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Unit Root Tests

Quick Reference

Module: panelbox.validation.unit_root (LLC, IPS, Fisher) and panelbox.diagnostics.unit_root (Hadri, Breitung) Purpose: Test whether panel data series are stationary or contain unit roots Stata equivalent: xtunitroot llc, xtunitroot ips, xtunitroot fisher, xtunitroot hadri, xtunitroot breitung R equivalent: plm::purtest()

Why Stationarity Matters

Non-stationary data creates serious problems for panel data analysis:

  • Spurious regression: Regressions between unrelated I(1) variables produce significant-looking but meaningless results (Granger & Newbold, 1974)
  • Invalid inference: Standard t-statistics and F-tests have non-standard distributions when variables contain unit roots
  • Inconsistent estimators: OLS estimates may not converge to true parameter values
  • Model misspecification: Ignoring unit roots leads to incorrect dynamic models

Testing for unit roots is a prerequisite before estimating panel models. If variables are I(1), you should either first-difference the data or test for cointegration.

Unit Root in Panel Data

Panel unit root tests extend the univariate ADF/PP framework to panel settings. The panel dimension provides more power because the cross-sectional variation supplements the time-series variation.

The general model is:

\[\Delta y_{it} = \rho_i y_{i,t-1} + \sum_{j=1}^{p_i} \phi_{ij} \Delta y_{i,t-j} + \alpha_i + \delta_i t + \varepsilon_{it}\]

where:

  • \(y_{it}\) is the variable of interest for entity \(i\) at time \(t\)
  • \(\rho_i\) is the autoregressive parameter (unit root if \(\rho_i = 0\))
  • \(\alpha_i\) is an entity-specific intercept (fixed effect)
  • \(\delta_i t\) is an entity-specific linear trend
  • \(p_i\) is the number of augmented lags

Five Tests Compared

PanelBox implements five panel unit root tests, each with different assumptions:

Test H₀ H₁ Root Type Heterogeneous? Unbalanced?
LLC Unit root All stationary Common \(\rho\) No No
IPS Unit root Some stationary Individual \(\rho_i\) Yes Limited
Fisher Unit root Some stationary Individual \(\rho_i\) Yes Yes
Hadri Stationary Some unit roots -- No No
Breitung Unit root All stationary Common \(\rho\) No No

Key Distinctions

Common vs. heterogeneous autoregressive parameter:

  • LLC and Breitung assume a common \(\rho\) across all entities. This is appropriate when entities share similar dynamics (e.g., exchange rates under a common monetary regime).
  • IPS and Fisher allow heterogeneous \(\rho_i\) per entity. This is more realistic when entities have different adjustment speeds (e.g., GDP across countries with different economic structures).

Hadri: Opposite Null Hypothesis

The Hadri test has the opposite null hypothesis from all other tests. Its H₀ is stationarity, not unit root. Rejecting H₀ in the Hadri test means evidence against stationarity -- the opposite of what rejecting H₀ means in LLC/IPS/Fisher/Breitung.

Trend Specification

All tests support deterministic trend specifications that control for entity-specific intercepts and trends:

Specification Code Description When to Use
No deterministic terms trend='n' No intercept or trend Returns, growth rates, already-demeaned data
Constant only trend='c' Entity-specific intercept Default; most common for levels
Constant + trend trend='ct' Intercept and linear trend Variables with deterministic trends (e.g., log GDP)

Note

Hadri and Breitung only support trend='c' and trend='ct' (no 'n' option).

Lag Selection

Lag selection controls the number of augmented terms \(\Delta y_{i,t-j}\) included to account for serial correlation in the errors:

  • Automatic (lags=None): Selected by AIC. Recommended for most applications.
  • Manual (lags=k): User-specified fixed lag length for all entities.
  • Per-entity (IPS only, lags=dict): Different lag orders per entity via a dictionary mapping entity to lag count.

A common rule of thumb for maximum lag length: \(p_{max} = \lfloor 12 (T/100)^{1/3} \rfloor\).

