July 18, 2026  Conditional Variances 

July 18, 2026  Omitted Variable Bias

We are interested in the effect of X on Y.  Z is a variable that effects X.  Think of the error term as a bucket of everything affecting $Y$ that you haven't put into the model.

If a confounder Z is left in that bucket, and Z also affects X, then X is correlated with the error term → biased causal estimate. 

It is not enough that $Z$ merely affects $Y$. For Z to create confounding of the X-Y relationship, it needs to be associated with X through the relevant causal structure.

For example:

$$Z\to Y$$

but

$$Z\nrightarrow X$$

does not create the same confounding problem.

The key problematic structure is:

X←Z→Y​

That's called a backdoor path.

Even if Z does not directly cause X but the two are correlated, then the error term in the regression of Y on X encompasses Z.  And Z is correlated with X. 

Suppose that we regress Y on only X.  In each case above, Z is encompassed in the error term and Z is correlated with X.  So omitting Z causes bias.  

Even if there is a mediator, omitting Z causes bias.

Regress Y on X.  Think about whether the error term is correlated with X.

Step 1: What causes $Y$?

Those are candidates for being represented in u.

Step 2: Which of those causes of $Y$ are associated with $X$?

Those are the ones that can create:

$$Cov(X,u)\ne 0.$$

Step 3: Why are they associated with $X$?

This is where the DAG becomes crucial. It could be because:

$$X\leftarrow Z\to Y$$

(confounding), or because of some more complicated path.

---

5. And this explains your collider example beautifully

Your DAG:

X←U1​→Z←U2​→Y

When estimating Y on X:

• U2​ causes $Y$ → relevant component of the error.
• U1​ causes X, not Y → not itself part of the Y-equation's error.
• Z causes neither X nor Y → not a cause of Y, so not part of the structural error.
• But X and Z are associated because of U1​.

And that last fact becomes dangerous only when you condition on $Z$, because conditioning on the collider makes U1​ and U2​ associated.

So:

Don’t just ask "Is Z correlated with X?"​

Instead ask:

"What is the causal path connecting X to the determinants of Y?"​

That's the deeper causal-inference way of thinking about the error term.

And one final refinement: variables that are not causes of $Y$ can still matter for causal identification if conditioning on them opens a path, exactly as Z does in your diagram. That's why "look at causes of Y" is an excellent starting point, but the DAG/backdoor paths give the complete rule.