What is the null hypothesis of the KPSS test?

H0: the series is stationary (level-stationary or trend-stationary). H1: the series has a unit root (non-stationary). A small LM statistic and a large p-value mean we fail to reject H0 — the series is stationary.

How do I interpret the KPSS LM statistic?

Compare the LM statistic to the critical values at 1%, 5%, and 10%. If LM is greater than the chosen critical value, reject H0 and conclude non-stationarity. If LM is below the critical value, the series is stationary at that level.

What is the difference between KPSS and ADF tests?

ADF tests for a unit root (H0: non-stationary). KPSS tests for stationarity (H0: stationary). They have opposite null hypotheses. Researchers often run both — when both agree, conclusions are robust.

Should I use level-stationary or trend-stationary KPSS?

Use level-stationary if your series has no obvious trend. Use trend-stationary if there is a clear linear trend you want to remove before testing for stationarity around that trend.

What lag should I choose for the KPSS test?

The default Schwert rule sets lag = floor(4 * (n/100)^(1/4)) for short-run lag, or floor(12 * (n/100)^(1/4)) for long-run lag. The tool uses the short-run rule by default and lets you override it.

How large a sample is needed for a KPSS test?

At least 30 observations are recommended; 50–200 gives reliable results. Very short series (n < 25) produce unstable LM statistics and unreliable conclusions.

What if the KPSS test rejects H0?

If the series is non-stationary, take first differences and re-run the KPSS test. Repeat until the differenced series is stationary. The number of differences needed is the d in ARIMA(p, d, q).

How do I report KPSS test results in APA format?

Report the LM statistic, lag length, the type of stationarity tested (level or trend), the critical values, and the conclusion. Example: 'KPSS test for level stationarity, LM = 0.241, lag = 4, was below the 5% critical value (0.463); we fail to reject the null of stationarity.'

Can I use this KPSS calculator for published research?

This tool is designed for education and exploratory analysis. For peer-reviewed work, verify results with R (tseries::kpss.test, urca::ur.kpss) or Python (statsmodels.tsa.stattools.kpss). Cite: STATS UNLOCK (2025). KPSS test calculator. https://statsunlock.com.

KPSS Test Calculator – Free Online Stationarity Test for Time Series | Stats Unlock

KPSS Test Calculator – Free Online Stationarity Test for Time Series | StatsUnlock

KPSS Test Calculator

Free online Kwiatkowski-Phillips-Schmidt-Shin (KPSS) stationarity test for time series — get the LM statistic, critical values, p-value range, four colourful diagnostic plots, and APA-format reporting in seconds.

Time Series Stationarity Free Online No Sign-up APA Output

Stationary

Trend

Random Walk

Seasonal

📥 1. Enter Your Time Series Data

Sample Dataset

Each row is one time series. The series name is editable. Comma-separated values are the default — example: 52, 48, 55, 61, 47, ...

Upload CSV or Excel file

Supports .csv, .txt, .xlsx, .xls — headers auto-detected. Click each numeric column you want to load — every selected column becomes a separate time series cluster.

Manual Entry — one value per row

⚙️ 2. Test Configuration

Test Type

Lag Selection

Custom Lag (l)

Significance Level (α)

🧮 8. Formulas & Technical Notes

A. Formulas Used

Step 1 — Detrend the series.

e_t = y_t − μ̂ (level test) e_t = y_t − (μ̂ + β̂·t) (trend test) Where: y_t = observed value at time t μ̂ = sample mean (level test) or OLS intercept (trend test) β̂ = OLS slope on time index (trend test only) e_t = residual at time t

Step 2 — Partial sums of residuals.

S_t = Σᵢ₌₁ᵗ e_i, t = 1, 2, …, n Where: S_t = cumulative sum of residuals up to time t n = sample size

Step 3 — Long-run variance estimator (Newey-West).

σ̂²(l) = (1/n)·Σ e_t² + (2/n)·Σⱼ₌₁ˡ wⱼ(l) · Σ e_t · e_{t-j} wⱼ(l) = 1 − j/(l+1) (Bartlett kernel) Where: σ̂²(l) = long-run variance with lag truncation l wⱼ(l) = Bartlett kernel weight for lag j l = bandwidth (lag truncation parameter)

Step 4 — KPSS LM statistic.