Quick Example: Battery of Tests

The recommended approach is to run multiple tests as a battery for robustness:

from panelbox.datasets import load_grunfeld
from panelbox.validation.unit_root.llc import LLCTest
from panelbox.validation.unit_root.ips import IPSTest
from panelbox.validation.unit_root.fisher import FisherTest
from panelbox.diagnostics.unit_root.hadri import hadri_test
from panelbox.diagnostics.unit_root.breitung import breitung_test

# Load data
data = load_grunfeld()

# --- Test battery on 'invest' variable ---

# 1. LLC: Common unit root (H0: unit root)
llc = LLCTest(data, "invest", "firm", "year", trend="c")
llc_result = llc.run()
print(f"LLC:      stat={llc_result.statistic:.4f}, p={llc_result.pvalue:.4f}")

# 2. IPS: Heterogeneous unit root (H0: unit root)
ips = IPSTest(data, "invest", "firm", "year", trend="c")
ips_result = ips.run()
print(f"IPS:      stat={ips_result.statistic:.4f}, p={ips_result.pvalue:.4f}")

# 3. Fisher-ADF: Combining p-values (H0: unit root)
fisher = FisherTest(data, "invest", "firm", "year", test_type="adf", trend="c")
fisher_result = fisher.run()
print(f"Fisher:   stat={fisher_result.statistic:.4f}, p={fisher_result.pvalue:.4f}")

# 4. Breitung: Debiased common root (H0: unit root)
breit_result = breitung_test(data, "invest", "firm", "year", trend="ct")
print(f"Breitung: stat={breit_result.statistic:.4f}, p={breit_result.pvalue:.4f}")

# 5. Hadri: Stationarity null (H0: stationary -- REVERSED!)
hadri_result = hadri_test(data, "invest", "firm", "year", trend="c")
print(f"Hadri:    stat={hadri_result.statistic:.4f}, p={hadri_result.pvalue:.4f}")

Testing Strategy

Step 1: Run the main battery

Run LLC + IPS + Fisher on the variable in levels. These all test H₀: unit root.

Step 2: Confirm with Hadri

Run the Hadri test (H₀: stationarity) to confirm the findings from Step 1:

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

Step 3: Test first differences

If variables are I(1) in levels, verify that first differences are I(0) to confirm integration order:

# First difference the variable
data["d_invest"] = data.groupby("firm")["invest"].diff()
data_diff = data.dropna(subset=["d_invest"])

# Test first differences (should be stationary)
llc_diff = LLCTest(data_diff, "d_invest", "firm", "year", trend="c")
result_diff = llc_diff.run()
print(f"LLC on first differences: p={result_diff.pvalue:.4f}")
# Expect: small p-value (reject unit root in differences)

Step 4: Next steps

  • If variables are I(0): Estimate models in levels (Fixed Effects, Random Effects, etc.)
  • If variables are I(1): Either first-difference the data or test for cointegration
  • If variables are cointegrated: Use Panel VECM or estimate long-run relationships

What If a Unit Root Is Detected?

  1. First-difference the data to achieve stationarity:

    data["dy"] = data.groupby("entity")["y"].diff()

  2. Use GMM estimators designed for dynamic panel data:

    from panelbox.gmm import DifferenceGMM

  3. Test for cointegration if multiple I(1) variables may share a long-run equilibrium:

    from panelbox.validation.cointegration.pedroni import PedroniTest
from panelbox.validation.cointegration.kao import KaoTest

  4. Include time effects to control for common trends across entities:

    from panelbox.models import FixedEffects

Software Comparison

Test PanelBox Stata R
LLC LLCTest xtunitroot llc plm::purtest(test="levinlin")
IPS IPSTest xtunitroot ips plm::purtest(test="ips")
Fisher FisherTest xtunitroot fisher plm::purtest(test="madwu")
Breitung breitung_test() xtunitroot breitung plm::purtest(test="Breitung")
Hadri hadri_test() xtunitroot hadri plm::purtest(test="hadri")

See Also

  • LLC Test -- Common unit root test (Levin-Lin-Chu)
  • IPS Test -- Heterogeneous unit root test (Im-Pesaran-Shin)
  • Fisher Test -- Combining individual p-values (Maddala-Wu)
  • Hadri Test -- Stationarity null test (reversed hypothesis)
  • Breitung Test -- Debiased common unit root test
  • Cointegration Tests -- Testing for long-run relationships between I(1) variables

References

  • Levin, A., Lin, C. F., & Chu, C. S. J. (2002). "Unit root tests in panel data: asymptotic and finite-sample properties." Journal of Econometrics, 108(1), 1-24.
  • Im, K. S., Pesaran, M. H., & Shin, Y. (2003). "Testing for unit roots in heterogeneous panels." Journal of Econometrics, 115(1), 53-74.
  • Maddala, G. S., & Wu, S. (1999). "A comparative study of unit root tests with panel data and a new simple test." Oxford Bulletin of Economics and Statistics, 61(S1), 631-652.
  • Hadri, K. (2000). "Testing for stationarity in heterogeneous panel data." Econometrics Journal, 3(2), 148-161.
  • Breitung, J. (2000). "The local power of some unit root tests for panel data." In Advances in Econometrics, Vol. 15, 161-177.
  • Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6th ed.). Springer. Chapter 12.