OLS as a Projection

Least squares answers a geometric question

For a scalar regressor $X$, the population linear projection

has slope

It is the best mean-squared-error linear summary of $Y$ using $1$ and $X$. This statement does not require the true conditional mean to be linear and does not make $X$ exogenous.

Small calculation

Here $\bar X=70$ and $\bar Y=55$. The sample slope is

Within this five-observation teaching sample, one score point is associated with about 0.67 additional credits. It is not evidence that manipulating the prior score would cause credits to rise.

Multiple regression is conditional projection

With scholarship receipt $D$ and pre-treatment controls $X$,

$\tau$ compares observations with different $D$ after removing linear differences associated with $X$. A causal interpretation requires that the residual treatment variation behaves as if unrelated to the relevant potential outcomes, plus treatment and outcome definitions, overlap and an appropriate functional form or estimand argument.

“Controlling for age and score” states a computation. It does not explain why all confounding paths are blocked.

Coefficients need units and support

Suppose log earnings are regressed on years of education:

The exact percentage difference for one additional year is $100(e^{0.08}-1)\approx8.33\%$; “8%” is the small-coefficient approximation. This remains an association unless schooling variation is credibly exogenous.

Do not extrapolate a line estimated over 11–16 years of education to 25 years merely because the formula returns a number.

Residuals expose misspecification, not causality alone

Inspect residuals against:

• fitted values and key regressors;
• time, location and treatment status;
• missingness and sample-selection indicators;
• influential observations and support boundaries.

Patterns can reveal nonlinearity, changing variance or data errors. A flat residual plot cannot prove no unmeasured confounding.

What adding a variable changes

Adding a control can change $\hat\tau$ because it:

1. removes confounded variation;
2. improves precision;
3. changes the population with complete data;
4. forces comparisons within weak-overlap cells;
5. conditions on a mediator or collider;
6. changes the functional-form approximation.

The phrase “the effect is robust to controls” is incomplete without explaining which mechanism each control addresses.

Quick check

The scholarship coefficient falls from 0.10 to 0.04 after prior score is added. Does that prove the remaining 0.04 is causal?

Next: FWL, Selection and Controls