Heteroskedasticity is the error that does not break your coefficients — it breaks your confidence in them. Run OLS on data where the spread of the errors changes across observations and the estimated line is still fine; what goes wrong is every standard error, t-statistic and p-value printed beside it. That is why robust standard errors have become the default in applied work: they leave the estimate alone and fix the inference. This page shows you exactly what heteroskedasticity does, how to spot it, and how the White correction repairs the damage.
1 · What heteroskedasticity is
The classical regression model assumes the error term has the same variance for every observation: Var(uᵢ) = σ² for all i. That assumption is homoskedasticity — equal scatter. Heteroskedasticity is its failure: the error variance depends on the observation, Var(uᵢ) = σᵢ², so some errors are systematically noisier than others.
It is the rule, not the exception, in cross-sectional data. Model household savings on income and the spread widens with income — a household earning very little has little room to vary, while a rich household might save aggressively or barely at all. Model firm profits on size, exam scores on study hours, city rents on floor area: in each, the dispersion around the line grows with the regressor. The single most common shape is exactly this fan.
Read the definitions carefully, because examiners test the distinction: homoskedasticity is about the error, not the regressor. The x values can be spread however they like. What matters is whether the vertical scatter around the line is constant.
2 · Why it matters — and why it doesn’t
Here is the fact that surprises most students, and the one your marking scheme wants stated cleanly.
OLS stays unbiased. Heteroskedasticity does not touch the coefficient estimates. The formula for the slope makes no use of the constant-variance assumption, so β̂ is still centred on the truth. If your only goal were a point estimate of the effect, you could ignore the problem entirely.
The standard errors become wrong. The classical SE formula — the one built into the default output — is derived assuming Var(uᵢ) = σ². When that fails, the formula computes the wrong number. Usually it understates the true sampling variability, so your t-statistics look too big and your p-values too small. You reject null hypotheses you should not, and your confidence intervals are too narrow. Every inference is quietly unreliable.
There is a third, milder cost. OLS is no longer efficient — it is no longer the minimum-variance linear unbiased estimator, because the Gauss–Markov theorem needs homoskedasticity. A weighted estimator could do better. But in modern practice the efficiency loss is usually tolerated, and the wrong-SE problem is fixed directly. That fix is the White correction in Section 4.
3 · Detecting it
Two routes, and good practice uses both.
Look at the residuals. Plot the residuals ûᵢ against the fitted values ŷᵢ (or against a suspect regressor). Under homoskedasticity the cloud is a band of roughly constant width. Under heteroskedasticity it fans, funnels or bulges. The diagram at the foot of this page is the two cases side by side — the eye catches the fan long before any test does.
Test formally. Two standard tests turn the picture into a number:
- The Breusch–Pagan test regresses the squared residuals on the original regressors. If the regressors explain the squared residuals, the error variance depends on them — heteroskedasticity.
- The White test does the same but adds squares and cross-products of the regressors, so it catches more general forms without assuming which shape.
Both use the same statistic: LM = n·R² from the auxiliary regression, compared against a chi-squared critical value with degrees of freedom equal to the number of regressors in that auxiliary. A large n·R² means the regressors predict the squared residuals, and you reject the null of homoskedasticity.
4 · The fix — robust standard errors
You do not need to know the exact form of the heteroskedasticity to correct for it. This is the insight that made heteroskedasticity-robust standard errors — also called White, Huber–Eicker, or “sandwich” standard errors — universal.
The classical variance of the slope assumes a single σ² and simplifies to σ̂²/Σ(xᵢ − x̄)². The robust version refuses that simplification and lets each observation carry its own squared residual as an estimate of its own variance. For a simple regression the robust variance of the slope is
Varrobust(β̂₁) = Σ(xᵢ − x̄)²·ûᵢ² ⁄ [Σ(xᵢ − x̄)²]².
Read it as a weighted average: observations far from x̄ (high leverage) that also have large residuals get the most weight, because those are the points whose noise most inflates the true sampling variance. Take the square root and you have the robust SE. Recompute every t-statistic with it and your inference is valid whether the errors are homoskedastic or not — the correction costs you nothing if they were homoskedastic all along, which is why it is safe to use by default.
Worked example — savings on income across ten households
You regress weekly savings y on weekly income x for ten households. Higher-income households vary far more in their saving, so you suspect heteroskedasticity.
| x (income) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| y (savings) | 3.66 | 5.35 | 8.15 | 9.55 | 12.84 | 13.54 | 18.23 | 16.23 | 24.43 | 18.02 |
Step 1 — Fit the line. OLS gives β̂₀ = 2.00 and β̂₁ = 2.00, with Σ(xᵢ − x̄)² = 82.5. So each extra unit of income is associated with two more units of savings.
Step 2 — Inspect the residuals. The residuals are tiny for the first few households (a few tenths) and grow to roughly ±5 by the richest — a textbook fan. Heteroskedasticity is present, so the default standard errors are suspect.
Step 3 — The classical standard error. Assuming a constant variance, SE(β̂₁) = 0.26, giving a t-statistic of 2.00 / 0.26 = 7.64. On this number the slope is overwhelmingly significant.
Step 4 — The robust standard error. Apply the sandwich formula, weighting each squared residual by its leverage. The White (HC1) robust SE comes out at 0.33 — about a quarter larger than the classical 0.26 — because the biggest residuals sit at the high-income, high-leverage households.
Step 5 — Redo the inference. The robust t-statistic is 2.00 / 0.33 = 6.02, down from 7.64. The slope is still significant here, but the honest measure of precision is meaningfully lower than the default output claimed.
