Endogeneity breaks OLS. When a regressor correlates with the error term, coefficient estimates are biased and inconsistent. Instrumental variables (IV) and two-stage least squares (2SLS) fix that — provided you have a valid instrument. If you are studying econometrics in London, this is the method your course uses to recover causal estimates from observational data. It is also the topic a quantitative economics tutor works through first when you hit the endogeneity chapter.
1 · The endogeneity problem in plain terms
OLS requires that every regressor is uncorrelated with the error term: E(u|X) = 0. Violate that and your estimates are inconsistent. The bias does not disappear as the sample grows.
Three common sources of endogeneity:
- Omitted variable bias. A confounder affects both X and Y. You leave it out, and X picks up its effect through the error term.
- Measurement error. Your regressor is measured with noise. The noise lands in the error term, and the regressor correlates with it.
- Simultaneity. Y causes X as much as X causes Y. Think supply and demand: price and quantity are determined together.
In each case, OLS is not identifying the causal parameter. It is estimating something else — a mixture of the true effect and the confounding correlation. Examiners test this distinction. You should be able to state the direction of bias from a given omitted variable.
2 · What makes a valid instrument
An instrument Z solves the endogeneity problem by isolating the exogenous variation in X. Two conditions must hold:
- Relevance: Z correlates with X, conditional on the other regressors. Cov(Z, X) ≠ 0. Weak instruments fail here.
- Exogeneity: Z is uncorrelated with the error term u. Cov(Z, u) = 0. Invalid instruments fail here.
The exogeneity condition is the harder one. You cannot test it directly — the error term is unobserved. You argue it from theory, institutional knowledge, or overidentification tests when you have more instruments than endogenous regressors.
Relevance you can test. The first-stage F-statistic tells you whether the instrument predicts the endogenous variable. A rule of thumb: F > 10 suggests the instrument is not weak. Below that, 2SLS bias approaches OLS bias.
3 · Two-stage least squares — how it works
2SLS runs two regressions.
Stage 1: Regress the endogenous regressor X on the instrument Z and all exogenous controls. Get the predicted values X̂.
Stage 2: Regress Y on X̂ and the exogenous controls. The coefficient on X̂ is the 2SLS estimator.
Why does this work? X̂ is a linear combination of Z and the controls. If Z is exogenous, X̂ is exogenous too — it contains only the part of X that is uncorrelated with the error. The stage 2 regression recovers the causal effect of X on Y using that clean variation.
The standard errors from stage 2 are wrong — they do not account for the fact that X̂ is estimated. Your software handles this automatically if you use the built-in IV command. Do not run 2SLS by hand and report the stage 2 standard errors. Examiners notice.
4 · The just-identified case — one instrument, one endogenous regressor
With one endogenous regressor and one instrument, the 2SLS estimator has a simple form:
β_IV = Cov(Z, Y) / Cov(Z, X)
The numerator is the reduced-form effect of Z on Y. The denominator is the first-stage effect of Z on X. The ratio scales the reduced form by the strength of the first stage.
This is the Wald estimator when Z is binary. If Z is a treatment assignment and X is actual treatment, β_IV = (E[Y|Z=1] − E[Y|Z=0]) / (E[X|Z=1] − E[X|Z=0]). The intention-to-treat effect divided by the compliance rate. That is the local average treatment effect (LATE) — the effect on compliers.
5 · Overidentification — testing when you have extra instruments
With more instruments than endogenous regressors, the model is overidentified. You can test whether the extra instruments are valid.
The Sargan test (or Hansen J-test under heteroskedasticity) works like this. Estimate the 2SLS model. Get the residuals. Regress the residuals on all instruments and controls. Under the null that all instruments are valid, N × R² from this regression follows a χ² distribution with degrees of freedom equal to the number of overidentifying restrictions.
A low p-value rejects the null. But a high p-value does not prove validity — the test has low power against weak instruments or instruments that fail in similar ways. It is a diagnostic, not a proof.
6 · Weak instruments — the practical threat
A weak instrument has low correlation with the endogenous regressor. The first-stage F-statistic is small. The consequences are severe:
- 2SLS bias approaches OLS bias as the instrument weakens.
- Standard errors blow up. Confidence intervals are wide and unreliable.
- The asymptotic approximations break down. Finite-sample distributions are non-normal.
The standard fix is to report the first-stage F-statistic. If F < 10, consider limited information maximum likelihood (LIML) instead of 2SLS — it is more robust to weak instruments. Or find a stronger instrument. There is no econometric substitute for a good research design.
Worked example — returns to schooling with a quarter-of-birth instrument
You want the causal effect of an extra year of schooling on log wages. OLS gives 0.08 (8% per year). But ability is an omitted variable — people with higher ability get more schooling and higher wages. OLS is biased upward.
You instrument schooling with quarter of birth (QOB). The idea: children born earlier in the year can leave school earlier due to compulsory schooling laws. QOB affects schooling but not wages directly.
Step 1 — Data setup. n = 10,000. Y = log hourly wage. X = years of schooling. Z = 1 if born in Q1 (Jan–Mar), 0 otherwise. Controls: age, age², region dummies.
Step 2 — First stage. Regress X on Z and controls. The coefficient on Z is −0.25: people born in Q1 have 0.25 fewer years of schooling on average. The F-statistic is 18.7 — above 10, so the instrument is not weak.
