Your coefficients are fine — but your standard errors are lying to you
Let’s start with an analogy. Imagine you’re trying to predict how much money people spend on food. For someone earning £1,000 a month, you can predict pretty confidently — they’ll spend maybe £200–£250 on food. Not much room to vary when you’re on a tight budget.
Now try predicting food spending for someone earning £20,000 a month. Could be £300. Could be £3,000. Fine dining every night or cooking at home — hard to say. The uncertainty is much wider.
That variation in the spread of your predictions as your explanatory variable changes? That’s heteroskedasticity. It literally means “different spread” (hetero = different, skedastic = spread).
The opposite — where the spread is roughly constant across all values of your variables — is homoskedasticity, which OLS assumes by default.
Now here’s the important bit: heteroskedasticity doesn’t make your regression coefficients wrong. Your estimate of “how much income affects food spending” is still fine. But it makes your standard errors wrong — which means every t-test, every p-value, every confidence interval you report is incorrect. And since hypothesis testing is how economists decide what matters, that’s a serious problem.
It’s everywhere. Here are the most common sources:
The OLS variance formula (what software uses to compute your standard errors) is:
What software assumes (homoskedastic): Var(β̂) = σ²(X’X)⁻¹
What’s actually true (heteroskedastic): Var(β̂) = (X’X)⁻¹(X’ΩX)(X’X)⁻¹
where Ω captures the different variances for each observation
When these two are different — and they are, under heteroskedasticity — your t-statistics are wrong. You might think you have a t-stat of 3.2 (highly significant!) when the true t-stat should be 1.4 (not significant). Or vice versa. The direction of the error depends on the pattern of heteroskedasticity.
The practical consequence: you can systematically over-report significant results (or miss real ones) purely because of heteroskedasticity in your residuals.
Before any formal test, just plot the residuals. The eye is surprisingly good at spotting heteroskedasticity.
You’re looking for a “fan shape” or “funnel shape” — where the spread of residuals grows (or shrinks) as the fitted values increase.
If the residuals look like a random cloud with roughly equal spread throughout — you probably have homoskedasticity. If they fan out — you have heteroskedasticity and need to address it.
The idea: if heteroskedasticity is present, then your squared residuals (êᵢ²) should be predictable from your regressors. If variance is constant, the squared residuals should be random noise — no systematic pattern. Breusch-Pagan tests this directly:
LM = n × R²auxiliary ~ χ²(k)
k = number of regressors. Critical value at 5%: χ²(2) = 5.99 for two regressors.
The Breusch-Pagan test looks for a simple linear relationship between variance and regressors. The White test goes further — it also includes squared terms and interaction terms, detecting more complex patterns of heteroskedasticity. It uses more degrees of freedom but is more general.
Both tests are widely used. Report both if you’re being thorough.
This is what most applied economists use today. You don’t change your regression at all — you just use a different formula for the standard errors that’s correct under heteroskedasticity. Halbert White (1980) derived this “sandwich” formula:
V̂Robust = (X’X)⁻¹ [Σᵢ êᵢ² xᵢxᵢ’] (X’X)⁻¹
The “sandwich” — (X’X)⁻¹ is the bread, Σ êᵢ²xᵢxᵢ’ is the filling
Same coefficients. Different (correct) standard errors. In Stata: just add , robust after your regression command. In R: use coeftest(model, vcov=vcovHC(model)).
In modern applied economics, most cross-section regressions use robust SEs as a default — not an afterthought.
If you know the structure of heteroskedasticity — say, variance is proportional to income — you can use WLS. The idea: give higher weight to observations you trust (low variance) and lower weight to noisy ones (high variance).
Mechanically, multiply every variable (including the constant) by √w where w is the weight. Then run OLS on the transformed data. If your weights are right, WLS gives you smaller standard errors than robust OLS — it’s actually efficient. If your weights are wrong, WLS can be worse than just using robust SEs.
For economic variables that are always positive (wages, prices, GDP, house prices), taking the logarithm often dramatically reduces heteroskedasticity. When variance is proportional to the mean (common in economic data), logging the variable gives roughly constant variance. Bonus: log specifications also have natural economic interpretations as elasticities and growth rates.
Samuel Preston (1975) documented one of the most famous relationships in health economics: richer countries have higher life expectancy. The “Preston curve” plots life expectancy against GDP per capita across countries.
If you run this regression in levels (not logs) across 120 countries, you get classic heteroskedasticity:
A White test on this regression gives LM > 20 with p < 0.001 — strongly rejecting homoskedasticity. Your standard errors in the levels regression are wrong.
How economists fix this:
Deaton (2013) used exactly these methods in his cross-country health research, noting that the choice between OLS, robust-SE OLS, and WLS can genuinely change your substantive conclusions about income and health — not just the precision.
Sources: Preston, S.H. (1975). Population Studies. Deaton, A. (2013). The Great Escape. Princeton University Press.
A regression of log(wages) on education and experience uses n = 500 observations. The Breusch-Pagan auxiliary regression has R² = 0.044. Is there evidence of heteroskedasticity at the 5% level?
LM = n × R² = 500 × 0.044 = 22.0
Under H₀, LM ~ χ²(2) (two regressors). Critical value at 5% = 5.99.
Since 22.0 ≫ 5.99, we strongly reject H₀. There is significant evidence of heteroskedasticity.
Action: Re-estimate using robust (White) standard errors. The coefficients stay the same but the SEs — and therefore the t-statistics and p-values — will change. Results declared significant under OLS may no longer be, or vice versa.
Why does heteroskedasticity not bias OLS coefficients, but does make the standard errors wrong? Explain in plain English.
Coefficients stay unbiased: OLS estimates the slope by finding the line that minimises squared errors. This process doesn’t require error variance to be constant — it just requires the errors to average out to zero (zero conditional mean). Heteroskedasticity affects how spread out the errors are, not their average. So OLS still finds the “right” slope on average — it’s unbiased.
Standard errors go wrong: Standard errors measure how uncertain your slope estimate is. The OLS formula for SEs assumes all observations contribute equally to your precision — it uses a single σ² for every observation. But if some observations have much higher variance than others (heteroskedasticity), this assumption is wrong. High-variance observations should contribute less to precision — but the OLS formula treats them equally. The result: your computed SEs may be too small (you think you’re more certain than you are) or too large. Either way, the t-statistics are wrong, and your hypothesis tests are unreliable.
A study of household food spending (Y) on income (X) shows a fan-shaped residual plot — scatter widens as X increases. (a) What is this problem called, and which OLS assumption does it violate? (b) Suggest two corrections and explain when each is more appropriate.
(a) This is heteroskedasticity — specifically, variance increasing with income (a common pattern). It violates Gauss-Markov Assumption 4: homoskedasticity (constant error variance, Var(εᵢ|X) = σ² for all i).
(b)
Correction 1 — Robust (White) Standard Errors: Keep the OLS estimates exactly as they are but replace standard errors with White-robust SEs. Best when: (i) you don’t know the exact form of heteroskedasticity; (ii) sample size is large enough for asymptotic properties to hold; (iii) you want to keep comparing results to previous studies. Limitation: less efficient than WLS if you correctly specify the variance structure.
Correction 2 — Log Transformation: Regress log(Y) on log(X) instead of levels. For spending data, variance often scales with income levels, so logging both variables produces roughly constant residual variance. Best when: (i) the economic relationship is naturally multiplicative (expenditure shares); (ii) interpreting elasticities (% change in spending per % change in income) is meaningful. Limitation: you can’t use log if Y or X can be zero or negative; changes the economic interpretation of coefficients.
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