Regression analysis is one of the most widely used tools in applied economics. However, obtaining coefficient estimates is only the first step. Economists must also determine whether the reported standard errors, confidence intervals, and hypothesis tests are reliable.
One of the most common violations of the classical regression assumptions is heteroskedasticity. Heteroskedasticity occurs when the variability of the regression errors changes across observations. In practice, larger farms, firms, households, or countries often exhibit greater variability than smaller ones.
Ignoring heteroskedasticity can lead to misleading conclusions about statistical significance. In this chapter, we learn how to identify heteroskedasticity, formally test for it, and correct inference using robust standard errors.
Do larger farms earn higher income?
Suppose we collect information from 200 farms and estimate the relationship between annual farm income and farm size.
Economic theory suggests that larger farms generally earn higher income. However, larger farms may also face greater uncertainty because they operate on a larger scale and are more exposed to weather shocks, market fluctuations, and production risks.
As a result, income variability may increase with farm size.
Many econometric applications involve observations with very different scales.
Examples include:
When the variability of outcomes increases with size, the assumption of constant error variance may be violated.
For example, a small farm may earn between 4,000 and 6,000 OMR annually, while a large farm may earn between 20,000 and 60,000 OMR. The average income increases with farm size, but so does the uncertainty surrounding income.
The classical linear regression model assumes that the variance of the error term is constant across all observations:
[ Var(u_i)=^2 ]
This assumption is known as homoskedasticity.
Heteroskedasticity occurs when the error variance changes across observations:
[ Var(u_i)^2 ]
In practical terms, some observations are measured with greater uncertainty than others.
The presence of heteroskedasticity does not necessarily bias the estimated regression coefficients. However, it can make standard errors unreliable, leading to incorrect confidence intervals and hypothesis tests.
To illustrate heteroskedasticity, we create a synthetic dataset in which larger farms have both higher income and greater income variability.
import numpy as np
import pandas as pd
np.random.seed(4107)
n = 200
land = np.random.uniform(5, 500, n)
error = np.random.normal(
0,
land * 20,
n
)
income = 5000 + 120 * land + error
farm_data = pd.DataFrame({
"Income": income,
"Land": land
})
farm_data.head()| Income | Land | |
|---|---|---|
| 0 | 24672.435314 | 138.996820 |
| 1 | 45479.198779 | 325.219332 |
| 2 | 21225.941326 | 116.010498 |
| 3 | 16007.881758 | 116.506283 |
| 4 | 11574.007515 | 58.509175 |
Before estimating a regression model, it is always useful to inspect the data visually.
import matplotlib.pyplot as plt
plt.figure(figsize=(8, 5))
plt.scatter(
farm_data["Land"],
farm_data["Income"],
alpha=0.7
)
plt.xlabel("Farm Size (hectares)")
plt.ylabel("Annual Income (OMR)")
plt.title("Farm Income and Farm Size")
plt.show()
The graph shows a clear positive relationship between farm size and income.
Larger farms tend to earn higher income. However, the spread of observations becomes wider as farm size increases. Small farms cluster relatively closely together, whereas large farms exhibit much greater variation.
This visual pattern suggests heteroskedasticity.
We estimate a simple linear regression model:
[ Income_i = _0 + _1 Land_i + u_i ]
where (Income_i) is annual farm income, (Land_i) is farm size, and (u_i) is the random error term.
import statsmodels.api as sm
X = sm.add_constant(farm_data["Land"])
y = farm_data["Income"]
model = sm.OLS(y, X).fit()
print(model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: Income R-squared: 0.902
Model: OLS Adj. R-squared: 0.901
Method: Least Squares F-statistic: 1815.
Date: Thu, 11 Jun 2026 Prob (F-statistic): 1.14e-101
Time: 06:54:14 Log-Likelihood: -2010.0
No. Observations: 200 AIC: 4024.
Df Residuals: 198 BIC: 4031.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 5031.2341 838.222 6.002 0.000 3378.246 6684.222
Land 120.8008 2.835 42.606 0.000 115.210 126.392
==============================================================================
Omnibus: 15.725 Durbin-Watson: 1.895
Prob(Omnibus): 0.000 Jarque-Bera (JB): 39.693
Skew: -0.248 Prob(JB): 2.40e-09
Kurtosis: 5.125 Cond. No. 622.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.Suppose the estimated slope coefficient equals 118.
