Hadri Test (Panel Stationarity)¶

Quick Reference

Function: panelbox.diagnostics.unit_root.hadri.hadri_test() H₀: All panels are stationary (\(\sigma^2_{u_i} = 0\) for all \(i\)) H₁: At least one panel has a unit root (\(\sigma^2_{u_i} > 0\) for some \(i\)) Stata equivalent: xtunitroot hadri variable, robust R equivalent: plm::purtest(x, test="hadri")

Reversed Null Hypothesis

The Hadri test has the opposite null hypothesis from LLC, IPS, Fisher, and Breitung. Here, H₀ is stationarity. Rejecting H₀ means evidence of unit roots, not stationarity. This reversal is the defining feature of the Hadri test.

What It Tests¶

The Hadri (2000) test is a panel extension of the KPSS framework. It tests whether all series in the panel are stationary against the alternative that at least some series contain a unit root.

Because it reverses the null hypothesis, the Hadri test is primarily used as a confirmation test: if LLC/IPS/Fisher reject H₀ (unit root) and Hadri does not reject H₀ (stationarity), there is strong evidence for stationarity.

Quick Example¶

from panelbox.datasets import load_grunfeld
from panelbox.diagnostics.unit_root.hadri import hadri_test

data = load_grunfeld()

# Hadri test with robust standard errors
result = hadri_test(data, "invest", "firm", "year",
                    trend="c", robust=True)
print(result.summary())

Output:

======================================================================
Hadri (2000) LM Test for Stationarity
======================================================================
H0: All series are stationary
H1: At least one series has a unit root

Specification: Constant only
Robust version: Yes
Number of entities (N): 10
Number of periods (T): 20

LM statistic: 0.4521
Z-statistic: 5.2847
P-value: 0.0000

Decision at 5% level: REJECT H0

Evidence against stationarity (at least one series has unit root)
======================================================================

Interpretation¶

Reversed Logic

Remember: in the Hadri test, rejecting H₀ means the data is not stationary (unit root evidence). This is the opposite of LLC/IPS/Fisher interpretation.

p-value	Decision	Meaning
\(p < 0.01\)	Strong rejection of H₀	Strong evidence against stationarity (unit roots present)
\(0.01 \leq p < 0.05\)	Rejection of H₀	Evidence against stationarity
\(0.05 \leq p < 0.10\)	Borderline	Weak evidence against stationarity
\(p \geq 0.10\)	Fail to reject H₀	Consistent with stationarity in all panels

Confirmation Strategy¶

Use the Hadri test alongside LLC/IPS/Fisher to reach a robust conclusion:

LLC/IPS/Fisher	Hadri	Conclusion
Not reject H₀ (unit root)	Reject H₀ (not stationary)	Unit root confirmed -- both tests agree
Reject H₀ (stationary)	Not reject H₀ (stationary)	Stationarity confirmed -- both tests agree
Reject H₀ (stationary)	Reject H₀ (not stationary)	Mixed evidence -- investigate further
Not reject H₀ (unit root)	Not reject H₀ (stationary)	Mixed evidence -- investigate further

Mathematical Details¶

Model Decomposition¶

The Hadri test decomposes the data generating process:

\[y_{it} = r_{it} + \beta_i t + \varepsilon_{it}\]

\[r_{it} = r_{i,t-1} + u_{it}\]

where:

\(r_{it}\) is a random walk component
\(u_{it}\) are random walk innovations with variance \(\sigma^2_{u_i}\)
\(\varepsilon_{it}\) is a stationary error

Under H₀: \(\sigma^2_{u_i} = 0\) for all \(i\) (no random walk component, series is stationary).

LM Statistic¶

The test statistic is an LM (Lagrange Multiplier) statistic based on the KPSS framework:

For each entity \(i\), regress \(y_{it}\) on deterministic terms and compute residuals \(\hat{\varepsilon}_{it}\)
Compute partial sums: \(S_{it} = \sum_{s=1}^{t} \hat{\varepsilon}_{is}\)
Entity-specific LM statistic:

\[LM_i = \frac{1}{T^2} \sum_{t=1}^{T} \frac{S_{it}^2}{\hat{\sigma}^2_{\varepsilon_i}}\]

Average across entities: \(\overline{LM} = \frac{1}{N} \sum_{i=1}^{N} LM_i\)

Standardization¶

The Z-statistic is:

\[Z = \frac{\sqrt{N}(\overline{LM} - \mu)}{\sigma} \xrightarrow{d} N(0, 1)\]

where \(\mu\) and \(\sigma\) are asymptotic moments that depend on the trend specification:

Specification	\(\mu\)	\(\sigma^2\)
Constant only (`trend='c'`)	\(1/6\)	\(1/45\)
Constant + trend (`trend='ct'`)	\(1/15\)	\(1/6300\)

The p-value is computed as \(1 - \Phi(Z)\) (right-tailed test: reject for large values).

