The KPSS test checks whether a time series is stationary. Its null hypothesis is stationarity, so a large statistic means the series is not stationary.

Fit a mean (level) or a straight line (trend), add up the residuals over time, and compare the size of those running sums with the long-run variance. Reject stationarity at 5% when η is above 0.463 for level, or above 0.146 for trend.

Worked example: 24 months of rainfall anomaly give η = 0.0856 with lag 2. That is below 0.463, so p > 0.10 and the series is level-stationary.

The KPSS test (Kwiatkowski, Phillips, Schmidt and Shin, 1992) is a test for stationarity. A stationary series has a constant mean, a constant variance, and the same correlation pattern at every point in time. Most forecasting models, such as ARIMA and VAR, assume this.

What it measures: the test removes a mean (level version) or a straight line (trend version) from the data. It then adds up the leftover residuals one step at a time. If the series is stationary, these running sums wander a little and come back to zero. If the series has a unit root, the sums drift far away. η is the size of that drift, scaled by the long-run variance.

Two versions:

• Level stationarity (η_μ): the series moves around a fixed mean.
• Trend stationarity (η_τ): the series moves around a fixed straight-line trend.

A worked one-liner: rainfall anomaly over 24 months, mean 13.54 mm, sum of squared running sums 67.34, long-run variance 1.366, so η = 67.34 ÷ (576 × 1.366) = 0.0856, well under 0.463.

Who uses it: economists before ARIMA, cointegration and VAR models; ecologists testing population counts and camera-trap indices; hydrologists checking streamflow; climate scientists checking temperature anomalies; finance analysts checking returns and spreads.

4.5 Reading the KPSS statistic

η has no units. It compares how far the running sums travel with how noisy the data are. Values near 0.05 to 0.15 are typical of clearly stationary series. Values above 1 usually mean a strong trend or a unit root.

4.6 Decision guide: KPSS with ADF

4.7 Level result vs trend result

If the level test rejects but the trend test does not, the series is trend-stationary. Remove the straight line with a linear regression (detrend), do not difference. If both reject, the series is difference-stationary: take first differences.

4.8 Common mistakes this calculator prevents

• Reading the p-value the ADF way round. Here a small p means non-stationary.
• Reporting p = 0.10 as if it were exact. The calculator shows p > 0.10 when η is below the 10% value.
• Forgetting to report the lag. The lag changes η, so the write-up templates always include it.
• Using level critical values for a trend test. The calculator switches tables automatically.

▶ Run the analysis above to auto-fill all five examples with your results.

This flowchart shows how the KPSS test is calculated, step by step, for the worked example: 24 months of rainfall anomaly (mm), tested for level stationarity. The same numbers appear in Sections 9, 10, 11 and 13.

Excel (Section 9), R (Section 10) and Python (Section 11) all return 0.0856 for this series with lag 2.

1. Load or paste your data. Paste values in time order, or load sample 1 (rainfall anomaly, 24 months).
2. Name the series. Click the name box and type a label; it is used in every report.
3. Or upload a file. Use the Upload tab for CSV or Excel and click the numeric column you want.
4. Choose level or trend. Level for a flat series, trend for a steadily rising or falling one.
5. Set alpha. 0.05 is standard. 0.10, 0.025 and 0.01 are available.
6. Pick the lag rule. Keep the short rule to match R; try the long rule to check sensitivity.
7. Choose the kernel. Keep Bartlett unless you have a reason to use rectangular.
8. Press Run KPSS Test. For sample 1 you get KPSS = 0.0856, lag 2, p > 0.10.
9. Read the verdict and charts. Check the running-sum chart and the assumption table.
10. Copy the write-up. Copy an APA, thesis or plain-language sentence from Section 5, or download the Doc or PDF report.

Excel has no KPSS function, and the Analysis ToolPak does not include one either. You can still build it with ordinary formulas in about five columns. The steps below reproduce η = 0.0856 for the rainfall example.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

For the trend version, put t in column D and use TREND in column B instead of AVERAGE. For a longer lag, add one more SUMPRODUCT row per lag and one more weighted term in the long-run variance formula. Blank cells break the shifted ranges, so remove gaps first.

