Panel Tests

Panel-level hypothesis testing addresses two distinct phases of the empirical workflow. Panel unit root tests detect non-stationarity in datasets with cross-sectional dependence, a prerequisite for correct specification of panel VARs and factor models. Panel VAR specification tests validate GMM instrument validity and select optimal lag orders after estimation. This page covers three second-generation panel unit root tests (PANIC, Pesaran CIPS, Moon-Perron) and three Panel VAR diagnostics (Hansen J-test, Andrews-Lu MMSC, lag selection).

• PANIC: Factor-based decomposition into common and idiosyncratic components (Bai & Ng 2004, 2010)
• Pesaran CIPS: Cross-sectionally augmented IPS test robust to common factors (Pesaran 2007)
• Moon-Perron: Factor-adjusted pooled AR(1) with bias correction (Moon & Perron 2004)
• Hansen J-test, Andrews-Lu MMSC, and MMSC-based lag selection for Panel VAR

using MacroEconometricModels, Random, DataFrames
Random.seed!(42)

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Quick Start

Recipe 1: PANIC test on a panel

# Stationary panel with one common factor
X = randn(100, 20)
result = panic_test(X; r=1)
report(result)

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Recipe 2: Pesaran CIPS test

X = randn(50, 20)
result = pesaran_cips_test(X; lags=1, deterministic=:constant)
report(result)

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Recipe 3: Moon-Perron test

X = randn(80, 15)
result = moon_perron_test(X; r=1)
report(result)

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Recipe 4: Panel VAR Hansen J-test and lag selection

N, T_total, m_vars = 50, 20, 3
data = zeros(N * T_total, m_vars)
for i in 1:N
    mu = randn(m_vars) * 0.5
    for t in 2:T_total
        idx = (i-1)*T_total + t
        data[idx, :] = mu + 0.5 * data[(i-1)*T_total + t - 1, :] + 0.2 * randn(m_vars)
    end
end
df = DataFrame(data, ["y1", "y2", "y3"])
df.id = repeat(1:N, inner=T_total)
df.time = repeat(1:T_total, outer=N)
pd = xtset(df, :id, :time)
model = estimate_pvar(pd, 2; steps=:twostep)
j = pvar_hansen_j(model)
report(j)

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PANIC

The PANIC (Panel Analysis of Nonstationarity in Idiosyncratic and Common components) test of Bai & Ng (2004, 2010) decomposes panel data into common factors and idiosyncratic residuals via principal components, then tests each component for unit roots separately. This separation is critical because standard panel unit root tests (IPS, LLC) assume cross-sectional independence, an assumption violated in macroeconomic panels where global shocks affect all units.

The panel follows a factor structure:

\[X_{it} = \lambda_i' F_t + e_{it}\]

where:

• $X_{it}$ is the observation for unit $i$ at time $t$
• $F_t$ is the $r \times 1$ vector of common factors
• $\lambda_i$ is the $r \times 1$ vector of unit-specific loadings
• $e_{it}$ is the idiosyncratic error

The test proceeds in three steps. First, estimate $\hat{F}_t$ and $\hat{\lambda}_i$ via PCA on the $T \times N$ panel. Second, run ADF tests on each estimated factor $\hat{F}_{j,t}$ to determine whether the common components are I(1). Third, run ADF tests on the defactored residuals $\hat{e}_{it}$ and pool the individual p-values into a standardized statistic:

\[P_a = \frac{\sum_{i=1}^N p_i - N/2}{\sqrt{N/12}} \xrightarrow{d} N(0,1)\]

where:

• $p_i$ is the ADF p-value for unit $i$'s idiosyncratic residual
• $N$ is the number of cross-sectional units
• The standardization uses the mean ($1/2$) and variance ($1/12$) of the uniform distribution

Under $H_0$ (all idiosyncratic components have unit roots), the individual p-values are approximately uniform, and $P_a$ converges to a standard normal. Large negative values of $P_a$ indicate rejection: many units have small ADF p-values, providing evidence of idiosyncratic stationarity.

# Stationary panel: idiosyncratic components are I(0)
X_stationary = randn(100, 20)
result_s = panic_test(X_stationary; r=1)
report(result_s)

# I(1) panel: cumulate to create unit roots
X_nonstat = cumsum(randn(100, 20), dims=1)
result_ns = panic_test(X_nonstat; r=1)
report(result_ns)

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The stationary panel produces a small pooled p-value, indicating that the defactored residuals are stationary. The I(1) panel produces a large p-value, consistent with the unit root null. The factor ADF statistics in result.factor_adf_stats reveal whether the common component itself is non-stationary.

