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Suppose $X$ and $Y$ are perfectly correlated, and we fit a model $Y=a+bX+\epsilon$. Is it possible that there would be omitted variable bias in this situation?

Intuitively, I think so, but I'm struggling to construct an example. If it is possible, how can we construct an example of this happening?

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Yes, because omitted variable bias depends on the underlying causal question you want to ask.

Suppose you are interested in explaining the causal effect of schooling on earnings (just to give an example I have already discussed elsewhere and need not repeat here, see e.g. Is the equation "$Y=\mathbb{E}[Y|X] + error$" an identity?, Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples? Omitted variable bias: which predictors do I need to include, and why?) in \$, ($Y$).

If you now regress earnings in \$ on earnings in cents ($X$), the variables will be perfectly dependent, yet you would surely not argue that someone earns, say, 3000$ because he earns 300,000 cents. The regression still suffers from omitted variable bias when your goal is to estimate the causal effect of schooling on earnings.

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Here's a trivial example

Let's say you think that scores on a qualification exam (X) are correlated with the wage for certain city jobs (Y)

You check and find that wage and salary are perfectly correlated! Every person who got a particular score on the test has the exact same wage (and vice versa).

But actually what's going on is that you wage is determined by how much the mafia boss running the city likes you (Z).

He decides how much to pay each you based on his whim, and then rigs your test to give you a particular score based on your salary to make it look (sort of) legit.

So X and Y are perfectly correlated, but only because of a confounding variable Z, which is causing both of them.

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