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Heteroskedasticity Tests

What Is Heteroskedasticity?

Heteroskedasticity occurs when the variance of the error terms is not constant:

\[\text{Var}(\varepsilon_{it}) \neq \sigma^2\]

In panel data, the most common form is groupwise heteroskedasticity, where the error variance differs across entities:

\[\sigma_i^2 \neq \sigma_j^2 \quad \text{for some } i \neq j\]

This means some entities (e.g., large firms) may have systematically larger or smaller residual variance than others (e.g., small firms).

Consequences of Ignoring Heteroskedasticity

  • OLS coefficient estimates remain consistent but are inefficient (not minimum variance)
  • Classical standard errors are biased (can be too small or too large)
  • t-tests, F-tests, and confidence intervals are unreliable
  • GLS estimators assume homoskedasticity and become suboptimal

Available Tests

PanelBox provides three complementary tests for detecting heteroskedasticity:

Test Hâ‚€ Type Best For
Modified Wald \(\sigma_i^2 = \sigma^2\) for all \(i\) Groupwise FE models
Breusch-Pagan Homoskedasticity LM (parametric) General; identifies source
White Homoskedasticity LM (model-free) No functional form assumed

When to Use Each Test

Use the Modified Wald test as your primary diagnostic. It is specifically designed for FE models and tests whether entity-level variances differ.

from panelbox.validation.heteroskedasticity.modified_wald import ModifiedWaldTest

test = ModifiedWaldTest(results)
result = test.run(alpha=0.05)

Use the White test when you want a model-free test with no assumptions about the form of heteroskedasticity.

from panelbox.validation.heteroskedasticity.white import WhiteTest

test = WhiteTest(results)
result = test.run(alpha=0.05, cross_terms=True)

Use the Breusch-Pagan test when you want to identify which regressors drive the heteroskedasticity.

from panelbox.validation.heteroskedasticity.breusch_pagan import BreuschPaganTest

test = BreuschPaganTest(results)
result = test.run(alpha=0.05)

from panelbox import FixedEffects
from panelbox.datasets import load_grunfeld
from panelbox.validation.heteroskedasticity.modified_wald import ModifiedWaldTest
from panelbox.validation.heteroskedasticity.breusch_pagan import BreuschPaganTest
from panelbox.validation.heteroskedasticity.white import WhiteTest

# Load data and estimate model
data = load_grunfeld()
fe = FixedEffects(data, "invest", ["value", "capital"], "firm", "year")
results = fe.fit()

# Step 1: Modified Wald test (FE-specific)
mw = ModifiedWaldTest(results)
mw_result = mw.run()
print(f"Modified Wald: chi2={mw_result.statistic:.3f}, p={mw_result.pvalue:.4f}")
print(f"  Variance ratio (max/min): {mw_result.metadata['variance_ratio']:.2f}")

# Step 2: White test (model-free)
white = WhiteTest(results)
w_result = white.run(cross_terms=True)
print(f"White test:    LM={w_result.statistic:.3f}, p={w_result.pvalue:.4f}")

# Step 3: Breusch-Pagan (parametric)
bp = BreuschPaganTest(results)
bp_result = bp.run()
print(f"Breusch-Pagan: LM={bp_result.statistic:.3f}, p={bp_result.pvalue:.4f}")

# Decision
if any(r.reject_null for r in [mw_result, w_result, bp_result]):
    print("\nHeteroskedasticity detected. Use robust standard errors:")
    results_robust = fe.fit(cov_type="robust")
    print(results_robust.summary())

What to Do If Heteroskedasticity Is Detected

Option 1: Robust Standard Errors (HC0--HC3)

The simplest correction -- adjusts SE without changing coefficient estimates:

results_robust = fe.fit(cov_type="robust")    # Default HC1
results_hc3 = fe.fit(cov_type="hc3")          # HC3 (recommended for small samples)

Option 2: Clustered Standard Errors

Also handles serial correlation within entities:

results_cluster = fe.fit(cov_type="clustered")

Option 3: Variable Transformation

If the variance is proportional to a variable (e.g., firm size):

import numpy as np

# Log transformation to stabilize variance
data["log_invest"] = np.log(data["invest"])
fe_log = FixedEffects(data, "log_invest", ["value", "capital"], "firm", "year")
results_log = fe_log.fit()

Interpreting Results

All heteroskedasticity tests return a ValidationTestResult:

result.test_name       # Name of the test
result.statistic       # Test statistic (chi-squared or Wald)
result.pvalue          # p-value
result.df              # Degrees of freedom
result.reject_null     # True if Hâ‚€ rejected
result.conclusion      # Human-readable conclusion
result.metadata        # Test-specific details
p-value Decision Action
< 0.01 Strong rejection Use robust or clustered SE
0.01 -- 0.05 Rejection Use robust SE
0.05 -- 0.10 Borderline Consider robust SE as precaution
> 0.10 Fail to reject Standard SE likely adequate

Practical Advice

Even when the test fails to reject, using robust standard errors is a common practice in applied work. The cost of robustness (slight efficiency loss) is small compared to the cost of incorrect inference from biased SE.

Software Equivalents

PanelBox Stata R
ModifiedWaldTest xttest3 Custom implementation
BreuschPaganTest estat hettest lmtest::bptest()
WhiteTest estat imtest, white skedastic::white()

See Also

References

  • Greene, W. H. (2018). Econometric Analysis (8th ed.). Pearson, Chapter 14.
  • Breusch, T. S., & Pagan, A. R. (1979). "A simple test for heteroscedasticity and random coefficient variation." Econometrica, 47(5), 1287-1294.
  • White, H. (1980). "A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity." Econometrica, 48(4), 817-838.
  • Baum, C. F. (2001). "Residual diagnostics for cross-section time series regression models." Stata Journal, 1(1), 101-104.
  • Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press, Chapter 10.