Panel Stationarity Tests: CADF and CIPS Explained

 When dealing with panel data, testing for stationarity (i.e., whether variables have a constant mean and variance over time) is crucial before estimation—especially in macroeconomic and financial studies.

Two advanced second-generation panel unit root tests used to check for stationarity while accounting for cross-sectional dependence are:

πŸ”Ή 1. CADF (Cross-sectionally Augmented Dickey-Fuller) Test

πŸ“Œ Developed by: Pesaran (2007)

CADF is an extension of the Augmented Dickey-Fuller (ADF) test that augments each ADF regression with cross-sectional averages to account for cross-sectional dependence.

πŸ” Model Form:

For each unit ii, the CADF regression is:

Ξ”yit=ai+biyi,t−1+ciyΛ‰t−1+diΞ”yΛ‰t+∑j=1pΟ•ijΞ”yi,t−j+Ξ΅it\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_{ij} \Delta y_{i,t-j} + \varepsilon_{it}

Where:

  • yΛ‰t\bar{y}_t is the cross-sectional average of all yity_{it},

  • bib_i tests for the unit root (null: bi=0b_i = 0).

πŸ”Ή 2. CIPS (Cross-sectionally Augmented IPS) Test

πŸ“Œ Also Developed by: Pesaran (2007)

CIPS is the panel version of the CADF test. It aggregates the individual CADF statistics into one panel test statistic.

πŸ“Š Formula:

CIPS=1N∑i=1NtiCADFCIPS = \frac{1}{N} \sum_{i=1}^{N} t_i^{CADF}

Where:

  • tiCADFt_i^{CADF} is the t-statistic from the CADF regression for unit ii,

  • The null hypothesis is: all series have a unit root (non-stationary),

  • The alternative is: at least some cross-sections are stationary.

πŸ§ͺ CIPS vs First-Generation Tests

Feature

First-Generation (e.g., IPS, LLC)

Second-Generation (CIPS, CADF)

Assumes Cross-Section Independence

✅ Yes

❌ No

Handles Common Shocks

❌

✅ Yes

Suitable for Financial Panels

❌

✅ Highly Suitable

Example

Levin-Lin-Chu, Im-Pesaran-Shin

Pesaran’s CADF & CIPS

✅ When to Use CADF/CIPS:

  • In macro or financial panel datasets where cross-sectional correlation is present (e.g., bank performance, regional inflation, country-level interest rates).

  • When standard IPS or LLC tests give inconclusive results due to cross-sectional dependence.

πŸ”Ž Interpreting Results:

·         If CIPS statistic < critical value → Reject null → Stationarity.

·         If CIPS statistic > critical value → Do not reject null → Non-stationary.

πŸ“ Summary Table:

Test

Accounts for Cross-Section Dependence

Based On

Output

Null Hypothesis

CADF

✅ Yes

ADF

Individual t-stats

Unit root for each unit

CIPS

✅ Yes

CADF average

Panel stat

All units have unit root

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