Omitted Variable Bias

Section one

Key concepts

The models, the formula, and the two cases, each stated as a claim.

Key concepts

Greek letters mean the truth; Latin letters mean your best guess.

Full model — every relevant variable is included, so the slopes are correct: β₁
Naïve model — variables are missing, so the slopes may be biased: b₁

Note the mapping the deck makes explicit: full model plays the role of a population statistic and naïve model the role of a sample statistic. But this is not sampling error — you can have the entire population and still be wrong if a variable is missing.

The main question.

If we have an omitted variable, will our estimate of program impact (b₁) sufficiently represent the true program impact (β₁)?

We will ALWAYS have omitted variables in observational studies. The real question is not whether it is there, but how much will it affect our estimates?

The three models.

Y = β₀ + β₁X₁ + β₂X₂ + ε₁

Y = b₀ + b₁X₁ + e

X₂ = α₀ + α₁X₁ + ε₂

The auxiliary regression puts the omitted variable on the left-hand side.

The take-away, in three lines.

bias = β₂α₁ — the product of two slopes: X₁ → X₂ and X₂ → Y
b₁ = β₁ + bias — the naïve slope is the true slope plus bias
slope = cov(x, y) / var(x) — the sign of a slope always follows the sign of the covariance

Bias is the indirect effect.

b₁ = direct effect + indirect effect
β₁ = direct effect
bias = b₁ − β₁ = indirect effect

The naïve slope absorbs everything that reaches Y through X₁, including the part that really belongs to X₂.

Two cases.

Case 1 — omitted variable correlated with the policy variable. The circles overlap; the shared region is discarded by the regression. b₁ ≈ A+B, β₁ ≈ A, so bias ≈ B. Biased.
Case 2 — omitted variable uncorrelated with the policy variable. No overlap. b₁ ≈ A and β₁ ≈ A, so bias = 0 · β₂ = 0. Unbiased.

A worked example.

naïve: b₁ = −0.433
full: β₁ = −0.377
auxiliary: α₁ = −0.0099, and β₂ = 5.65

5.65 · −0.0099 = −0.056, and −0.433 − (−0.377) = −0.056. The two routes agree. In the multi-model table the bias reaches 51% — the naïve model overestimates program impact by half.

Section two

Key relationships

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

Bias needs two things to be true at once

This is the whole lecture compressed into one line:

bias = α₁ · β₂

A product is zero if either factor is zero, so omitted variable bias requires both:

α₁ ≠ 0 — the omitted variable is correlated with your policy variable
β₂ ≠ 0 — the omitted variable actually affects Y

Fail either condition and the omission is harmless. This is why the deck can be relaxed about the fact that observational studies always omit something: most omitted variables are irrelevant on at least one of the two channels.

It also tells you what to worry about. The dangerous omitted variable is not the one most strongly related to your outcome — it is the one related to your outcome and to your treatment.

You can sign the bias without measuring anything

Because bias is a product of two slopes, its direction is the product of two signs. That is a genuinely useful skill, and it works even when the omitted variable is unmeasurable:

α₁ (X₁→X₂)	β₂ (X₂→Y)	Bias	Naïve slope is
+	+	+	too large
−	−	+	too large
+	−	−	too small
−	+	−	too small

So even if you cannot fix the problem, you can often say which way your estimate is wrong — and therefore whether your finding survives. If you estimate a positive effect and reason that the bias is positive, your true effect is smaller than reported.

The deck’s warning attaches here: if the naïve slope is too large it can make it look significant when it’s not.

This is the mirror image of p-06

Put the two lectures side by side and they are the same diagram read in opposite directions:

	p-06: adding a control	p-07: omitting a variable
Uncorrelated with X₁	shrinks residual, slope unchanged, SE smaller	no bias
Correlated with X₁	eats the overlap, slope changes, SE larger	bias

Notice that the same variable is harmless in both directions or harmful in both directions. A control uncorrelated with your policy variable neither biases you when omitted nor distorts you when added — it only ever affects precision. A correlated one always does both.

