Lesson M06.L01: The Nature and Consequences of Heteroskedasticity

Field	Detail
Module	06 — Heteroskedasticity
Level	Graduate
Duration	30–35 min
Learning Objective	Define heteroskedasticity precisely; understand which OLS properties are retained and which are lost; identify likely sources in economic data; diagnose heteroskedasticity from residual plots.
Provenance	Wooldridge Introductory Econometrics, Ch 8.1

Explanation

Recall: The Gauss-Markov Assumptions

The classical linear regression model (MLR.1–MLR.6) requires five core assumptions for OLS to be the Best Linear Unbiased Estimator (BLUE). The assumption relevant to this module is:

MLR.5 (Homoskedasticity):

\[\text{Var}(u_i \mid \mathbf{x}_i) = \sigma^2 \quad \text{for all } i = 1, \ldots, n\]

The variance of the unobserved error is constant across all observations, regardless of the values of the regressors.

Definition: Heteroskedasticity

Heteroskedasticity is the violation of MLR.5 — the error variance is not constant:

\[\text{Var}(u_i \mid \mathbf{x}_i) = \sigma_i^2\]

where $\sigma_i^2$ varies across observations. The variance may depend on one or more of the regressors (e.g. $\sigma_i^2 = \sigma^2 x_i$), on omitted variables, or on observation-level characteristics.

Etymology: from Greek hetero (different) + skedasis (dispersion). Opposite: homoskedasticity (homo = same).

Consequences for OLS

Understanding exactly what breaks — and what doesn't — is critical for applied work.

What OLS Retains Under Heteroskedasticity

Unbiasedness ($E[\hat{\boldsymbol{\beta}}] = \boldsymbol{\beta}$): requires only MLR.1–MLR.4. Since heteroskedasticity is a violation of MLR.5, not MLR.1–MLR.4, OLS estimates remain unbiased.

Consistency ($\hat{\boldsymbol{\beta}} \xrightarrow{p} \boldsymbol{\beta}$ as $n \to \infty$): also requires only MLR.1–MLR.4. OLS is still consistent under heteroskedasticity.

Key takeaway: Heteroskedasticity does not bias your coefficient estimates.

What OLS Loses Under Heteroskedasticity

Efficiency (Gauss-Markov fails): OLS is no longer BLUE. There exist other linear unbiased estimators (specifically, Weighted Least Squares) with smaller variance. OLS is inefficient.
Standard errors are wrong: The standard OLS formula for the variance of $\hat{\boldsymbol{\beta}}$:

\[\widehat{\text{Var}}_{OLS}(\hat{\boldsymbol{\beta}}) = \hat{\sigma}^2 (\mathbf{X}'\mathbf{X})^{-1}\]

assumes $\text{Var}(\mathbf{u}) = \sigma^2 \mathbf{I}$. Under heteroskedasticity, $\text{Var}(\mathbf{u}) = \boldsymbol{\Sigma} \neq \sigma^2 \mathbf{I}$. The OLS standard errors are therefore biased and inconsistent — they may be too large or too small.

t-tests and F-tests are invalid: Since $t$-statistics and $F$-statistics are constructed from OLS standard errors, they no longer have the correct sampling distributions. Confidence intervals are misleading. Hypothesis tests have incorrect size (rejection rates).

Common Sources of Heteroskedasticity

Heteroskedasticity arises naturally in many economic datasets:

Context	Why heteroskedasticity arises
Household expenditure data	Higher-income households have more discretion over spending → more variable expenditure patterns.
Firm-level cross-sectional data	Larger firms have larger absolute errors (revenue, employment); the scale of "noise" grows with firm size.
Wage data	Higher earners have more variable wages due to bonuses, commissions, equity compensation.
Country-level panel data	Larger, more diverse economies have more variable macroeconomic outcomes.
Aggregated data	When individual observations are averages over groups of different sizes, the variance of the group mean is $\sigma^2/n_g$, which varies with group size $n_g$.