The interpretation is:
On average, each additional hectare of land is associated with approximately 118 OMR higher annual farm income.
This interpretation focuses on the average relationship. The regression coefficient does not describe the variability of income around that relationship.
A residual plot is often the simplest way to detect heteroskedasticity.
residuals = model.resid
fitted = model.fittedvalues
plt.figure(figsize=(8, 5))
plt.scatter(
fitted,
residuals,
alpha=0.7
)
plt.axhline(
y=0,
linestyle="--"
)
plt.xlabel("Fitted Values")
plt.ylabel("Residuals")
plt.title("Residuals versus Fitted Values")
plt.show()
If the homoskedasticity assumption holds, the residuals should display a roughly constant vertical spread across all fitted values.
Instead, the residual plot resembles a funnel. Residuals become increasingly dispersed as fitted values increase. This pattern is one of the most common indicators of heteroskedasticity.
Visual inspection is useful, but economists often supplement graphs with formal statistical tests.
The Breusch-Pagan test evaluates whether the variance of the residuals changes systematically across observations.
Null hypothesis:
[ H_0: ]
Alternative hypothesis:
[ H_1: ]
from statsmodels.stats.diagnostic import het_breuschpagan
bp_test = het_breuschpagan(
model.resid,
model.model.exog
)
labels = [
"LM Statistic",
"LM p-value",
"F Statistic",
"F p-value"
]
for name, value in zip(labels, bp_test):
print(name, value)LM Statistic 24.852888240654703
LM p-value 6.187626749033513e-07
F Statistic 28.095649549796637
F p-value 3.0665931032837663e-07A small p-value indicates evidence against the null hypothesis of constant variance.
If the p-value is below 0.05, heteroskedasticity is typically considered statistically significant.
Many students believe that heteroskedasticity changes coefficient estimates. This is usually incorrect.
The primary problem is that heteroskedasticity affects standard errors.
As a result:
Consequently, economists focus on correcting inference rather than changing coefficient estimates.
One of the most common solutions is to estimate robust standard errors.
Robust standard errors adjust the estimated variability of the coefficients without changing the coefficient estimates themselves.
robust_model = model.get_robustcov_results()
print(robust_model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: Income R-squared: 0.902
Model: OLS Adj. R-squared: 0.901
Method: Least Squares F-statistic: 1964.
Date: Thu, 11 Jun 2026 Prob (F-statistic): 1.00e-104
Time: 06:54:14 Log-Likelihood: -2010.0
No. Observations: 200 AIC: 4024.
Df Residuals: 198 BIC: 4031.
Df Model: 1
Covariance Type: HC1
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 5031.2341 490.888 10.249 0.000 4063.194 5999.274
Land 120.8008 2.726 44.312 0.000 115.425 126.177
==============================================================================
Omnibus: 15.725 Durbin-Watson: 1.895
Prob(Omnibus): 0.000 Jarque-Bera (JB): 39.693
Skew: -0.248 Prob(JB): 2.40e-09
Kurtosis: 5.125 Cond. No. 622.
==============================================================================
Notes:
[1] Standard Errors are heteroscedasticity robust (HC1)comparison = pd.DataFrame({
"OLS Standard Error": model.bse,
"Robust Standard Error": robust_model.bse
})
comparison| OLS Standard Error | Robust Standard Error | |
|---|---|---|
| const | 838.221999 | 490.888159 |
| Land | 2.835285 | 2.726142 |
The coefficient estimates remain unchanged because the regression line itself is unchanged. However, the estimated uncertainty surrounding the coefficients may increase or decrease.
The purpose of robust standard errors is to provide more reliable inference when heteroskedasticity is present.
Many modern empirical studies routinely report robust standard errors.
Reasons include:
For cross-sectional data, robust standard errors are often considered good practice.
Do not assume that heteroskedasticity biases OLS coefficients. The main concern is usually unreliable standard errors and misleading inference.
Students often focus only on coefficient estimates and p-values. Residual plots frequently reveal important problems that summary tables cannot.
The main concern is usually unreliable standard errors rather than biased coefficients.
Large samples do not prevent heteroskedasticity from occurring.
Logarithmic transformations sometimes reduce heteroskedasticity but should not be applied without economic justification.
When heteroskedasticity is suspected, robust standard errors are often preferable.
Heteroskedasticity involves changing error variance across observations. In the next chapter, we examine a different problem that arises in time-series data: autocorrelation.