FWL, Selection and Controls

Frisch–Waugh–Lovell reveals the comparison

In

the coefficient $\hat\tau$ can be obtained in three steps:

1. regress $D$ on $X$ and keep residual $\tilde D$;
2. regress $Y$ on $X$ and keep residual $\tilde Y$;
3. regress $\tilde Y$ on $\tilde D$.

Thus adjusted OLS uses treatment variation unexplained by the controls. If $\tilde D$ is nearly zero for some profiles, the estimate relies on narrow support and numerical extrapolation.

The adjustment changes which units supply the comparison. It does not verify that $X$ is sufficient for exchangeability.

Omitted-variable bias has a direction

Suppose the true model is

but ability $A$ is omitted. If the regression of $A$ on $D$ has slope $\delta$, the short-regression coefficient is

If ability raises earnings by £4,000 per unit and scholarship recipients average 0.5 units higher, omitted ability contributes £2,000 to the raw scholarship difference. This calculation clarifies a plausible bias; it does not establish that ability is the only omission.

Precision variables versus confounders

In a randomised experiment, pre-treatment outcome predictors can reduce standard errors, but randomisation already supplies identification. In an observational design, controls must support an exchangeability argument.

Keep these roles separate:

Flexible adjustment still needs design

Interactions, splines or machine learning may represent $E[Y\mid D,X]$ better. They cannot recover counterfactual outcomes where there is no treatment overlap, and they cannot determine whether $X$ contains post-treatment variables.

Report:

• treatment propensity or overlap by important profiles;
• effective sample after missing controls;
• whether controls were defined before outcome inspection;
• sensitivity to plausible unmeasured confounding;
• a design-based estimate where available.

Quick check

A postcode control improves prediction but nearly determines scholarship receipt. Is more precise adjustment automatically better?

Next: Inference and Clustering