Visual Detection: Residual Plots

The primary diagnostic tool is a plot of OLS residuals $\hat{u}_i$ (or $\hat{u}_i^2$) against:

Fitted values $\hat{y}_i$: should show a random scatter with roughly constant spread. A fan shape (spread increases with $\hat{y}$) indicates heteroskedasticity.
A suspected regressor $x_j$: if $\text{Var}(u_i)$ depends on $x_j$, the residual spread will widen or narrow systematically with $x_j$.

Patterns to look for:

Fan shape (left to right): variance increases with $x$ or $\hat{y}$ — classic heteroskedasticity.
Funnel shape (right to left): variance decreases with $x$ — less common.
Butterfly shape: variance peaks at intermediate values of $x$.

Residual plots are informal but essential. Formal tests (Breusch-Pagan, White) follow in M06.L02.

Worked Example

Setting: Household expenditure on food ($food\_exp_i$, $/month) regressed on monthly income ($income_i$, $/month).

\[\widehat{food\_exp_i} = 83.4 + 0.109\, income_i \quad (n = 40)\]

Residual plot description:

At low incomes ($income < \$500$/month), residuals cluster tightly around zero (spread ≈ ±$30). At high incomes ($income > \$2000$/month), residuals are dispersed widely (spread ≈ ±$200). The residual plot shows a clear fan shape opening to the right.

Interpretation:

$\text{Var}(u_i \mid income_i)$ increases with $income_i$. This makes economic sense: low-income households must spend a relatively fixed amount on food (basic subsistence); high-income households have much more discretion (gourmet vs. budget food, eating out, etc.), generating higher variance in food expenditure.

Consequence: OLS standard error on the income coefficient ($SE = 0.032$) is unreliable. The $t$-statistic ($t = 0.109/0.032 = 3.41$) may be inflated or deflated depending on the direction of the bias in the SE. Inference based on the naive OLS SE is not trustworthy.

Common Misconception

"Heteroskedasticity biases the OLS coefficient estimates."

Correction: This is false. Heteroskedasticity violates MLR.5 but leaves MLR.1–MLR.4 intact. Unbiasedness depends only on MLR.1–MLR.4. Your $\hat{\boldsymbol{\beta}}$ estimates are still unbiased and consistent — they are just inefficient (larger variance than necessary) and they come with incorrect standard errors. The problem is with inference (t-tests, F-tests, confidence intervals), not with the point estimates. A student who writes "heteroskedasticity biases my slope estimates" has misidentified the problem; the correct statement is "heteroskedasticity invalidates my standard errors and hypothesis tests."

Practice Prompts

Prompt 1 — Identification Which of the following datasets would you most suspect of exhibiting heteroskedasticity, and why?

(a) A time-series dataset of monthly inflation rates in Australia, 1990–2023. (b) A cross-sectional dataset of 500 manufacturing firms with revenue as the dependent variable and employment as the regressor. (c) An experimental dataset of test scores from 30 students in a randomised controlled study with identical class sizes.

Prompt 2 — Consequences A researcher estimates a model of household savings using OLS and obtains the coefficient on income as $\hat{\beta}_{income} = 0.18$ with $SE = 0.04$ ($t = 4.5$). A diagnostician then detects significant heteroskedasticity in the residuals.

(a) Is the researcher's estimate $\hat{\beta}_{income} = 0.18$ still reliable? Why or why not? (b) Is the $t$-statistic of 4.5 still reliable? Why or why not? (c) What should the researcher do next?

Prompt 3 — Residual plot Describe what you would expect to see in a residual plot ($\hat{u}_i$ vs $\hat{y}_i$) if:

(a) The model is correctly specified and homoskedastic. (b) Variance increases with fitted values. (c) Variance is large at intermediate fitted values and small at extremes.

In each case, what would you conclude?

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), §8.1
Stata: rvfplot after regress draws residual vs. fitted plot; rvpplot varname plots residuals vs a specific variable
R: plot(lm_object, which = 1) draws residual vs. fitted plot; plot(lm_object, which = 3) draws scale-location plot
Greene, Econometric Analysis (8th ed.), Ch 9 — for a more rigorous treatment of the GLS framework