Cross-Sectional Dependence Tests¶

What Is Cross-Sectional Dependence?¶

Cross-sectional dependence (CD) occurs when the error terms are correlated across different entities at the same point in time:

\[\text{Cov}(\varepsilon_{it}, \varepsilon_{jt}) \neq 0 \quad \text{for } i \neq j\]

This means the residuals of entity \(i\) and entity \(j\) move together contemporaneously, violating the standard assumption of cross-sectional independence.

Common Sources¶

Common macroeconomic shocks: recessions, policy changes, or financial crises that affect all entities simultaneously
Spatial proximity: neighboring regions or countries that share economic linkages
Institutional connections: firms in the same industry, banks in the same network
Omitted common factors: unobserved variables that affect all cross-sectional units

Consequences of Ignoring Cross-Sectional Dependence

Entity-clustered standard errors are insufficient (they only handle within-entity serial correlation)
Standard errors may be severely downward biased, leading to spurious significance
Coefficient estimates may be inconsistent if CD results from omitted common factors
Unit root and cointegration tests lose their properties

Available Tests¶

PanelBox provides three complementary tests for cross-sectional dependence:

Test	H₀	Distribution	Best For
Pesaran CD	No CD	N(0,1)	Large N, any T (default)
Breusch-Pagan LM	No CD	\(\chi^2(N(N-1)/2)\)	Small N, large T
Frees	No CD	z-statistic	Non-parametric alternative

Decision Guide¶

graph TD
    A[Panel Dimensions?] --> B{N > 30?}
    B -->|Yes| C[Use Pesaran CD Test]
    B -->|No| D{T > N?}
    D -->|Yes| E[Use Breusch-Pagan LM Test]
    D -->|No| F[Use Pesaran CD or Frees Test]
    C --> G{Reject H₀?}
    E --> G
    F --> G
    G -->|Yes| H[CD Detected: Use DK-SE, PCSE, or Spatial Models]
    G -->|No| I[No CD: Standard or Clustered SE OK]

Quick Comparison¶

from panelbox import FixedEffects
from panelbox.datasets import load_grunfeld
from panelbox.validation.cross_sectional_dependence.pesaran_cd import PesaranCDTest
from panelbox.validation.cross_sectional_dependence.breusch_pagan_lm import BreuschPaganLMTest
from panelbox.validation.cross_sectional_dependence.frees import FreesTest

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

# Run all three tests
pesaran = PesaranCDTest(results).run()
bp_lm = BreuschPaganLMTest(results).run()
frees = FreesTest(results).run()

print("Test              | Statistic | p-value  | Reject H₀?")
print("------------------|-----------|----------|----------")
print(f"Pesaran CD        | {pesaran.statistic:9.3f} | {pesaran.pvalue:.4f}  | "
      f"{'Yes' if pesaran.reject_null else 'No'}")
print(f"Breusch-Pagan LM  | {bp_lm.statistic:9.3f} | {bp_lm.pvalue:.4f}  | "
      f"{'Yes' if bp_lm.reject_null else 'No'}")
print(f"Frees             | {frees.statistic:9.3f} | {frees.pvalue:.4f}  | "
      f"{'Yes' if frees.reject_null else 'No'}")

# Average correlation from Pesaran test
print(f"\nAvg. abs. correlation: {pesaran.metadata['avg_abs_correlation']:.3f}")

What to Do If Cross-Sectional Dependence Is Detected¶

Option 1: Driscoll-Kraay Standard Errors (Recommended)¶

Robust to cross-sectional dependence, serial correlation, and heteroskedasticity:

results_dk = fe.fit(cov_type="driscoll_kraay")
print(results_dk.summary())

Option 2: Panel-Corrected Standard Errors (PCSE)¶

Beck and Katz (1995) standard errors designed for panels with CD:

results_pcse = fe.fit(cov_type="pcse")

Option 3: Time Fixed Effects¶

Common shocks may be absorbed by time dummies:

fe_tw = FixedEffects(
    data, "invest", ["value", "capital"], "firm", "year",
    time_effects=True
)
results_tw = fe_tw.fit()

# Re-test after including time effects
pesaran_tw = PesaranCDTest(results_tw).run()
print(f"CD after time effects: {pesaran_tw.statistic:.3f}, p={pesaran_tw.pvalue:.4f}")

Option 4: Spatial Panel Models¶

If the cross-sectional dependence has a spatial structure:

from panelbox.models.spatial import SpatialLag

sar = SpatialLag(data, "invest", ["value", "capital"], "firm", "year", W=W)
results_spatial = sar.fit()

Interpreting Results¶

All cross-sectional dependence tests return a ValidationTestResult:

result.test_name       # Name of the test
result.statistic       # Test statistic
result.pvalue          # p-value
result.df              # Degrees of freedom (None for Pesaran CD, Frees)
result.reject_null     # True if H₀ rejected
result.conclusion      # Human-readable conclusion
result.metadata        # Test-specific details (correlations, entity counts)

Average Correlation Guidelines¶

| \(|\bar{\rho}|\) | Dependence Strength | Recommended Action | |----------------|---------------------|---------------------| | < 0.1 | Negligible | Clustered SE sufficient | | 0.1 -- 0.3 | Moderate | Use Driscoll-Kraay SE | | 0.3 -- 0.5 | Strong | Use PCSE or spatial models | | > 0.5 | Very strong | Potential model misspecification; consider common factors |

Frees Test¶

The Frees (1995) test is a non-parametric test for cross-sectional dependence based on Spearman rank correlations. It is robust to non-normality and outliers.

from panelbox.validation.cross_sectional_dependence.frees import FreesTest

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

print(f"z-statistic: {result.statistic:.3f}")
print(f"P-value:     {result.pvalue:.4f}")

# Frees-specific metadata
meta = result.metadata
print(f"Q_F statistic:       {meta['q_frees_statistic']:.4f}")
print(f"Expected under H₀:  {meta['expected_qf_under_h0']:.4f}")
print(f"Critical value (5%): {meta['critical_values']['alpha_0.05']:.4f}")
print(f"Mean |rank corr.|:   {meta['mean_abs_rank_correlation']:.4f}")

The Frees test is especially useful when:

You suspect non-normal residuals
You want a test robust to outliers
The Pesaran CD test may lack power due to positive and negative correlations canceling out

Software Equivalents¶

PanelBox	Stata	R (plm)
`PesaranCDTest`	`xtcsd, pesaran abs`	`plm::pcdtest(, test="cd")`
`BreuschPaganLMTest`	`xtcsd, xttest2`	`plm::pcdtest(, test="lm")`
`FreesTest`	`xtcsd, frees`	`plm::pcdtest(, test="sclm")`

References¶

Pesaran, M. H. (2004). "General diagnostic tests for cross section dependence in panels." University of Cambridge Working Paper, No. 0435.
Breusch, T. S., & Pagan, A. R. (1980). "The Lagrange Multiplier test and its applications to model specification in econometrics." Review of Economic Studies, 47(1), 239-253.
Frees, E. W. (1995). "Assessing cross-sectional correlation in panel data." Journal of Econometrics, 69(2), 393-414.
De Hoyos, R. E., & Sarafidis, V. (2006). "Testing for cross-sectional dependence in panel-data models." Stata Journal, 6(4), 482-496.
Beck, N., & Katz, J. N. (1995). "What to do (and not to do) with time-series cross-section data." American Political Science Review, 89(3), 634-647.