Cross-Sectional Dependence (CSD) Test in Panel Data Analysis

 When using panel data, especially in macroeconomics, banking, or finance, cross-sectional dependence (CSD) is a critical concern. It refers to the situation where error terms (or shocks) across different cross-sectional units (like firms, countries, or banks) are correlated.

✅ Ignoring CSD leads to biased test statistics and inconsistent estimators, especially in unit root, cointegration, and regression models.

🧠 Why Test for CSD?

  • Traditional first-generation panel unit root tests (e.g., IPS, LLC) assume cross-sectional independence.

  • If cross-units are correlated (due to common shocks like inflation, policy changes, etc.), these tests are invalid.

  • Second-generation tests (e.g., CADF, CIPS, system GMM) are designed to handle CSD—but only after CSD is confirmed.

✅ Popular Cross-Sectional Dependence Tests

Test

Developed by

Suitable for

Description

Breusch-Pagan LM

Breusch & Pagan (1980)

Small T, large N

Tests pairwise correlations of residuals

Pesaran's CD test

Pesaran (2004)

Large N, small/medium T

Simple average of pairwise correlations

Bias-corrected scaled LM

Pesaran, Ullah, and Yamagata (2008)

Medium/large N and T

Improves on LM when N is large

Friedman’s test

Friedman (1937)

General

Non-parametric rank-based approach

🔍 1. Pesaran’s CD Test – Most Common

Test Statistic:

CD=2TN(N−1)∑i=1N−1∑j=i+1Nρ^ijCD = \sqrt{\frac{2T}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \hat{\rho}_{ij}

Where:

  • ρ^ij\hat{\rho}_{ij} = sample correlation of residuals between cross-sectional units ii and jj

Hypotheses:

  • H₀: No cross-sectional dependence (errors are independent)

  • H₁: Cross-sectional dependence exists

Interpretation:

  • p-value < 0.05 → Reject H₀ → Significant CSD

  • p-value > 0.05 → Fail to reject H₀ → No significant CSD

📊 Result Table Example

Test

Statistic

p-value

Conclusion

Pesaran CD

6.12

0.000

Cross-sectional dependence exists

Breusch-Pagan LM

92.45

0.002

CSD present

Friedman

5.31

0.001

CSD present

📝 When to Apply and Next Steps

✅ Apply CSD test:

·         After estimating your model (FE, RE, pooled OLS)

·         Before using first-generation unit root or cointegration tests

🔁 If CSD is detected:

·         Use second-generation panel tests:

o    CIPS/CADF for unit root

o    Westerlund for cointegration

o    System GMM or FGLS for estimation

 

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