Omitted variable bias is the reason multiple regression exists. Leave out a variable that both matters and moves with your regressor, and OLS quietly hands you the wrong slope — confidently, with a small standard error attached. Knowing when that happens, in which direction, and by how much is the single most examined skill in an intermediate econometrics course, and the one an econometrics tutor for LSE students rebuilds first. It comes down to one formula and one sign table.
1 · Why multiple regression
A simple regression of y on one variable x₁ answers a raw question: how does y move when x₁ moves, with everything that travels alongside x₁ tagging along. Multiple regression answers the question you usually want: how does y move when x₁ moves holding the other included variables fixed.
In the model y = β₀ + β₁x₁ + β₂x₂ + u, the coefficient β₁ is a ceteris paribus effect — the change in y from one more unit of x₁ with x₂ held constant. Each coefficient is estimated from the variation in its own regressor that the other regressors cannot explain — the effect of x₁ after x₂ has been partialled out.
Which raises the obvious threat. What if a variable that belongs in the model is not there to be held constant?
2 · The omitted variable bias formula
Call the model with both regressors the long regression and the one that omits x₂ the short regression. Omit x₂ and its influence does not vanish — it is forced into the error term. If x₂ is correlated with x₁, the error is now correlated with the regressor, and the zero-conditional-mean assumption fails. OLS on the short regression converges not on β₁ but on
β₁ + β₂δ₁,
where δ₁ is the slope from the auxiliary regression of the omitted x₂ on the included x₁. The bias term β₂δ₁ has a clean reading: how much x₂ matters (β₂) times how strongly x₂ tracks x₁ (δ₁).
Two conditions must both hold for bias:
- the omitted variable affects y (β₂ ≠ 0), and
- the omitted variable is correlated with the included regressor (δ₁ ≠ 0).
Kill either and the bias is zero. Examiners love asking for both conditions — stating only one loses the mark.
3 · Signing the bias
You rarely know β₂ and δ₁ numerically, but you can usually sign them — and the sign of the bias is just the product of the signs. That gives the 2×2 table in the diagram below: when the two signs match, the bias is upward; when they differ, it is downward.
Run any proposed regression through it. Wages on education, omitting ability: ability raises wages (+) and correlates positively with schooling (+) — upward bias, the return to education is overstated. Crime on police numbers, omitting local crime risk: risk raises crime (+) and high-risk areas hire more police (+) — upward bias again. Before trusting any slope, ask what is missing, sign its two links, and read off the direction of the damage.
Worked example — the return to education, with ability left out
Suppose the true model of hourly wages is wage = 2 + 1.50 educ + 0.40 score, where educ is years of schooling and score is an ability test score. The true causal return to a year of schooling is £1.50, holding ability fixed.
Step 1 — The auxiliary link. In the population, abler people stay in school longer: score = 5 + 2 educ. So δ₁ = 2.
Step 2 — Omit the score. You only observe wages and schooling, so you run the short regression of wage on educ alone.
Step 3 — What OLS actually estimates. Substitute the auxiliary link into the true model: wage = 2 + 1.50 educ + 0.40(5 + 2 educ) = 4 + 2.30 educ. The short regression converges on a slope of 2.30, not 1.50.
Step 4 — Read off the bias. Bias = β₂δ₁ = 0.40 × 2 = +0.80. Check: 1.50 + 0.80 = 2.30; the intercept absorbs the rest (2 + 0.40 × 5 = 4).
Step 5 — Sign-check with the table. β₂ > 0 (ability raises wages) and δ₁ > 0 (ability tracks schooling): signs match, so the bias is upward — the highlighted cell in the diagram.
Step 6 — Interpretation. The short regression overstates the return to education by 0.80/1.50 ≈ 53%, because it hands education the credit for ability. Nothing in the OLS output warns you: the fit is fine, the standard error is small, and the number is wrong. The cure is to include the variable if you can measure it, or to reach for the instrumental-variables designs covered elsewhere on this site.
