Cross-sectionally Augmented Dickey-Fuller (CADF) Test
Also known as: Cross-Sectionally Augmented ADF, Panel CADF Test, Pesaran Panel Unit Root Test, CADF Birim Kök Testi
The Cross-sectionally Augmented Dickey-Fuller (CADF) test, introduced by Pesaran (2007), is a second-generation panel unit-root test designed to handle cross-sectional dependence among panel units. Unlike first-generation panel unit-root tests that assume cross-sectional independence, the CADF test augments individual ADF regressions with cross-sectional averages of lagged levels and first differences, making it suitable for macro-panels and cross-country studies where common factors drive co-movement.
Key highlights
- Robust to cross-sectional dependence without requiring explicit factor model estimation
- Allows for heterogeneous slope coefficients across panel units
- Individual CADF statistics enable unit-specific stationarity diagnostics
- Computationally straightforward; standard OLS on augmented regressions
Intuition
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How it works
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When to use it
Use the CADF test when you have a balanced or near-balanced macro-panel — typically 10 to 200 units observed over 20 or more time periods — and you suspect cross-sectional dependence driven by global or regional common factors. It is appropriate after a cross-section dependence test (e.g., Pesaran CD test) rejects independence. The test assumes a single unobserved common factor with heterogeneous loadings. For panels with multiple or strongly correlated common factors, the Moon-Perron or Bai-Perron factor-augmented panel tests may be preferred. Individual CADF statistics can also be inspected to identify which specific units drive non-stationarity.
Strengths & limitations
- Robust to cross-sectional dependence without requiring explicit factor model estimation
- Allows for heterogeneous slope coefficients across panel units
- Individual CADF statistics enable unit-specific stationarity diagnostics
- Computationally straightforward; standard OLS on augmented regressions
- Assumes a single common factor; may be misspecified under multiple strong factors
- Requires a sufficiently long time dimension (T) relative to N for reliable size and power
- Critical values depend on whether a constant or trend is included and on N and T, requiring case-specific tables
- Low power against near-unit-root alternatives, especially in small samples
Common pitfalls
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Applications
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Frequently asked
How does the CADF test differ from first-generation panel unit-root tests like LLC or IPS?
First-generation tests (Levin-Lin-Chu, Im-Pesaran-Shin) assume cross-sectional independence, so they are severely size-distorted when panel units share common shocks. The CADF test augments each unit's regression with cross-sectional averages to proxy for common factors, restoring correct size under dependence without requiring the researcher to specify or estimate the factor structure explicitly.
What is the difference between individual CADF statistics and the CIPS statistic?
The CADF statistic is a unit-specific t-ratio testing whether a single panel member is non-stationary after controlling for cross-sectional dependence. The CIPS statistic is the simple average of all individual CADF statistics and provides a single panel-wide test of the null that all units have a unit root, compared against Pesaran's (2007) panel critical values.
What lag order should be chosen for the CADF regression?
Pesaran (2007) recommends selecting the lag truncation order p using standard information criteria (AIC or BIC) applied to each individual ADF regression, or using a maximum lag derived from a rule of thumb such as p_max = floor(12(T/100)^(1/4)). Consistent lag selection is important because over-augmentation reduces power while under-augmentation leaves residual serial correlation.
Sources
- 1.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.
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ScholarGate. (2026, June 2). CADF Test. ScholarGate. https://scholargate.app/econometrics/cadf-test