Options

PANICResult Return Values

The PanelData dispatch panic_test(pd::PanelData; kwargs...) extracts the first variable from each group, constructs a balanced $T \times N$ matrix, and passes it to the matrix method.

Pesaran CIPS

The Cross-sectionally Augmented IPS (CIPS) test of Pesaran (2007) augments individual ADF regressions with cross-sectional averages to account for dependence driven by a single unobserved common factor. Unlike PANIC, this test does not require explicit factor estimation: the cross-section average serves as a proxy for the common factor.

The Cross-sectionally Augmented Dickey-Fuller (CADF) regression for unit $i$ is:

\[\Delta y_{it} = a_i + b_i y_{i,t-1} + c_i \bar{y}_{t-1} + d_i \Delta\bar{y}_t + \sum_{j=1}^{p} \phi_j \Delta y_{i,t-j} + \varepsilon_{it}\]

where:

• $\Delta y_{it} = y_{it} - y_{i,t-1}$ is the first difference
• $b_i$ is the coefficient of interest; $H_0: b_i = 0$ for all $i$
• $\bar{y}_t = N^{-1}\sum_{i=1}^{N} y_{it}$ is the cross-sectional average
• $a_i$ is a unit-specific intercept (when deterministic=:constant)
• $p$ is the number of augmenting lags

The CIPS statistic averages the individual CADF t-statistics with truncation:

\[\text{CIPS} = N^{-1} \sum_{i=1}^{N} \tilde{t}_i\]

where $\tilde{t}_i = \max(-6.19, \min(6.19, t_i))$ are the truncated CADF statistics. The truncation at $\pm 6.19$ ensures that the CIPS statistic has finite moments even when some individual regressions produce extreme t-values.

Critical values depend on $N$, $T$, and the deterministic specification. The test rejects when the CIPS statistic falls below the critical value (left-tailed test).

# Stationary panel
X = randn(50, 20)
result_const = pesaran_cips_test(X; lags=1, deterministic=:constant)
report(result_const)

# With trend deterministic
result_trend = pesaran_cips_test(X; lags=1, deterministic=:trend)
report(result_trend)

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A CIPS statistic below the 5% critical value rejects the null of a panel unit root. The critical_values dictionary provides thresholds at the 1%, 5%, and 10% levels for the nearest tabulated $(N, T)$ combination.

Options

PesaranCIPSResult Return Values

Moon-Perron

The Moon & Perron (2004) test constructs factor-adjusted panel unit root statistics by projecting out estimated common factors from the data and then pooling unit-level AR(1) regressions. The approach is semi-parametric: it estimates the factor space via PCA but does not require parametric assumptions on the factor dynamics.

The test produces two modified t-statistics:

\[t_a^* = \frac{\sqrt{N}\, T\, (\hat{\rho}_{pool} - 1) - B_a}{S_a}, \quad t_b^* = \frac{\sqrt{N}\, \bar{t} - B_b}{S_b}\]

where:

• $\hat{\rho}_{pool}$ is the pooled AR(1) coefficient from de-factored data
• $\bar{t}$ is the average of individual t-statistics
• $B_a, B_b$ are bias corrections based on the ratio of long-run to short-run variance
• $S_a, S_b$ are variance corrections using fourth moments of the long-run variance

Both statistics converge to $N(0,1)$ under $H_0$: all units have unit roots. The bias corrections account for the serial correlation induced by the de-factoring step. The test rejects for large negative values (left-tailed).

# Stationary panel
X = randn(80, 15)
result = moon_perron_test(X; r=1)
report(result)

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Both p-values test the same null hypothesis using different pooling approaches. When both reject at the 5% level, the evidence against the panel unit root is strong. When only one rejects, the evidence is moderate.

Options

MoonPerronResult Return Values

Panel Unit Root Summary

The convenience function panel_unit_root_summary runs all three panel unit root tests (PANIC, Pesaran CIPS, Moon-Perron) and prints a consolidated report. This is the recommended entry point for pre-estimation diagnostics on panel data.

X = randn(100, 20)
panel_unit_root_summary(X; r=1)

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The output displays each test's specification, statistics, p-values, and conclusion in sequence. If any individual test fails (e.g., due to insufficient observations), the function prints the error and continues with the remaining tests.

