This section provides technical details about the validation tests performed on your panel data model. Each test is designed to detect specific violations of panel data assumptions.
Tests the null hypothesis that Random Effects estimates are consistent. Rejection suggests systematic differences between FE and RE estimates, favoring Fixed Effects.
Test Statistic: H = (βFE - βRE)' [Var(βFE) - Var(βRE)]-1 (βFE - βRE)
Distribution: χ² with K degrees of freedom
Interpretation: If p-value < 0.05, use Fixed Effects
Alternative to Hausman test based on auxiliary regression. Tests whether entity means of time-varying regressors are correlated with the unobserved effect.
Approach: Estimate Random Effects including entity means of X as additional regressors
Test Statistic: Wald test on coefficients of entity means
Interpretation: Rejection indicates correlation, favoring Fixed Effects
Tests for first-order autocorrelation in panel data models. Specifically designed for Fixed Effects models with small T.
Null Hypothesis: No first-order autocorrelation
Test Statistic: F-test on correlation of lagged residuals
Distribution: F(1, N-1)
Durbin-Watson analogue for panel data. Tests for serial correlation in the idiosyncratic error component.
Test Statistic: BW = ΣiΣt(êit - êi,t-1)² / ΣiΣtêit²
Range: 0 to 4 (2 indicates no autocorrelation)
Tests for heteroskedasticity in panel data residuals. Detects whether error variance depends on explanatory variables.
Null Hypothesis: Homoskedastic errors
Approach: Regress squared residuals on regressors
Test Statistic: LM = N·R² ~ χ²(K)
Tests for groupwise heteroskedasticity in Fixed Effects models. Allows variance to differ across entities.
Null Hypothesis: σ²1 = σ²2 = ... = σ²N
Test Statistic: Wald statistic ~ χ²(N)
Tests for weak cross-sectional dependence in large panels. Robust to non-stationarity and structural breaks.
Test Statistic: CD = √(2T/(N(N-1))) Σi<j ρ̂ij
Distribution: N(0,1) asymptotically
Advantages: Valid for large N, valid for dynamic models
Nonparametric test for cross-sectional independence. Based on squared rank correlation of residuals across entities.
Null Hypothesis: No cross-sectional correlation
Test Statistic: Frees = T·mean(R²ij) where R is rank correlation
Critical values: Tabulated by Frees (1995, 2004)
Tests use standard significance levels: 1% (***), 5% (**), 10% (*). A p-value below the chosen significance level (typically 5%) leads to rejection of the null hypothesis.
When conducting multiple tests, consider adjusting significance levels (e.g., Bonferroni correction) to control for family-wise error rate. However, validation tests are typically exploratory, so strict corrections may be overly conservative.
Some tests have low power in small samples. With large N or T, even minor deviations from assumptions may be detected as statistically significant. Always consider economic significance alongside statistical significance.
Failing a test doesn't necessarily invalidate your model. Many violations can be addressed through robust standard errors, alternative estimators, or model re-specification. Consult the Recommendations tab for specific guidance.