η = (1/n²) · Σₜ₌₁ⁿ S_t² / σ̂²(l) Where: η = KPSS LM (Lagrange Multiplier) statistic S_t = partial sums (Step 2) σ̂²(l) = long-run variance (Step 3) n = sample size

Step 5 — Schwert lag rule (default).

l_short = ⌊4 · (n/100)^(1/4)⌋ l_long = ⌊12 · (n/100)^(1/4)⌋

Step 6 — Asymptotic critical values (Kwiatkowski et al., 1992, Table 1):

Type	α = 0.10	α = 0.05	α = 0.025	α = 0.01
Level (μ)	0.347	0.463	0.574	0.739
Trend (μ+βt)	0.119	0.146	0.176	0.216

B. Technical Notes

Null hypothesis (H₀): the series is stationary (level- or trend-stationary).
Alternative (H₁): the series has a unit root (non-stationary).
Decision rule: reject H₀ if η > CV(α). Large η ⇒ non-stationary.
Bandwidth choice matters: too-small l overstates LM; too-large l understates LM. Schwert's rule is the conventional default.
Tables vs interpolation: p-values are interpolated linearly between the four published critical values; outside the range we report p < 0.01 or p > 0.10.
Pair with ADF: KPSS and ADF have opposite nulls. When both agree the conclusion is robust; when they disagree, inspect plots and consider differencing.

📚 9. When to Use the KPSS Test

This free KPSS test calculator is designed for time series analysts, econometricians, and data scientists who need a quick stationarity test calculator with effect-size context, p-value interpretation, and APA-format output for hypothesis testing in research reports.

Decision Checklist

✓ Your data are a single, evenly-spaced time series (one observation per period).
✓ You want to confirm stationarity before fitting AR, MA, ARMA, ARIMA, or regression models.
✓ Sample size n ≥ 30; ideally n ≥ 50.
✓ You can specify whether to test level- or trend-stationarity based on visual inspection.
✗ Do NOT use on multivariate panel data — use a panel unit root test (Im-Pesaran-Shin, Levin-Lin-Chu).
✗ Do NOT use without inspecting the time plot first — KPSS cannot detect structural breaks.
✗ Do NOT rely on KPSS alone for critical decisions — pair with the ADF or Phillips-Perron test.

Real-World Examples

Econometrics: Test whether monthly inflation rates are stationary before fitting an ARMA model for forecasting.
Finance: Verify that daily stock returns (not prices) are stationary before applying GARCH volatility models.
Climate science: Check whether de-trended global temperature anomalies are stationary around their long-run mean.
Ecology: Confirm that a 50-year animal-count series is trend-stationary before testing for periodicity.
Operations: Test whether daily call-centre volume is level-stationary before forecasting staffing needs.

Sample Size Guidance

Minimum: n ≥ 25 (results unstable below this).
Recommended: 50 ≤ n ≤ 500 for reliable LM statistics.
Very large n (n > 1000): very small departures from stationarity become "significant" — always inspect plots.

Related Tests — Decision Tree

Single time series → Stationarity check
   ├─ KPSS (H₀: stationary)        ← THIS TEST
   ├─ ADF / PP    (H₀: unit root)  ← opposite null; run both
   └─ Zivot-Andrews (H₀: unit root, allows one structural break)

Panel data → Im-Pesaran-Shin / Levin-Lin-Chu
Cointegration → Engle-Granger / Johansen

📖 10. How to Use This KPSS Test Calculator (10 Steps)

Enter Your Data — paste comma-separated values into Series 1 (e.g., 52, 48, 55, 61, 47, ...), upload a CSV/Excel file, or use the manual-entry tab. Each cluster row is one time series.
Choose a Sample Dataset — the dropdown offers White Noise, Random Walk, AR(1) with Trend, Inflation Rate, and Stock Returns. Dataset 1 is pre-loaded so you can run immediately.
Pick Test Type — Level Stationarity (no trend) or Trend Stationarity (linear trend removed first).
Select Lag — Schwert short (default), long, or custom integer. The lag controls the long-run variance estimator's bandwidth.
Set Alpha — typically 0.05.
Click Run KPSS Test — results appear in under a second.
Read the Summary Cards — LM statistic (η), critical value at chosen α, p-value range, and the verdict (Stationary / Non-Stationary).
Inspect All 4 Plots — time series with mean, partial sums S_t, ACF, and rolling mean/std.
Read the Interpretation — six paragraphs covering result, p-value, effect, practical meaning, limitations, and next steps.
Export — Download Doc (.txt) or Download PDF for journal submission.