Panel VAR Specification Tests

After estimating a Panel VAR by GMM, three diagnostics validate the specification: the Hansen J-test for instrument validity, Andrews-Lu MMSC criteria for model selection, and MMSC-based lag order selection. These tests apply to GMM-estimated models (estimate_pvar with :onestep or :twostep), not to FE-OLS models.

The following data generation pattern is used throughout this section:

N, T_total, m_vars = 50, 20, 3
data = zeros(N * T_total, m_vars)
for i in 1:N
    mu = randn(m_vars) * 0.5
    for t in 2:T_total
        idx = (i-1)*T_total + t
        data[idx, :] = mu + 0.5 * data[(i-1)*T_total + t - 1, :] + 0.2 * randn(m_vars)
    end
end
df = DataFrame(data, ["y1", "y2", "y3"])
df.id = repeat(1:N, inner=T_total)
df.time = repeat(1:T_total, outer=N)
pd = xtset(df, :id, :time)
nothing # hide

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Hansen J-Test

The Hansen (1982) J-test evaluates whether the overidentifying restrictions in a GMM model are satisfied. In Panel VAR, the instrument set (lagged levels for first-difference GMM, or lagged levels and differences for system GMM) typically exceeds the number of parameters, creating overidentifying conditions.

The J-statistic is:

\[J = N \, \bar{g}' \, W \, \bar{g} \sim \chi^2(q - k)\]

where:

• $\bar{g} = N^{-1} \sum_{i=1}^{N} Z_i' e_i$ is the average moment condition
• $W$ is the optimal weighting matrix
• $q$ is the number of instruments
• $k$ is the number of parameters per equation
• $q - k$ are the degrees of freedom (overidentifying restrictions)

Under $H_0$ (all moment conditions are valid), the J-statistic follows a chi-squared distribution. Rejection indicates that some instruments are invalid, pointing to misspecification or endogeneity problems.

N2, T2, m2 = 50, 20, 3
data2 = zeros(N2 * T2, m2)
for i in 1:N2
    mu = randn(m2) * 0.5
    for t in 2:T2
        idx = (i-1)*T2 + t
        data2[idx, :] = mu + 0.5 * data2[(i-1)*T2 + t - 1, :] + 0.2 * randn(m2)
    end
end
df2 = DataFrame(data2, ["y1", "y2", "y3"])
df2.id = repeat(1:N2, inner=T2)
df2.time = repeat(1:T2, outer=N2)
pd2 = xtset(df2, :id, :time)
model = estimate_pvar(pd2, 2; steps=:twostep)
j = pvar_hansen_j(model)
report(j)

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A p-value above 0.05 means the test fails to reject instrument validity. This is a necessary but not sufficient condition: the J-test has low power when the number of instruments is large relative to the sample size.

PVARTestResult Return Values

Andrews-Lu MMSC

The Andrews & Lu (2001) Model and Moment Selection Criteria extend information criteria to the GMM setting. They penalize the J-statistic by the number of overidentifying restrictions, enabling comparison across models with different lag orders or instrument sets.

The three criteria are:

\[\text{MMSC-BIC} = J - (q - k) \ln n\]

\[\text{MMSC-AIC} = J - 2(q - k)\]

\[\text{MMSC-HQIC} = J - Q(q - k) \ln \ln n\]

where:

• $J$ is the Hansen J-statistic
• $q - k$ is the number of overidentifying restrictions
• $n$ is the total number of observations
• $Q = 2.1$ is the default Hannan-Quinn constant

Lower values indicate a better-fitting specification. MMSC-BIC penalizes most heavily and tends to select parsimonious models.

mmsc = pvar_mmsc(model)
report(mmsc)

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Lag Selection

The pvar_lag_selection function estimates Panel VAR models for lag orders $p = 1, \ldots, p_{max}$ and compares them using Andrews-Lu MMSC criteria. This automates the model comparison workflow and returns the optimal lag order under each criterion.

sel = pvar_lag_selection(pd2, 4)
report(sel)

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The table field contains a $p_{max} \times 4$ matrix with columns for the lag order and the three MMSC values. The models vector stores all estimated PVARModel objects for further analysis.

Complete Example

This example demonstrates a full panel diagnostics workflow: test for unit roots with all three second-generation tests, estimate a Panel VAR, and validate the specification with GMM diagnostics.