Step 6 — Interpret. The estimate never moved: β̂₁ = 2.00 under both standard errors, because heteroskedasticity does not bias OLS. Only the standard error changed — and the classical one understated the uncertainty by around a quarter. Here the conclusion survives; when a t-statistic sits near the critical value, the same correction is what turns a “significant” result into an insignificant one. That is why you report robust standard errors as a matter of course, not only when a test flags a problem.
Not sure when the output is lying to you? Recognising that the estimate is fine but the standard errors are not — and fixing it in one line — is the applied skill that separates a good empirical answer from a wrong one. A one-on-one econometrics tutor builds the residual plot, the White test and the robust correction into a routine you run without thinking. Book a trial session.
Practice
Q1. In a simple regression you are told Σ(xᵢ − x̄)² = 100 and Σ(xᵢ − x̄)²·ûᵢ² = 900. Compute the heteroskedasticity-robust standard error of the slope.
Q2. An estimate is β̂₁ = 0.6. Its classical standard error is 0.20 and its robust standard error is 0.40. Compute both t-statistics and state, at the 5% level (critical value 1.96), whether the coefficient is significant under each. What do you conclude?
Q3. With n = 100 observations you run a White test: regressing the squared residuals on x and x² gives an auxiliary R² of 0.15. Compute the test statistic and decide, at the 5% level, whether there is heteroskedasticity. (The relevant chi-squared critical value, with 2 degrees of freedom, is 5.99.)
Answers.
Q1: Robust Var(β̂₁) = 900 / 100² = 0.09, so the robust SE = √0.09 = 0.30.
Q2: Classical t = 0.6 / 0.20 = 3.0, which exceeds 1.96 — “significant”. Robust t = 0.6 / 0.40 = 1.5, which is below 1.96 — not significant. The verdict flips: the apparent significance was an artefact of a standard error that ignored the heteroskedasticity.
Q3: LM = n·R² = 100 × 0.15 = 15. Since 15 > 5.99, reject the null of homoskedasticity — there is evidence of heteroskedasticity, so robust standard errors should be used.
Key takeaways
- Heteroskedasticity means the error variance changes across observations, Var(uᵢ) = σᵢ². In cross-sections it usually shows up as residuals that fan with a regressor or the fitted value.
- It leaves OLS unbiased — the coefficients are fine — but makes the classical standard errors wrong, usually too small, so t-tests over-reject.
- Detect it with a residual-vs-fitted plot and confirm with a Breusch–Pagan or White test, using the statistic n·R² from the auxiliary regression.
- Fix it with robust (White/sandwich) standard errors, which correct the inference without changing the estimate and are safe to use by default.
- Heteroskedasticity is an inference problem, not an estimation one — the number to distrust is the standard error, never the coefficient.
Why Bristol students choose our econometrics tutoring
- Course-matched to your unit: whether you are on Econometrics I, Econometrics II or a time-series option, our tutors teach heteroskedasticity in the notation and software your course uses — Stata’s robust option, R’s sandwich estimator, or the matrix derivation for a theory paper.
- Software and theory together: sessions cover both the intuition and the exact commands, so you can read robust output in a problem set and derive the sandwich estimator in an exam.
- Exam-focused: our tutors know which distinctions the marks turn on — unbiased versus efficient, estimate versus standard error — and drill the ones examiners reward.
FAQ
Q: Does heteroskedasticity bias the OLS coefficients?
A: No. OLS remains unbiased and consistent under heteroskedasticity — the coefficient estimates are fine. What becomes invalid is the classical standard errors, and therefore the t-statistics, p-values and confidence intervals built on them.
Q: What is the difference between the Breusch–Pagan and White tests?
A: Both test for heteroskedasticity by regressing the squared residuals on explanatory variables and using the statistic n·R². Breusch–Pagan uses the original regressors; the White test also includes their squares and cross-products, so it detects more general forms of heteroskedasticity at the cost of more degrees of freedom.
Q: What are robust standard errors?
A: Heteroskedasticity-robust (White, Huber–Eicker, or “sandwich”) standard errors estimate the sampling variability of the coefficients without assuming constant error variance. They use each observation’s own squared residual, so they remain valid under heteroskedasticity and are the standard fix in applied work.
Q: When should I use robust standard errors?
A: In cross-sectional work it is standard to use them by default. They are valid whether or not heteroskedasticity is present — if the errors happen to be homoskedastic, robust and classical standard errors are similar — so there is little cost to reporting them routinely.
Q: What is the difference between robust standard errors and weighted least squares?
A: Robust standard errors keep OLS estimates and only correct the inference. Weighted least squares (or feasible GLS) instead re-weights the data to restore efficiency, giving different, lower-variance estimates — but it requires you to model the form of the variance, which robust standard errors avoid.
Q: How does heteroskedasticity differ from autocorrelation?
A: Heteroskedasticity is non-constant variance of the errors, common in cross-sections. Autocorrelation is correlation between errors across observations, common in time series. Both leave OLS unbiased but invalidate the classical standard errors; the time-series analogue of the White fix is the Newey–West standard error.
Book an econometrics tutor in Bristol
Heteroskedasticity is where applied econometrics starts to feel real: the estimate is right, the output lies, and knowing the difference is the whole skill. A one-on-one session builds the residual plot, the White test and the robust correction into a routine you run without thinking. Tell us your unit and whether you work in Stata, R or by hand, and we will match you with the right tutor this week.
