Causality, omitted variable bias

This might be a basic question, but I want to be sure that what I'm doing is right. I have a model that suggests that variable X causes both Y and Z. When I regress Y on X, or Z on X, I get positive and significant coefficients as expected.

Now, when I regress Z on Y, I still get a positive significant coefficient.

Question 1: is this an omitted variable bias?

Question 2: is it legitimate to regress Z on Y and X to test whether the relationship between Z and Y is spurious?

Question 3: if it is legitimate and if I get positive significant coefficients on both Y and X what does that mean? Does it mean "X causes Y and Z, but Y still has marginal explanatory power on Z"?

Many thanks, Dave

You need to distinguish the causal graph from the regression coefficients here. Something is only 'spurious' if it does not identify the causal effect of interest, and this depends on the graph structure you have assumed, not on any regression coefficients.

As an example (and restricting ourselves to causal DAG structures with no hidden variables) assume X causes Y and X causes Z. Then even if Z does not cause Y you will be able to regress the Y on Z and get a non-zero coefficient, so that doesn't tell you much. Conditioning on X in a regression of Y on Z is the right thing to do if you want to know what the causal effect of Z is on Y assuming that X causes both Y and Z and that Z causes Y rather than vice versa. If, on the other hand, Y causes Z, then despite there being no causal effect to estimate you will again get a non-zero regression coefficient.

It all depends on which variables are connected by causal arrows and which direction those arrows point. It's sometimes useful to simulate data with the relevant structure and run the regressions to get a feel for what can happen.

There are some situations where causal structure can be inferred from regressing things on other things and finding zero coefficients, but they are fairly limited. A nice overview can be found in chapter 25 of Shalizi's draft textbook (ch.21-24 are also worth reading). Leaving aside discovery, the basic theoretical framework can be found in compressed form in Pearl's review paper, and as a more leisurely exposition in the references here.

Unfortunately this means that the answer to each of your three questions is "it depends" (on the graph), but the references above should hopefully point you towards what you would have to assume to interpret things they way you're considering.

• Many thanks. My model suggests that $X$ causes $Y$ and $Z$, and I don't know whether $Y$ causes $Z$. I wasn't sure whether I could regress $Z$ on $Y$ and $X$, since $X$ causes $Y$. The way I read your answer is that, this is ok. The estimates from this regression suggests $Z$ is related to both $Y$ and $X$. Since, my model predicts that $X$ causes $Z$, I take the significant coefficient of $X$ as consistent with this prediction. Finally, if I get you right, the significant coefficient of $Y$ suggests a relationship, but the direction of the causality (if there is) is not known. – Dave Mar 5 '13 at 13:33
• Pretty much. If you maintain the assumptions that X causes Y, X causes Z and the direction of possible causality is from Y to Z then (in the absence of any paths connecting Y to Z except via common effects), conditioning on X identifies the causal effect. That is, the coefficient on Y in a regression of Z on Y and X is interpretable as the causal effect of Y on Z. There are more caveats here, but that's the gist of it. (Note that if Y actually caused X and X caused Z - just one arrow flip - then this would not work and you'd want to leave X out of the regression altogether!) – conjugateprior Mar 5 '13 at 20:31

I think that causality is about some underlying process for how something "works". Regression is much more to do with association and correlation. So you need much more than significant coefficients to claim that X "causes" anything. You need to be able to use your theory to make quantitative and/or qualitative predictions about what happens to Y and Z when X varies. If you can't do this without using your data, then causality is out of reach. The data can only tell you whether or not your predictions are accurate. This means that to get a definitive answer on causality, you really need to be able to make sharp predictions.

In terms of your specific questions, if X causes both Z and Y, then we can infer that Z and Y are both related to each other via the inference path $Y\to X \to Z$. For example, if $Y\approx X\beta$ and $Z\approx X\alpha$ means that $Z\approx Y\frac{\alpha}{\beta}$. This is the omitted variable problem. As far as testing significance goes, I think you should really be going back to whatever theory you've obtained your causal hypothesis from and see ask those same three questions of your theory. For example could the causal sequence go X causes Y and Z, but Y also causes Z. Could Z or Y feedback onto X in a causal cycle? Its all about testing the predictions made from your theory. I think a legitimate test is one that has meaning in terms of your theory.

Forgive the assumption on my part, but your question is a bit vague/abstract, and it sounds more like you've found associations and/or relationships that "make sense" after looking at the estimates. While this is a good way to validate your regression results, you need to give evidence of being able to predict the result from the theory without using the data. Otherwise I think you are simply confusing association with causation.

• Very useful comments, thanks. The theory predicts that $\beta>0$ and $\alpha>0$. So, I was expecting to get these results before the analysis. $Y$ is determined after $X$, and $Z$ after $Y$, which rules out the possibility of a feedback onto $X$. I guess what I'm really trying to understand is the following: Can a regression model have on the right hand side two variables one of which causes the other? Or would that be econometrically problematic? – Dave Mar 5 '13 at 13:05