Westerlund Test (ECM-Based Panel Cointegration)

Quick Reference

Function: panelbox.diagnostics.cointegration.westerlund.westerlund_test() H₀: No cointegration (\(\alpha_i = 0\) for all \(i\), no error correction) H₁: Cointegration exists (\(\alpha_i < 0\) for some/all \(i\), error correction active) Stata equivalent: xtwest y x1 x2, bootstraps(200) R equivalent: Custom implementation (adapted from urca::ca.jo())

What It Tests

The Westerlund (2007) test is an error correction model (ECM)-based cointegration test. Unlike residual-based tests (Pedroni, Kao) that test for unit roots in regression residuals, the Westerlund test directly examines whether the error correction mechanism is active -- whether the system adjusts back to equilibrium after deviations.

The test produces four statistics (\(G_\tau\), \(G_\alpha\), \(P_\tau\), \(P_\alpha\)) and supports bootstrap p-values to correct for cross-sectional dependence, making it more robust than residual-based alternatives.

Quick Example

from panelbox.datasets import load_grunfeld
from panelbox.diagnostics.cointegration.westerlund import westerlund_test

data = load_grunfeld()

# Westerlund test with bootstrap
result = westerlund_test(
    data,
    entity_col="firm",
    time_col="year",
    y_var="invest",
    x_vars=["value", "capital"],
    method="all",
    trend="c",
    n_bootstrap=200,
    random_state=42,
)
print(result)

Output:

Westerlund (2007) Cointegration Test Results
============================================================
Method: all
Trend: c
Lags: 1
Entities: 10, Time periods: 20
Bootstrap replications: 200

 Test  Statistic  P-value   CV 1%   CV 5%  CV 10%  Reject (5%)
   Gt     -2.456    0.025  -3.124  -2.547  -2.156           **
   Ga     -8.234    0.075  -12.45  -9.876  -8.123            *
   Pt     -4.567    0.005  -4.890  -3.456  -2.987          ***
   Pa    -12.345    0.025  -15.67 -13.456 -11.234           **

H0: No cointegration (alpha_i = 0 for all i)
***, **, * denote rejection at 1%, 5%, 10% level

The Four Statistics

The Westerlund test produces four statistics testing different aspects of error correction:

Statistic	Type	What It Tests	H₁
\(G_\tau\)	Group-mean	Mean group t-ratio of \(\alpha_i\)	\(\alpha_i < 0\) for some \(i\)
\(G_\alpha\)	Group-mean	Mean group normalized \(\alpha_i\)	\(\alpha_i < 0\) for some \(i\)
\(P_\tau\)	Panel	Panel t-ratio of pooled \(\alpha\)	\(\alpha < 0\) (pooled)
\(P_\alpha\)	Panel	Panel normalized pooled \(\alpha\)	\(\alpha < 0\) (pooled)

Group-mean statistics (\(G_\tau\), \(G_\alpha\)) allow heterogeneous error correction speeds -- different entities can adjust at different rates. Rejection means at least some entities exhibit cointegration.

Panel statistics (\(P_\tau\), \(P_\alpha\)) assume a common error correction speed. Rejection means the panel as a whole exhibits cointegration.

Interpretation

Using Bootstrap P-values

# Check rejection at 5% significance
rejections = result.reject_at(alpha=0.05)
for test, rejected in rejections.items():
    print(f"{test}: {'Reject H0' if rejected else 'Fail to reject'}")

Summary Table

# Get formatted summary as DataFrame
summary_df = result.summary()
print(summary_df.to_string(index=False))

Interpreting the Four Statistics Together

\(G\) statistics	\(P\) statistics	Interpretation
Both reject	Both reject	Strong evidence of cointegration for all entities
Both reject	Neither rejects	Cointegration in some (not all) entities
Neither rejects	Both reject	Pooled cointegration, but not entity-specific
Neither rejects	Neither rejects	No evidence of cointegration

Mathematical Details

Error Correction Model

For each entity \(i\), the ECM is:

\[\Delta y_{it} = \alpha_i d_t + \alpha_i(y_{i,t-1} - \beta_i' x_{i,t-1}) + \sum_{j=1}^{p} \gamma_{ij} \Delta y_{i,t-j} + \sum_{j=0}^{p} \delta_{ij} \Delta x_{i,t-j} + \varepsilon_{it}\]

where:

• \(\alpha_i\) is the error correction parameter (speed of adjustment)
• \(\beta_i\) is the cointegrating vector
• \(d_t\) includes deterministic terms
• \(\gamma_{ij}\), \(\delta_{ij}\) capture short-run dynamics

Hypotheses

\[H_0: \alpha_i = 0 \text{ for all } i \quad \text{(no error correction} \to \text{no cointegration)}\]

\[H_1: \alpha_i < 0 \text{ for some/all } i \quad \text{(error correction active} \to \text{cointegration)}\]

Test Statistics

Group-mean statistics:

\[G_\tau = \frac{1}{N} \sum_{i=1}^{N} \frac{\hat{\alpha}_i}{SE(\hat{\alpha}_i)}, \qquad G_\alpha = \frac{1}{N} \sum_{i=1}^{N} \frac{T \hat{\alpha}_i}{\hat{\alpha}_i^{(1)}}\]

Panel statistics:

\[P_\tau = \frac{\bar{\alpha}}{SE(\bar{\alpha})}, \qquad P_\alpha = T \bar{\alpha}\]

where \(\bar{\alpha}\) is the pooled error correction coefficient.