Step 3 — Reduced form. Regress Y on Z and controls. The coefficient on Z is −0.015: people born in Q1 earn 1.5% less on average.
Step 4 — 2SLS estimate. β_IV = −0.015 / −0.25 = 0.06. Each extra year of schooling raises wages by 6%. That is lower than the OLS estimate of 8%, consistent with positive ability bias in OLS.
Step 5 — Standard error. The 2SLS standard error is 0.025. The 95% confidence interval is [0.011, 0.109]. The effect is statistically significant at the 5% level.
Step 6 — Interpretation. The IV estimate is a local average treatment effect. It measures the return to schooling for people whose schooling is affected by QOB — those who would leave school earlier if born in Q1. That is a specific subpopulation: less educated, likely lower ability. The LATE may differ from the average treatment effect for the whole population.
Step 7 — Check. The OLS estimate of 0.08 lies inside the IV confidence interval [0.011, 0.109], so although the point estimates suggest positive ability bias (0.08 vs 0.06), the difference is not statistically significant here — the IV interval is wide. The IV estimate is plausibly causal, but it applies to a specific group. Generalising requires an argument about treatment effect homogeneity.
Getting stuck on the LATE interpretation? That distinction — what the IV estimate actually identifies — is the most common exam trap in applied econometrics. A one-on-one quantitative economics tutor works through it with you, whether you are in a lecture hall in London or joining online. Book a trial session.
Practice
Q1. An instrument Z has Cov(Z, X) = 2.5 and Cov(Z, Y) = 1.0. The reduced-form regression of Y on Z gives a coefficient of 0.4. What is the IV estimate of the effect of X on Y?
Q2. In a just-identified IV model with n = 500, the first-stage F-statistic is 4.2. Is this instrument weak? What problem does this cause for 2SLS?
Q3. An overidentified IV model has 3 instruments for 1 endogenous regressor. The Sargan test statistic is 7.8. At the 5% significance level, do you reject the null of instrument validity? (The critical value from χ²(2) is 5.99.)
Answers: Q1: β_IV = 1.0 / 2.5 = 0.4. Q2: F = 4.2 < 10, so the instrument is weak. 2SLS bias approaches OLS bias, standard errors are inflated, and asymptotic approximations break down. Q3: 7.8 > 5.99, so reject the null. The overidentifying restrictions are not valid at the 5% level.
Key takeaways
- Endogeneity (omitted variables, measurement error, simultaneity) makes OLS inconsistent. IV recovers causal estimates using an exogenous instrument.
- A valid instrument must be relevant (Cov(Z, X) ≠ 0) and exogenous (Cov(Z, u) = 0). Relevance is testable; exogeneity must be argued.
- 2SLS runs two regressions: first-stage predicts X from Z, second-stage regresses Y on predicted X. Standard errors must be corrected.
- The IV estimate is a local average treatment effect — it applies to compliers, not the whole population.
- Weak instruments (first-stage F < 10) bias 2SLS toward OLS and inflate standard errors. Report the F-statistic and consider LIML.
Why London students choose our quantitative economics tutoring
- One-on-one format: every session is private and built around your course, from your lecture slides to your problem sets and your department’s notation for the 2SLS estimator.
- Applied econometrics specialists: our tutors teach IV as your programme teaches it — from the path diagram through the first-stage F-test, the Sargan test, and the LATE interpretation.
- Exam-first preparation: sessions work through past papers with marking schemes in view, because the “instrument exogeneity cannot be tested” distinction is where marks are won and lost.
FAQ
Q: What is an instrumental variable?
A: A variable Z that affects the endogenous regressor X but has no direct effect on the outcome Y except through X. It isolates the exogenous variation in X so you can estimate the causal effect.
Q: How do I know if my instrument is valid?
A: Relevance you test with the first-stage F-statistic. Exogeneity you argue from theory, institutional knowledge, or — if you have extra instruments — an overidentification test like Sargan or Hansen J.
Q: What is the weak instruments problem?
A: When Z has low correlation with X, the first-stage F-statistic is small (below 10). 2SLS becomes biased toward OLS, standard errors blow up, and confidence intervals are unreliable. Report the F-statistic and consider LIML.
Q: What is the local average treatment effect (LATE)?
A: The IV estimate identifies the treatment effect for compliers — people whose treatment status is changed by the instrument. It does not necessarily equal the average treatment effect for the whole population.
Q: Can I test instrument exogeneity?
A: Not directly — the error term is unobserved. With exactly one instrument per endogenous regressor, you cannot test exogeneity at all. With extra instruments, the Sargan test checks whether the overidentifying restrictions hold.
Q: Do I need to know matrix algebra for IV?
A: At intermediate level, yes — the 2SLS estimator is β_2SLS = (X′P_Z X)⁻¹X′P_Z Y, where P_Z = Z(Z′Z)⁻¹Z′. If your course is less matrix-heavy, say so when booking and we will match you with a tutor who teaches the intuition first.
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Endogeneity is the dividing line between descriptive and causal econometrics. IV and 2SLS are the tools that cross it. A one-on-one session turns the path diagram into a framework you can apply to any dataset. Tell us your university and module, and we will match you with the right tutor this week.