That gives you the decision rule the two lectures jointly imply:

Correlated with X₁ and affects Y → you must include it, and you will pay in standard errors.
Uncorrelated with X₁ but affects Y → include it if you can; it is free precision.
Does not affect Y → it does not matter either way.

p-06 showed the cost. p-07 shows the reason to pay it. Together they are the accuracy-versus- precision trade from p-05, made concrete.

Why bias is not the same as sampling error

Slide 2 makes a point worth dwelling on. You are used to Greek-versus-Latin meaning population-versus-sample, and here it means full-model-versus-naïve. The analogy is deliberate but the mechanism is completely different.

Sampling error shrinks as n grows. Omitted variable bias does not. You can have census data on every unit in the population and still get the wrong slope, because the problem is not that you have too few rows — it is that you have too few columns.

This is the clearest illustration of the p-05 distinction: more data buys you efficiency, never unbiasedness. No sample size fixes a missing variable.

The formula that cannot be used, and the one that can

Slide 13 is honest about something most treatments gloss over: bias = b₁ − β₁ is the definition, and it is useless in practice — if you knew β₁ you would not be estimating it.

What makes the lecture practical is the other route, bias = α₁β₂, because both of those can sometimes be reasoned about from theory, prior studies, or partial data even when the full model cannot be run. That is the difference between a definition and a tool.

The path diagram is the thing to remember

If you keep one image from this deck, keep the triangle:

X₁ → Y is the direct effect, β₁
X₁ → X₂ → Y is the indirect path, α₁ then β₂
the naïve regression cannot tell the two routes apart, so it reports their sum

Every question about omitted variable bias reduces to: is there a second route from my treatment to my outcome, and how strong is it?

How to read the slides

Slides	What they are doing	What to take away
2	Notation	Greek = truth, Latin = best guess. Bias is not sampling error.
3–4	The main question	Omission is universal; magnitude is the real question.
5–8	p-06 recapped	The two control types, restated as a setup.
9–11	The two cases	Correlated → bias. Uncorrelated → none.
12	The multi-model table	Bias of 51% in the worked example.
13	The definition	b₁ − β₁. True, and unusable.
14–17	Direct and indirect effects	The path diagram. X₂ goes on the left in the auxiliary regression.
19	Worked calculation	Both routes give −0.056. Check the arithmetic yourself.
20	The take-away	Three numbered lines. If you memorise one slide, this one.
21	Why it matters	Case 1 too large, Case 2 too small; significance can be manufactured.
22–26	When does OVB occur?	The two Venn cases, with the algebra beside each.

This deck has no closing checklist. Slide 20 is the summary.

Section three

What should be clear in my mind?

The deck ends on its take-away rather than a question list.

What omitted variable bias is. The naïve slope absorbs the indirect effect of the omitted variable, so b₁ = β₁ + bias.
The two conditions. Bias requires the omitted variable to be correlated with the policy variable and to affect the outcome. Either alone is harmless.
How to compute it two ways. b₁ − β₁ (definition, needs the truth) or α₁β₂ (product of two estimable slopes).
How to sign it. Multiply the signs of the two paths to know whether your naïve estimate is too large or too small.
Why more data does not help. Bias is a specification problem, not a sampling problem.

Key takeaways

bias = α₁β₂. Two slopes multiplied — the effect of your treatment on the omitted variable, times the effect of the omitted variable on the outcome.
Both channels must be open. Zero on either one means zero bias.
The naïve slope is direct plus indirect effect. Regression cannot separate routes it cannot see.
You can often sign the bias without measuring the variable, which tells you whether your result would survive correction.
Bias does not shrink with sample size. It is a missing-column problem, not a missing-row problem.
The dangerous omitted variable is correlated with your treatment, not merely predictive of your outcome.
p-06 and p-07 are one decision. Correlated controls cost precision; omitting them costs accuracy. The choice is about the world, not the data.