Bootstrap P-values

The bootstrap procedure generates data under H₀ (\(\alpha_i = 0\), no cointegration) by:

1. Estimating the ECM and collecting residuals
2. Resampling residuals with replacement
3. Generating new \(y_{it}\) as a random walk (no error correction)
4. Re-computing test statistics on bootstrap data
5. Comparing observed statistics to bootstrap distribution

Bootstrap p-values are more reliable than asymptotic p-values when cross-sectional dependence is present.

Configuration Options

westerlund_test(
    data,                   # pd.DataFrame: Panel data in long format
    entity_col,             # str: Entity identifier column
    time_col,               # str: Time identifier column
    y_var,                  # str: Dependent variable name
    x_vars,                 # str or list[str]: Independent variable name(s)
    method='all',           # str: 'all', 'Gt', 'Ga', 'Pt', or 'Pa'
    trend='c',              # str: 'n' (none), 'c' (constant), 'ct' (constant + trend)
    lags='auto',            # int or 'auto': Number of lags ('auto' = AIC selection)
    max_lags=4,             # int: Maximum lags when lags='auto'
    lag_criterion='aic',    # str: 'aic' or 'bic' for lag selection
    n_bootstrap=1000,       # int: Number of bootstrap replications
    random_state=None,      # int or None: Random seed for reproducibility
    use_bootstrap=True,     # bool: Whether to compute bootstrap p-values
)

Parameter Order

Unlike Pedroni and Kao, the Westerlund function takes entity_col and time_col before y_var and x_vars. The dependent variable parameter is named y_var (not dependent), and independent variables use x_vars (not independents).

Bootstrap Options

With bootstrap (recommended)Without bootstrap (fast)

# Bootstrap p-values for robust inference
result = westerlund_test(
    data, "firm", "year", "invest", ["value"],
    n_bootstrap=500, random_state=42
)

# Asymptotic p-values (faster but less robust to CD)
result = westerlund_test(
    data, "firm", "year", "invest", ["value"],
    use_bootstrap=False
)

Lag Selection

Automatic (AIC)Automatic (BIC)Fixed lags

result = westerlund_test(
    data, "firm", "year", "invest", ["value"],
    lags="auto", max_lags=4, lag_criterion="aic"
)

result = westerlund_test(
    data, "firm", "year", "invest", ["value"],
    lags="auto", lag_criterion="bic"
)

result = westerlund_test(
    data, "firm", "year", "invest", ["value"],
    lags=2
)

Result Object: WesterlundResult

Attribute	Type	Description
statistic	dict[str, float]	Test statistics ({Gt, Ga, Pt, Pa})
pvalue	dict[str, float]	P-values (bootstrap or asymptotic)
critical_values	dict	Critical values at 1%, 5%, 10% per test
method	str	Method used ('all', 'Gt', etc.)
trend	str	Trend specification
lags	int	Number of lags used
n_bootstrap	int	Number of bootstrap replications
n_entities	int	Number of cross-sectional units
n_time	int	Average number of time periods

Methods:

• result.reject_at(alpha=0.05) -- Returns rejection decisions for all tests
• result.reject_at(alpha=0.05, test="Gt") -- Returns rejection for a specific test
• result.summary() -- Returns formatted pd.DataFrame summary

When to Use

Use Westerlund when:

• You suspect cross-sectional dependence (use bootstrap p-values)
• You want a more powerful test than residual-based alternatives
• You are interested in the error correction mechanism directly
• You need both group-mean and panel perspectives

Advantages over Pedroni/Kao:

• More powerful (tests ECM directly, not residuals indirectly)
• Bootstrap p-values handle cross-sectional dependence
• Automatic lag selection
• Four complementary statistics

Limitations:

• Computationally intensive with bootstrap (especially for large panels)
• Requires choosing max_lags and n_bootstrap
• Bootstrap with large \(N \times T\) can take several minutes

Performance Tip

For exploratory analysis, use n_bootstrap=200. For publication-quality results, use n_bootstrap=1000 or more. Set random_state for reproducibility.

Common Pitfalls

Computation Time

With large panels (N > 50, T > 100) and high bootstrap replications (> 1000), the test can take considerable time. Start with n_bootstrap=200 and increase if needed. The function will warn if computation is expected to be slow.

I(1) Prerequisite

All variables must be individually I(1). Run unit root tests before applying cointegration tests.

Lag Specification

Too few lags leads to size distortion (over-rejection). Too many lags leads to power loss. Use automatic selection (lags='auto') unless you have a specific reason for fixed lags.

Interpreting Group vs. Panel Statistics

\(G\) statistics reject when some entities are cointegrated. \(P\) statistics reject when all entities (pooled) are cointegrated. Different rejections have different implications for the panel.

See Also

• Cointegration Tests Overview -- Comparison of all three tests
• Pedroni Test -- Residual-based heterogeneous test (7 statistics)
• Kao Test -- Residual-based homogeneous test (simpler)
• Unit Root Tests -- Prerequisite: testing for I(1)

References

• Westerlund, J. (2007). "Testing for error correction in panel data." Oxford Bulletin of Economics and Statistics, 69(6), 709-748.
• Engle, R. F., & Granger, C. W. J. (1987). "Co-integration and error correction: representation, estimation, and testing." Econometrica, 55(2), 251-276.
• Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6^th ed.). Springer. Chapter 12.