# --- Step 1: Generate panel data with a common factor ---
N_ce, T_obs = 30, 60
r_true = 1
F = cumsum(randn(T_obs, r_true), dims=1)          # I(1) common factor
Lambda = randn(N_ce, r_true)                        # unit-specific loadings
e = randn(T_obs, N_ce) * 0.5                        # stationary idiosyncratic errors
X = F * Lambda' + e                                  # T x N panel

# --- Step 2: PANIC test ---
panic_result = panic_test(X; r=1)
report(panic_result)

# --- Step 3: Pesaran CIPS test ---
cips_result = pesaran_cips_test(X; lags=1, deterministic=:constant)
report(cips_result)

# --- Step 4: Moon-Perron test ---
mp_result = moon_perron_test(X; r=1)
report(mp_result)

# --- Step 5: Run all three tests together ---
panel_unit_root_summary(X; r=1)

# --- Step 6: Panel VAR estimation and specification tests ---
N_pvar, T_pvar, m_v = 50, 20, 3
pvar_data = zeros(N_pvar * T_pvar, m_v)
for i in 1:N_pvar
    mu = randn(m_v) * 0.5
    for t in 2:T_pvar
        idx = (i-1)*T_pvar + t
        pvar_data[idx, :] = mu + 0.5 * pvar_data[(i-1)*T_pvar + t - 1, :] + 0.2 * randn(m_v)
    end
end
df_ce = DataFrame(pvar_data, ["y1", "y2", "y3"])
df_ce.id = repeat(1:N_pvar, inner=T_pvar)
df_ce.time = repeat(1:T_pvar, outer=N_pvar)
pd_ce = xtset(df_ce, :id, :time)

# Estimate PVAR(2) by two-step GMM
model_ce = estimate_pvar(pd_ce, 2; steps=:twostep)

# Hansen J-test: validate overidentifying restrictions
j = pvar_hansen_j(model_ce)
report(j)

# MMSC criteria for this specification
mmsc = pvar_mmsc(model_ce)
report(mmsc)

# Lag selection: compare p = 1, ..., 4
sel = pvar_lag_selection(pd_ce, 4)
report(sel)

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The PANIC test separates the common factor (I(1)) from the stationary idiosyncratic errors, providing a nuanced view that the composite tests (CIPS, Moon-Perron) cannot. The Panel VAR diagnostics confirm that the GMM instruments are valid and identify the preferred lag order.

Common Pitfalls

1. Too few cross-sectional units. Panel unit root tests rely on $N \to \infty$ asymptotics. With $N < 20$, all three tests (PANIC, CIPS, Moon-Perron) have reduced size and power. Use at least 20 cross-sectional units for reliable inference.

2. Sensitivity of automatic factor selection. The :auto option for r in PANIC and Moon-Perron uses the Bai-Ng IC2 criterion, which can be sensitive to the signal-to-noise ratio. Always compare results with different fixed values of r (e.g., r=1, r=2, r=3) to check robustness.

3. CIPS truncation with unbalanced panels. The CADF t-statistics are truncated at $\pm 6.19$ to ensure finite moments. In heavily unbalanced panels where some units have very short time series, truncation binds frequently, reducing the test's discriminatory power. Ensure adequate time dimension for all units.

4. J-test low power with many instruments. In Panel VAR, the number of GMM instruments grows quadratically with the time dimension. When $q \gg k$, the J-test almost never rejects, even when some instruments are invalid. Non-rejection does not validate the instrument set. Use system_instruments=false and shorter lag windows to keep the instrument count manageable.

References

• Andrews, D. W. K., & Lu, B. (2001). Consistent Model and Moment Selection Procedures for GMM Estimation with Application to Dynamic Panel Data Models. Journal of Econometrics, 101(1), 123-164. DOI

• Bai, J., & Ng, S. (2002). Determining the Number of Factors in Approximate Factor Models. Econometrica, 70(1), 191-221. DOI

• Bai, J., & Ng, S. (2004). A PANIC Attack on Unit Roots and Cointegration. Econometrica, 72(4), 1127-1177. DOI

• Bai, J., & Ng, S. (2010). Panel Unit Root Tests with Cross-Section Dependence: A Further Investigation. Econometric Theory, 26(4), 1088-1114. DOI

• Hansen, L. P. (1982). Large Sample Properties of Generalized Method of Moments Estimators. Econometrica, 50(4), 1029-1054. DOI

• Moon, H. R., & Perron, B. (2004). Testing for a Unit Root in Panels with Dynamic Factors. Journal of Econometrics, 122(1), 81-126. DOI

• Pesaran, M. H. (2007). A Simple Panel Unit Root Test in the Presence of Cross-Section Dependence. Journal of Applied Econometrics, 22(2), 265-312. DOI