Nonlinear ADF Unit Root Test (KSS Test)
Nonlinear Augmented Dickey-Fuller Unit Root Test · Also known as: KSS test, nonlinear unit root test, ESTAR unit root test, Kapetanios-Shin-Snell test
The Nonlinear ADF unit root test, most prominently operationalized by Kapetanios, Shin, and Snell (2003), extends the classical Augmented Dickey-Fuller test to detect mean reversion that occurs via an Exponential Smooth Transition Autoregressive (ESTAR) process. It tests the null of a unit root against a nonlinear stationary alternative, capturing adjustment dynamics that the standard linear ADF test misses.
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When to use it
Use the Nonlinear ADF test when theory or prior empirical evidence suggests that the series may be stationary but through nonlinear mean reversion — common for purchasing power parity, interest rate parity, and commodity price series. It is most appropriate for moderately long time series (T >= 100 observations) where the transition dynamics can be identified. Do not use it as a routine substitute for the standard ADF test; apply it when you have subject-matter reason to believe adjustment speed varies with distance from equilibrium. Avoid it when the sample is short, when the series exhibits structural breaks rather than smooth transitions, or when the nonlinear alternative is a threshold (SETAR) rather than a smooth transition process — in those cases, a threshold unit root test or a structural break unit root test is more appropriate.
Strengths & limitations
- Detects mean reversion masked by nonlinear adjustment dynamics that the linear ADF test cannot identify.
- Grounded in a clear theoretical model (ESTAR process) with a tractable Taylor-approximation derivation.
- Straightforward to implement: standard OLS on a simple auxiliary regression with tabulated critical values.
- Extends naturally to demeaned or demeaned-and-detrended series with separate critical value tables.
- Widely applied in purchasing power parity, real interest rate parity, and commodity market studies.
- Low power in short samples (T < 100): the cubic-term auxiliary regression loses power quickly as sample size decreases.
- Sensitive to correct model specification — failing to account for serial correlation through inadequate augmentation inflates size.
- Critical values are simulation-based and may not match all sample sizes perfectly; asymptotic tables assume large T.
- Designed specifically for the ESTAR alternative; if the true nonlinearity is a sharp threshold (SETAR), power against it is lower than that of a dedicated threshold unit root test.
Frequently asked
How does the Nonlinear ADF test differ from the standard ADF test?
The standard ADF test regresses the first difference of a series on its lagged level, testing for a zero coefficient under the unit-root null versus a negative coefficient (linear mean reversion). The KSS nonlinear ADF replaces the lagged level with its cube, which approximates the ESTAR transition function. This detects mean reversion that accelerates with distance from equilibrium, a pattern the linear ADF misses.
Why is the cubed lagged level used instead of the level itself?
The cubic term arises from a first-order Taylor expansion of the ESTAR smooth transition function around the unit-root null (gamma = 0). This linearizes the otherwise nonlinear ESTAR model into a tractable OLS regression while retaining the key nonlinear information.
Can I use the same critical values as in the standard ADF test?
No. The KSS test statistic has a non-standard distribution that is different from and more negative than the standard Dickey-Fuller distribution. You must use the simulation-based critical values provided in Kapetanios, Shin, and Snell (2003), which vary by whether you demean only or also detrend.
What sample size is recommended for the Nonlinear ADF test?
The test has reasonable power for T >= 100 observations. With shorter series the cubic auxiliary regression loses power substantially, and failure to reject should be interpreted cautiously.
When should I use a structural break unit root test instead?
If your series has a sudden, sharp shift in level or trend (e.g., due to a policy change or crisis), a structural break test such as Zivot-Andrews or Lumsdaine-Papell is more appropriate. The Nonlinear ADF is designed for gradual, smooth adjustment dynamics, not discrete breaks.
Sources
- Kapetanios, G., Shin, Y., & Snell, A. (2003). Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics, 112(2), 359-379. DOI: 10.1016/S0304-4076(02)00202-6 ↗
- Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366), 427-431. DOI: 10.2307/2286348 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Augmented Dickey-Fuller Unit Root Test. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-adf-unit-root-test
Which method?
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