Confounding · omitted variables
Build-a-Bias
Make coffee look productive. The truth stays put.
Coffee gets the credit.
Let U be an unobserved productivity trait. U can affect both coffee consumption X and earnings Y. How far can the naive slope drift from the true coffee effect?
X = aU + V
Y = βX + bU + ε
Y = βX + bU + ε
U, V and ε are mutually independent, mean zero, variance one. No sampling noise is shown.
True causal slope0.00
Naive regression slope1.00
Omitted-variable bias1.00
Check the algebra.
The naive population slope is β + b·Cov(X,U)/Var(X) = β + ba/(a² + 1). Holding β fixed while changing a or b changes the bias. With a = 0 or b = 0, this particular omitted-variable bias vanishes.
If U were observed and appropriately controlled, the population coefficient on X would recover β in this model. Actual studies require a defensible adjustment set and data; adding arbitrary controls is no guarantee.
Read the method: Nick Huntington-Klein · The Effect, ch. 13: Regression ↗