❓ 11. Frequently Asked Questions

Q1. What is the KPSS test and when should I use it?

The KPSS test (Kwiatkowski-Phillips-Schmidt-Shin, 1992) checks whether a time series is stationary. Unlike ADF, the null hypothesis here is stationarity. Use it before fitting any AR, MA, ARMA, ARIMA, or time-series regression model.

Q2. What is a p-value, and how do I interpret it for the KPSS test?

The p-value here is the probability of seeing an LM statistic this large (or larger) if the true series were stationary. A small p (< α) means the data are inconsistent with stationarity, so we reject H₀ and conclude the series is non-stationary.

Q3. What does statistical significance mean for the KPSS test?

"Significant" here means rejecting stationarity. Even tiny non-stationarity becomes "significant" in very long series, so always confirm with a time plot and the rolling-statistics chart on this page.

Q4. What is the LM statistic and what value indicates non-stationarity?

The KPSS LM statistic η is a scaled sum of squared partial sums of residuals. Compare it to the critical values: at α = 0.05, level-test CV is 0.463 and trend-test CV is 0.146. If η exceeds the CV, the series is non-stationary.

Q5. What assumptions does the KPSS test require?

The series must be evenly spaced, finite-variance, and free of structural breaks. Heavy-tailed errors and very small samples (n < 25) can distort the LM statistic. If a structural break exists, use the Zivot-Andrews test instead.

Q6. How large a sample do I need?

At least 30 observations; 50–500 is the reliable range. Very short series produce unstable LM values; very long series become hyper-sensitive to tiny departures from stationarity.

Q7. Should I run a one-tailed or two-tailed KPSS test?

The KPSS test is inherently one-sided — we reject only when η is large (in the right tail). There is no two-tailed version, because small η simply means the data look stationary.

Q8. How do I report KPSS results in APA 7th format?

Report the LM statistic, lag, test type, critical value, and conclusion. Section 6 of this page auto-fills five reporting templates (APA, Thesis, Plain-Language, Conference Abstract, Pre-Registration) with your numbers — copy and paste.

Q9. Can I use this calculator for published research?

This tool is intended for education and exploratory analysis. For peer-reviewed work, verify with R (tseries::kpss.test or urca::ur.kpss) or Python (statsmodels.tsa.stattools.kpss). Cite STATS UNLOCK (2025). KPSS test calculator. https://statsunlock.com.

Q10. What should I do if the KPSS test rejects H₀?

Take first differences (Δy_t = y_t − y_t−1) and re-run KPSS. If still non-stationary, difference again. The number of differences needed equals the integration order d in ARIMA(p, d, q). Most economic series are I(1) — one difference is enough.

📚 12. References

The following references support the statistical methods used in this KPSS test calculator, covering p-value interpretation, time series stationarity, and best practices in hypothesis testing.

Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1–3), 159–178. https://doi.org/10.1016/0304-4076(92)90104-Y
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. https://doi.org/10.1080/01621459.1979.10482531
Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75(2), 335–346. https://doi.org/10.1093/biomet/75.2.335
Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708. https://doi.org/10.2307/1913610
Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics, 7(2), 147–159. https://doi.org/10.1080/07350015.1989.10509723
Hobijn, B., Franses, P. H., & Ooms, M. (2004). Generalizations of the KPSS-test for stationarity. Statistica Neerlandica, 58(4), 483–502. https://doi.org/10.1111/j.1467-9574.2004.00272.x
Hamilton, J. D. (1994). Time series analysis. Princeton University Press. https://press.princeton.edu/books/hardcover/9780691042893
Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: Forecasting and control (5th ed.). Wiley. wiley.com
Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: principles and practice (3rd ed.). OTexts. https://otexts.com/fpp3/
Cumming, G. (2014). The new statistics: Why and how. Psychological Science, 25(1), 7–29. https://doi.org/10.1177/0956797613504966
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
Trapletti, A., & Hornik, K. (2024). tseries: Time Series Analysis and Computational Finance (R package). https://CRAN.R-project.org/package=tseries
Seabold, S., & Perktold, J. (2010). statsmodels: Econometric and statistical modeling with Python. Proceedings of the 9th Python in Science Conference. statsmodels.org
NIST/SEMATECH. (2013). e-Handbook of statistical methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/