Could you sign the bias — up or down — before running a single regression? Naming both conditions and reading the direction off the two signs is the exam skill OVB questions test, and the OLS output never warns you when the slope is wrong. Drilling that judgement is exactly what a one-on-one econometrics tutor does with your own problem sets. Book a trial session.
Practice
Q1. The true model has β₁ = 2. An omitted variable has β₂ = −3 and auxiliary slope δ₁ = 0.5. Compute the bias and the value the short-regression slope converges on. Is the bias upward or downward?
Q2. A demand study regresses quantity on price; the true price slope is −0.8. Product quality is omitted: quality raises quantity demanded (β₂ = 1.2) and higher-quality products carry higher prices (δ₁ = 0.5). Compute the bias and the short-regression slope, and explain what the researcher will wrongly conclude about price sensitivity.
Q3. An omitted variable has a large effect on y (β₂ = 5) but is uncorrelated with the included regressor (δ₁ = 0). The true β₁ = 3. Compute the bias and the short-regression slope, and state the lesson.
Answers. Q1: bias = (−3)(0.5) = −1.5, so the short slope converges on 2 − 1.5 = 0.5 — a downward bias (signs differ). Q2: bias = (1.2)(0.5) = +0.6, so the short slope converges on −0.8 + 0.6 = −0.2. The estimate is biased toward zero: demand looks far less price-sensitive than it is, because expensive goods in the sample are also the high-quality ones people still buy. Q3: bias = (5)(0) = 0; the short slope still converges on 3. Omitting a variable only biases OLS if it is both relevant and correlated with the included regressor — condition 2 fails here, so the estimate survives.
Key takeaways
- Multiple regression estimates ceteris paribus effects: each coefficient uses only the variation in its regressor that the other regressors cannot explain.
- The OVB formula: the short regression converges on β₁ + β₂δ₁, where δ₁ is the slope of the omitted variable on the included one. Bias needs both β₂ ≠ 0 and δ₁ ≠ 0.
- Sign the bias with the 2×2 table: signs match → upward; signs differ → downward.
- The output won’t warn you: a biased regression fits well and prints small standard errors — the defence is thinking about what’s missing.
Why LSE students choose our econometrics tutoring
- Matched to your course’s notation: short and long regressions, the partialling-out result, and the OVB formula are taught the way your lecture notes and problem sets write them, so nothing needs translating in the exam.
- Problem-set to proof: sessions move between computing a bias numerically, signing it in words, and deriving the formula — the three forms exam questions take.
- One-on-one, exam-first: your tutor works from your own past papers and marks written answers the way the graders do.
FAQ
Q: What is omitted variable bias?
A: The systematic error in an OLS slope when a variable that affects the outcome and is correlated with an included regressor is left out. The estimate converges on β₁ + β₂δ₁ instead of β₁.
Q: What two conditions produce omitted variable bias?
A: The omitted variable must (1) affect the dependent variable and (2) be correlated with the included regressor. Both are required; if either fails, there is no bias.
Q: How do I work out the direction of the bias?
A: Multiply the signs: the omitted variable’s effect on y times its correlation with the included regressor. Matching signs bias the estimate upward; opposite signs, downward.
Q: Does adding more variables always help?
A: No. Controls that belong in the model reduce bias, but adding outcomes of the treatment (“bad controls”) or near-duplicates of existing regressors can introduce new bias or inflate standard errors.
Q: Can omitted variable bias be fixed without measuring the omitted variable?
A: Sometimes. A time-invariant omitted factor is removed by panel-data fixed effects; a valid instrument recovers the causal slope via IV/2SLS. Both have their own pages on this site.
Book an econometrics tutor for LSE students — in London or online
Omitted variable bias is where econometrics stops being formulas and starts being judgement — and judgement is coachable. One-on-one sessions train the formula, the sign table and the written argument on your course’s own problem sets, in London or online. Tell us your course and exam date, and we’ll match you with the right tutor this week.
