Nature & Consequences of Heteroscedasticity

ECON 3209 · Week 12, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you will be able to:

  1. define heteroscedasticity and contrast it with homoskedasticity
  2. identify examples from cross-sectional economic data
  3. explain which OLS properties survive and which fail under heteroscedasticity
  4. show why standard errors become biased when error variance is not constant
  5. use Python plots to visualize a funnel-shaped residual pattern

What Is Heteroscedasticity?

In a homoskedastic model,

\[Var(\varepsilon_i \mid X_i) = \sigma^2\]

Under heteroscedasticity,

\[Var(\varepsilon_i \mid X_i) = \sigma_i^2\]

so the spread of the errors changes with the regressors.

Why Is It Common in Cross-Section Data?

  • Richer households often show greater absolute variation in expenditure.
  • Larger farms may have more variable profits than small farms.
  • In cooperative member income data, variance can increase with wealth or landholding.

Cross-sectional data often contain units of very different scale, and scale differences naturally create unequal variance.

What Happens to OLS?

If the zero conditional mean assumption still holds, OLS coefficients remain unbiased.

But under heteroscedasticity: - OLS is no longer efficient among linear unbiased estimators. - Usual homoskedastic standard errors are biased. - t-tests and confidence intervals can be misleading.

Why Standard Errors Are the Main Problem

The usual textbook variance formula assumes constant error variance.

When that assumption fails, software that uses the homoskedastic formula may understate or overstate uncertainty.

So the biggest danger is often false confidence in hypothesis tests.

Python Demo: Simulate Heteroscedastic Data

Python Demo: Residual Funnel Plot

Visual Signatures to Watch For

  • Funnel shape: residual spread widens or narrows.
  • Megaphone pattern: variability grows with scale.
  • Cluster-specific spread: one subgroup is more variable than another.

Residual plots are informal, but they are often the first useful diagnostic.

Python Demo: OLS Coefficient Unbiasedness in Simulation

Python Demo: Residual Variance by Wealth Group

Practical Consequences for Applied Work

  • Significant coefficients can become insignificant once robust SEs are used.
  • Policy recommendations may look more certain than they really are.
  • Forecast intervals may be poorly calibrated.

Heteroscedasticity is mainly an inference problem unless the variance structure itself is of substantive interest.

🏋️ Exercise

  1. Simulate income and wealth data where the error variance increases with wealth.
  2. Estimate an OLS regression of income on wealth.
  3. Plot residuals against fitted values.
  4. Compare residual variance across four wealth groups.

Summary

✅ Heteroscedasticity means the variance of the error term changes across observations.

✅ It is very common in cross-sectional data such as income, farm size, and expenditure data.

✅ OLS coefficients can remain unbiased, but usual standard errors become unreliable.

✅ Residual plots often reveal funnel or megaphone patterns.

✅ The main danger is incorrect inference, not necessarily biased coefficient estimates.

Next Lecture

Lecture 2 — Detection Tests

We will cover: - Breusch-Pagan and White tests - reading residual plots - Python implementation in statsmodels - interpreting p-values and test outputs