# Rule of thumb to rule out reverse causality in the OLS model

Let' say I have a regression model:

$y=a+b*x+error$

Suppose $x$ is income and $y$ is consumption. The hypothesis is that higher income leads to higher consumption and hence, the coefficient on $x$ should be positive, other things remaining the same.Let's also say the estimated coefficient is 0.60. This model obviously suffers from omitted variable bias. Please ignore this issue. My question:

a) Does this model suffer from reverse causality? In other words, is it the case that the relationship is because higher consumption is driving down income? My first guess is that this is not the case because coefficient is positive which means correlation between income and consumption is positive. See here.

b) Given (a), can I use this as a rule-of-thumb to rule out the reverse causality in this case? Is this generalizable to other cases with two variables?

Thanks. P.S. One can also just use correlation rather than running a simple regression as mentioned earlier.

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In an economic sense, $y \equiv x$, at least if $y$ is total consumption. Even if $y$ is partial consumption, I would be worried about endogeneity resulting from simultaneity. Is there an instrument you could use instead of income? – gregmacfarlane Aug 15 '14 at 16:30
Thanks gmacfarlane. Are you implying that the estimated coefficient can not be used as a rule-of thumb to detect endogeneity? – Metrics Aug 15 '14 at 16:49
Yes, the coefficient value has no relationship with the correlation between $x$ and the error terms. – gregmacfarlane Aug 15 '14 at 23:42

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.

• 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”.

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+1 your answer came as I was typing my comment. – gregmacfarlane Aug 15 '14 at 16:31
Thanks Dan. But, my question was not about solving the problem. My question was, whether the simple correlation or simple regression can be used as a rule-of-thumb to rule out reverse-causality. – Metrics Aug 15 '14 at 16:44
Sorry. Didn't notice. Look at the edited answer. There is no rule of thumb but you can test for simultaneity using the Hausman test! – Plissken Aug 15 '14 at 17:02
Thanks Dan for the answer. – Metrics Aug 17 '14 at 15:35