DF-GLS Test: GLS-Detrended Dickey-Fuller Unit-Root Test
Also known as: Elliott-Rothenberg-Stock test, ERS unit-root test, GLS-detrended Dickey-Fuller test, DF-GLS birim kök testi
The DF-GLS test, introduced by Elliott, Rothenberg, and Stock (1996), is a modified augmented Dickey-Fuller procedure that applies generalized least squares (GLS) detrending before the standard unit-root regression. By removing deterministic components under a local alternative rather than the null hypothesis, the test achieves near-optimal power for detecting stationarity in time series, making it the preferred unit-root test in applied econometrics when a trend or intercept is present.
Key highlights
- Substantially higher power than the ADF test against near-unit-root alternatives, particularly in small samples.
- Near-optimal in the sense of Elliott et al. (1996): it approaches the Gaussian power envelope for unit-root tests.
- Flexible lag selection via MAIC controls size distortions from residual autocorrelation.
- Available in virtually every major econometric software package (Stata, EViews, R, Python).
Intuition
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How it works
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When to use it
Use the DF-GLS test when your time series appears to have a deterministic intercept or linear trend and you need a more powerful alternative to the ADF test. It is especially valuable in small to moderate samples (T < 150) where the ADF test is known to have low power. The test assumes errors are weakly dependent and homoskedastic; under severe heteroskedasticity or structural breaks consider complementary tests such as KPSS or the Zivot-Andrews test. When multiple series are tested, combine with the KPSS to cross-validate stationarity conclusions.
Strengths & limitations
- Substantially higher power than the ADF test against near-unit-root alternatives, particularly in small samples.
- Near-optimal in the sense of Elliott et al. (1996): it approaches the Gaussian power envelope for unit-root tests.
- Flexible lag selection via MAIC controls size distortions from residual autocorrelation.
- Available in virtually every major econometric software package (Stata, EViews, R, Python).
- Critical values depend on whether a constant or a constant-plus-trend is included; misspecification of deterministics can distort inference.
- Like all unit-root tests, it cannot distinguish a unit root from a near-integrated process without very large samples.
- Assumes no structural break in the deterministic component; breaks can severely reduce power.
- One-sided test structure means it does not directly test for explosive roots (α > 0).
Common pitfalls
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Applications
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Frequently asked
How does DF-GLS differ from the standard ADF test?
Both tests run an augmented Dickey-Fuller regression and use its t-statistic, but DF-GLS first removes the deterministic component by GLS using a near-unit-root design point rather than OLS under the unit-root null. This pre-filtering substantially raises power against stationary alternatives close to the unit circle, which is where macroeconomic series typically lie.
Which lag-length selector should I use with DF-GLS?
Elliott et al. (1996) and subsequent Monte Carlo studies recommend the modified Akaike information criterion (MAIC) proposed by Ng and Perron (2001). Standard AIC and BIC tend to under-select lags when applied to GLS-detrended series, causing over-rejection of the null. Most software packages implement MAIC as an option alongside automatic lag selection.
Should I include a constant, a trend, or both in the DF-GLS test?
The choice should reflect the data-generating process of the series. For variables expected to fluctuate around a fixed mean (e.g., interest rate spreads), include only a constant (c̄ = −7). For variables with a deterministic trend (e.g., log real GDP), include a constant and trend (c̄ = −13.5). Misspecification of this component changes both the local alternative and the critical values, potentially invalidating inference.
Sources
- 1.Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836.
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Cite this page
ScholarGate. (2026, June 2). DF-GLS Test. ScholarGate. https://scholargate.app/econometrics/df-gls