Does throwing in an irrelevant regressor bias anything?

Question

Omitting a relevant variable biases your coefficients. The natural follow-up: what if you overcorrect and toss in an extra regressor $z$ that has no true effect on $y$ ( $\beta_z = 0$ )?

Does including an irrelevant variable bias your estimates? What is the cost, if any?

Accepted Answer

No bias. OLS stays unbiased and consistent when you include an irrelevant regressor. The bias formula needs a nonzero coefficient on the extra variable: bias \propto \betaz, and here \betaz = 0, so there is nothing for the other coefficients to absorb. Including a variable that genuinely belongs at zero simply estimates its coefficient near zero and leaves the rest alone in expectation. The cost is variance. Adding a regressor that is correlated with your real predictors inflates the standard errors of the coefficients you care about. In the two-regressor case, the variance of the slope on x …

Does throwing in an irrelevant regressor bias anything?

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