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LLC Test (Levin-Lin-Chu)

Quick Reference

Class: panelbox.validation.unit_root.llc.LLCTest H₀: All panels contain unit roots (common \(\rho = 0\)) H₁: All panels are stationary (common \(\rho < 0\)) Stata equivalent: xtunitroot llc variable, lags(p) trend(none|demean|trend) R equivalent: plm::purtest(x, test="levinlin")

What It Tests

The Levin-Lin-Chu (LLC) test examines whether all panels in the dataset share a common unit root. It assumes that the autoregressive parameter \(\rho\) is the same across all entities, making it a homogeneous panel unit root test.

If the test rejects H₀, there is evidence that all panels are stationary. If it fails to reject, the data is consistent with a unit root process in all panels.

Quick Example

from panelbox.datasets import load_grunfeld
from panelbox.validation.unit_root.llc import LLCTest

data = load_grunfeld()

# Basic LLC test with constant
llc = LLCTest(data, "invest", "firm", "year", trend="c")
result = llc.run()
print(result)

Output:

======================================================================
Levin-Lin-Chu Panel Unit Root Test
======================================================================
Test statistic:    -3.2145
P-value:           0.0007
Lags:              1
Observations:      180
Cross-sections:    10
Deterministics:    Constant

H0: All panels contain unit roots
H1: All panels are stationary

Conclusion: Reject H0: Evidence against unit root (panels are stationary)
======================================================================

Interpretation

p-value Decision Meaning
\(p < 0.01\) Strong rejection of H₀ Strong evidence of stationarity across all panels
\(0.01 \leq p < 0.05\) Rejection of H₀ Evidence of stationarity
\(0.05 \leq p < 0.10\) Borderline Weak evidence; consider additional tests
\(p \geq 0.10\) Fail to reject H₀ Consistent with unit root in all panels

Mathematical Details

Model

The LLC test is based on the augmented Dickey-Fuller regression applied to each panel:

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

where:

  • \(\rho\) is the common autoregressive parameter (same for all \(i\))
  • \(\alpha_i\) are entity-specific intercepts
  • \(\delta_i t\) are entity-specific time trends (when trend='ct')
  • \(p_i\) is the lag order (can be entity-specific via AIC)

Procedure

The test follows a three-step procedure:

  1. Orthogonalize: For each entity \(i\), regress \(\Delta y_{it}\) and \(y_{i,t-1}\) on the augmented lags and deterministic terms. Collect the residuals \(\tilde{e}_{it}\) and \(\tilde{v}_{it}\).

  2. Normalize: Divide \(\tilde{e}_{it}\) and \(\tilde{v}_{it}\) by each entity's long-run standard deviation \(\hat{\sigma}_i\) to remove heteroskedasticity across entities.

  3. Pool and test: Run the pooled regression \(\tilde{e}_{it} / \hat{\sigma}_i = \rho \, \tilde{v}_{it} / (\hat{\sigma}_i \sqrt{T_i}) + \text{error}\) and compute the adjusted t-statistic.

Test Statistic

The adjusted t-statistic converges to a standard normal distribution under H₀:

\[t_{\rho}^{*} \xrightarrow{d} N(0, 1) \quad \text{as } N, T \to \infty\]

The p-value is computed from the standard normal CDF (left-tailed test).

Configuration Options

LLCTest(
    data,                   # pd.DataFrame: Panel data in long format
    variable,               # str: Variable to test for unit root
    entity_col,             # str: Entity identifier column
    time_col,               # str: Time identifier column
    lags=None,              # int or None: Number of augmented lags (None = auto via AIC)
    trend='c',              # str: 'n' (none), 'c' (constant), 'ct' (constant + trend)
)

Trend Specifications

# For data without intercept or trend (e.g., returns)
llc = LLCTest(data, "returns", "firm", "year", trend="n")
result = llc.run()

# Default — entity-specific intercept
llc = LLCTest(data, "invest", "firm", "year", trend="c")
result = llc.run()

# For variables with deterministic trends (e.g., log GDP)
llc = LLCTest(data, "log_gdp", "country", "year", trend="ct")
result = llc.run()

Result Object: LLCTestResult

Attribute Type Description
statistic float Adjusted t-statistic
pvalue float P-value from standard normal
lags int Number of lags used
n_obs int Number of observations
n_entities int Number of cross-sectional units
test_type str "LLC"
deterministics str Description of trend specification
null_hypothesis str "All panels contain unit roots"
alternative_hypothesis str "All panels are stationary"
conclusion str Test conclusion at 5% level

When to Use

Use LLC when:

  • Entities share similar dynamics (common \(\rho\) is plausible)
  • The panel is balanced (same \(T\) for all entities)
  • You need a simple, well-known test as a baseline

Examples of appropriate settings:

  • Exchange rates for countries in the same monetary zone
  • Stock returns in the same market
  • Inflation rates for similar economies

Common Pitfalls

Balanced Panel Required

The LLC test requires a balanced panel. If your panel is unbalanced, a warning is issued. Consider using the Fisher test instead, which handles unbalanced panels naturally.

Homogeneity Assumption

The LLC test assumes all panels share the same \(\rho\). If entities have heterogeneous adjustment speeds, the test may have low power or produce misleading results. Use the IPS test for heterogeneous panels.

Cross-Sectional Independence

The LLC test assumes cross-sectional independence. Common shocks (e.g., global financial crises) violate this assumption and can lead to size distortions (over-rejection). The Breitung test is more robust to cross-sectional dependence.

See Also

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.
  • Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6th ed.). Springer. Chapter 12.