for simultaneity bias. What we are testing is whether or not the OLS estimates are consistent or not.
Try to look at: Campbell, John Y. and Mankiw, N. Gregory. 1989. “Consumption, Income and Interest Rates: Reinterpreting the Time Series Evidence”.
Best Answer
True underlying causality is very difficult to test, this being said two of the most used tests for causality are:
$$y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \cdots + a_my_{t-m} + b_px_{t-p} + \cdots + b_qx_{t-q} + u_t$$
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.
$$H=(\beta_{OLS}-\beta_{IV})'\big(\operatorname{Var}(\beta_{IV})-\operatorname{Var}(\beta_{OLS})\big)^{-1}(\beta_{OLS}-\beta_{IV})$$
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.