Taxonomy of Control Variables

Section one

Key concepts

The classification and the four resulting models, each stated as a claim.

Key concepts

A good estimate of program impact is accurate and precise.

Accurate (“unbiased”) = no omitted variable bias
Precise (“efficient”) = small standard errors

These are separate properties with separate causes, and different control variables buy you different ones.

The full model has one policy variable and two kinds of control.

Y = B₀ + B₁X₁ + B₂X₂ + B₃X₃ + e

X₁ = the policy variable; B₁ = program impact
X₂ = a control correlated with X₁ — a competing hypothesis
X₃ = a control uncorrelated with X₁ — a pure predictor of Y

Type A: the control is uncorrelated with the policy variable.

Explains extra Y
Smaller standard errors
More precise estimates

Type B: the control is correlated with the policy variable.

Removes bias from B₁
More accurate estimates

Dropping each type produces a two-by-two.

	X₃ included	X₃ omitted
X₂ included	Unbiased & Precise	Unbiased & Imprecise
X₂ omitted	Biased & Precise	Biased & Imprecise

A — y = b₀ + b₁x₁ + b₂x₂ + b₃x₃ — unbiased and precise
B — y = b₀ + b₁x₁ + b₂x₂ — unbiased, but large standard errors
C — y = b₀ + b₁x₁ + b₃x₃ — biased, with false confidence
D — y = b₀ + b₁x₁ — biased and imprecise

Section two

Key relationships

The mental map. How the pieces connect, and how to read the deck.

Two jobs, two kinds of variable

This deck is a short one and it does one thing: it sorts control variables by which quality problem they solve.

A control that is correlated with your treatment is a competing hypothesis. Including it is how you rule out the rival explanation, which is what makes your estimate accurate.
A control that is uncorrelated with your treatment but predicts the outcome is a noise reducer. Including it is how you shrink the residual, which is what makes your estimate precise.

The classification depends on the control’s relationship to X₁, not to Y. Both types predict Y — that is what makes them controls at all. What separates them is whether they also move with your treatment.

Case C is the one to be afraid of

Three of the four cells are honest failures. Case C is a dishonest one.

If you include the noise-reducing control and omit the competing hypothesis, you get tight confidence intervals around a biased estimate. The model looks excellent — small standard errors, strong significance, high R² — and the number it is confident about is wrong.

The deck says this plainly: the uncorrelated control results in small standard error, which can give false confidence when the absence of X₂ results in omitted variable bias.

Compare case D, the naïve model. It is also biased, but its standard errors are large, so it advertises its own uncertainty. Adding a type-A control to a mis-specified model does not fix anything — it just removes the warning label.

This is the practical reason R² and significance are poor guides to model quality. Neither one can see bias.

Precision is cheap; accuracy is a claim about the world

There is an asymmetry between the two fixes that is worth naming.

Adding a type-A control is a technical improvement. You can hunt for such variables in your data, add them, and read the standard errors to see if it worked. Nothing about your causal argument changes.

Adding a type-B control is a substantive claim. You are asserting that this particular alternative explanation exists and needs to be ruled out. No diagnostic in the regression output will tell you which competing hypotheses you have forgotten — that comes from theory and knowledge of the setting.

Which is why the exam question at the end of the deck is about matching diagrams to descriptions rather than computing anything. The skill being tested is reading a specification, not running one.

How this deck sits between p-06 and p-07

p-06 showed what happens to slopes and standard errors when you add each type of control.
p-07 showed what happens to bias when you omit one.
This deck puts both moves in one grid and names the resulting model quality.

If you understood those two lectures, this one should feel like a summary. If it does not, the grid is the place to start over.

How to read the slides

Slides	What they are doing	What to take away
1	The two quality criteria	Accurate and precise are different goals.
2	The taxonomy	Type A buys precision; Type B buys accuracy.
3	Four candidate models	Try to rank them before turning the page.
4–8	Each case explained	Note that case C is called out as “complicated.”
9, 11	Exam-style matching	Class size, SES, teacher quality. Do these.
10	The two-by-two	The summary slide. Four cells, four verdicts.

Section three

What should be clear in my mind?

Four things to be able to answer without notes.

What makes a regression estimate good. Two things, independently: unbiased (no omitted variable bias) and efficient (small standard errors).
Which control buys which. Correlated with the treatment → removes bias. Uncorrelated with the treatment → reduces standard errors.
How to classify a control. Look at its relationship to your policy variable. Its relationship to Y is what makes it a control; its relationship to X₁ is what makes it one type or the other.
Why “biased and precise” is the dangerous cell. Small standard errors around the wrong estimate look like a good model and are not.

Key takeaways

Accuracy and precision are separate goals with separate remedies.
Type A (uncorrelated) controls buy precision; Type B (correlated) controls buy accuracy.
The classification is by correlation with the treatment, not with the outcome.
Biased & precise is worse than biased & imprecise, because it hides its own error behind tight intervals.
No regression diagnostic detects a missing competing hypothesis. That judgement comes from theory.
Adding noise-reducing controls to a mis-specified model makes it look better without making it better.