Nonlinear KPSS Test
Nonlinear Kwiatkowski-Phillips-Schmidt-Shin Test · Also known as: KPSS nonlinearity test, nonlinear stationarity test, flexible Fourier KPSS, NL-KPSS
The nonlinear KPSS test extends the classic Kwiatkowski-Phillips-Schmidt-Shin stationarity test by modelling unknown smooth structural breaks in the deterministic trend using a Fourier approximation. Under the null hypothesis the series is stationary around a flexible nonlinear trend, guarding against spurious unit-root findings caused by regime shifts or gradual transitions.
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When to use it
Use the nonlinear KPSS test when you suspect a time series is stationary but the presence of smooth structural breaks or gradual regime shifts might cause a standard KPSS test to reject stationarity falsely. It is particularly suited to macroeconomic and financial series — GDP, inflation, exchange rates, interest rates — that are known to undergo long, gradual transitions. Do not use it as a replacement for a conventional unit root test battery; it tests stationarity under the null, and results should be triangulated with ADF-type tests. It is also less appropriate when breaks are abrupt rather than smooth, or when the sample is very short (fewer than roughly 80 observations) because critical values depend on asymptotic theory.
Strengths & limitations
- Maintains correct size in the presence of unknown smooth structural breaks that would inflate the rejection rate of the standard KPSS test.
- Requires no prior specification of the break date, number of breaks, or transition speed — the Fourier approximation handles them flexibly.
- Directly comparable to the classic KPSS framework, making it a natural drop-in robustness check.
- Handles both level-stationary and trend-stationary null hypotheses.
- Performs well in Monte Carlo experiments under gradual transition alternatives.
- Shares the standard KPSS weakness: because stationarity is the null, the test cannot confirm a unit root — it can only fail to reject stationarity.
- Power against abrupt (instantaneous) structural breaks is lower than against smooth breaks; sharp changes are better handled by Zivot-Andrews or Clemente-Montañés-Reyes tests.
- Critical values are non-standard and depend on the chosen Fourier frequency and deterministic specification, requiring simulation or pre-tabulated tables.
- Selecting the Fourier frequency by minimising RSS introduces a pre-testing step that can affect size in small samples.
Frequently asked
How does the nonlinear KPSS test differ from the standard KPSS test?
The standard KPSS test allows only for a constant or linear trend under the null. The nonlinear version adds Fourier sine and cosine terms to capture smooth breaks in the deterministic component, so rejection of the null is less likely to be driven by unmodelled structural change rather than a true unit root.
What does it mean that stationarity is the null hypothesis?
Unlike ADF or Phillips-Perron tests (where the null is a unit root), the KPSS family tests whether the data are stationary. Failure to reject means the evidence is consistent with stationarity; rejection means a unit root cannot be ruled out. The two families are complementary: a series with conflicting verdicts may have a near-unit-root or be influenced by structural breaks.
How is the optimal Fourier frequency chosen?
The frequency k that minimises the residual sum of squares from the Fourier regression over a grid (usually k = 1 to 5) is selected. Because this introduces a pre-test, some practitioners also report results for all candidate frequencies for transparency.
Can the test handle multiple structural breaks?
Yes, implicitly. A single Fourier frequency captures one dominant smooth cycle, but using multiple Fourier components (higher n in the expansion) can approximate multiple smooth breaks. In practice, one or two frequencies are usually sufficient for macroeconomic data.
What sample size is adequate for this test?
The test is based on asymptotic theory and performs reasonably from about 80-100 observations. For smaller samples, bootstrap critical values are recommended to maintain correct size.
Sources
- Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409. DOI: 10.1111/j.1467-9892.2006.00478.x ↗
- Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599. DOI: 10.1111/j.1468-0084.2011.00662.x ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Kwiatkowski-Phillips-Schmidt-Shin Test. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-kpss-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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