Configuration Options¶

hadri_test(
    data,                   # pd.DataFrame: Panel data in long format
    variable,               # str: Variable to test
    entity_col='entity',    # str: Entity identifier column
    time_col='time',        # str: Time identifier column
    trend='c',              # str: 'c' (constant) or 'ct' (constant + trend)
    robust=True,            # bool: Use heteroskedasticity-robust variance
    alpha=0.05,             # float: Significance level
)

Robust vs. Non-Robust¶

Robust (recommended)Non-robust

# Heteroskedasticity-robust version using Newey-West estimator
result = hadri_test(data, "invest", "firm", "year",
                    trend="c", robust=True)

Uses a Bartlett kernel with automatic bandwidth selection for the long-run variance estimate.

# Standard (homoskedastic) version
result = hadri_test(data, "invest", "firm", "year",
                    trend="c", robust=False)

Assumes homoskedastic errors within each entity.

Result Object: `HadriResult`¶

Attribute	Type	Description
`statistic`	`float`	Z-statistic (standardized LM)
`pvalue`	`float`	P-value from standard normal (right-tailed)
`reject`	`bool`	Whether to reject H₀ at the specified \(\alpha\)
`lm_statistic`	`float`	Raw LM statistic before standardization
`individual_lm`	`np.ndarray`	LM statistics for each entity
`n_entities`	`int`	Number of cross-sectional units
`n_time`	`int`	Number of time periods
`trend`	`str`	Trend specification (`'c'` or `'ct'`)
`robust`	`bool`	Whether robust version was used

When to Use¶

Use Hadri when:

You want to confirm stationarity findings from other tests
The economic theory suggests variables should be stationary, and you want to test this as H₀
You want a complete battery covering both unit root and stationarity null hypotheses

Complete Battery Example¶

from panelbox.datasets import load_grunfeld
from panelbox.validation.unit_root.llc import LLCTest
from panelbox.validation.unit_root.ips import IPSTest
from panelbox.diagnostics.unit_root.hadri import hadri_test

data = load_grunfeld()

# H0: unit root tests
llc = LLCTest(data, "invest", "firm", "year", trend="c")
llc_result = llc.run()

ips = IPSTest(data, "invest", "firm", "year", trend="c")
ips_result = ips.run()

# H0: stationarity test (Hadri)
hadri_result = hadri_test(data, "invest", "firm", "year",
                           trend="c", robust=True)

print("=== Unit Root Battery ===")
print(f"LLC (H0: unit root):      p={llc_result.pvalue:.4f}")
print(f"IPS (H0: unit root):      p={ips_result.pvalue:.4f}")
print(f"Hadri (H0: stationarity): p={hadri_result.pvalue:.4f}")

# Interpret
if llc_result.pvalue < 0.05 and not hadri_result.reject:
    print("Conclusion: Stationarity confirmed")
elif llc_result.pvalue >= 0.05 and hadri_result.reject:
    print("Conclusion: Unit root confirmed")
else:
    print("Conclusion: Mixed evidence -- investigate further")

Common Pitfalls¶

Balanced Panel Required

The Hadri test requires a balanced panel (same \(T\) for all entities). If the panel is unbalanced, the function raises a ValueError.

Sensitivity to Cross-Sectional Dependence

The Hadri test is sensitive to cross-sectional dependence. When entities are correlated (common shocks), the test tends to over-reject the stationarity null, falsely indicating unit roots. Always test for cross-sectional dependence first and interpret with caution.

Trend Misspecification

Choosing the wrong trend specification can distort results. If the data has a deterministic trend but you use trend='c', the test may falsely reject stationarity. Conversely, including an unnecessary trend (trend='ct' for trendless data) reduces power.

References¶

Hadri, K. (2000). "Testing for stationarity in heterogeneous panel data." Econometrics Journal, 3(2), 148-161.
Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). "Testing the null hypothesis of stationarity against the alternative of a unit root." Journal of Econometrics, 54(1-3), 159-178.
Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6^th ed.). Springer. Chapter 12.