9.4 Why your answer might differ across software

The KPSS statistic itself is the same formula everywhere. Differences almost always come from the lag rule. Always state the lag you used.

9.5 Charting in Excel

Select column C and insert a Line chart. This is the running-sum plot from chart 3 of this calculator. A line that drifts away and does not come back is the visual sign of non-stationarity.

9.6 Error messages

R has two standard KPSS functions: kpss.test() in the tseries package and ur.kpss() in urca. The script below runs both, adds an ADF test, prints a plain-English verdict, and saves one 300 dpi figure.

10.1 The complete script

R, tseries + urca, saves kpss_plot.png

# KPSS stationarity test with ADF cross-check
# StatsUnlock - https://statsunlock.com/kpss-test-calculator/
library(tseries)
library(urca)
set.seed(1)

# ---- 1. Your data (time order, earliest first) ----
# y <- read.csv("data.csv")$value
# y <- y[!is.na(y)]
y <- c(12,15,11,14,16,13,10,14,17,12,15,13,
       11,16,14,12,15,13,17,12,14,11,15,13)
null_type <- "Level"   # or "Trend"

# ---- 2. KPSS (tseries) ----
k <- kpss.test(y, null = null_type, lshort = TRUE)
print(k)   # KPSS Level = 0.0856, Truncation lag parameter = 2

# ---- 3. Same test in urca, gives critical values ----
u <- ur.kpss(y, type = if (null_type == "Level") "mu" else "tau", lags = "short")
print(summary(u))

# ---- 4. ADF cross-check ----
a <- adf.test(y)
print(a)

# ---- 5. Plain-English verdict ----
cv5 <- if (null_type == "Level") 0.463 else 0.146
eta <- unname(k$statistic)
cat(sprintf("\nKPSS = %.4f (lag %d). 5%% critical value = %.3f.\n", eta, k$parameter, cv5))
if (eta > cv5) {
  cat("Reject stationarity: difference or detrend before modelling.\n")
} else {
  cat("Do not reject: the series is consistent with stationarity.\n")
}

# ---- 6. One 300 dpi figure: series + running sums ----
e <- if (null_type == "Level") y - mean(y) else resid(lm(y ~ seq_along(y)))
S <- cumsum(e)
png("kpss_plot.png", width = 7, height = 5, units = "in", res = 300)
par(mfrow = c(2, 1), mar = c(4, 4, 2, 1))
plot(y, type = "o", pch = 16, col = "#16a34a", xlab = "Time", ylab = "Value",
     main = sprintf("KPSS %s = %.4f", null_type, eta))
lines(y - e, col = "#dc2626", lty = 2, lwd = 2)
plot(S, type = "l", lwd = 2, col = "#7c3aed", xlab = "Time", ylab = "Running sum S(t)")
abline(h = 0, lty = 3)
dev.off()

10.2 What each line does

10.3 What the figure shows

The top panel is the series with the fitted mean or trend. The bottom panel is the running sum. For the rainfall data it stays within about ±3 and returns to zero, which is why η is small.

10.4 Problems and fixes

10.5 One-liners

tseries::kpss.test(y, null = "Trend")
tseries::kpss.test(diff(y))          # test the differenced series
forecast::ndiffs(y, test = "kpss")  # how many differences are needed

In Python the KPSS test lives in statsmodels. The script below sets the lag to match R and this calculator, runs ADF beside it, prints a verdict and saves one 300 dpi figure.

11.1 The complete script

Python 3, statsmodels, saves kpss_plot.png

# KPSS stationarity test with ADF cross-check
# StatsUnlock - https://statsunlock.com/kpss-test-calculator/
import warnings
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.stattools import kpss, adfuller
np.random.seed(1)

# ---- 1. Your data (time order, earliest first) ----
# import pandas as pd
# y = pd.read_csv("data.csv")["value"].dropna().to_numpy()
y = np.array([12,15,11,14,16,13,10,14,17,12,15,13,
              11,16,14,12,15,13,17,12,14,11,15,13], dtype=float)
y = y[~np.isnan(y)]
regression = "c"   # "c" = level, "ct" = trend

# ---- 2. Lag rule that matches R tseries ----
T = len(y)
lag = int(4 * (T / 100) ** 0.25)   # = 2 for T = 24

# ---- 3. KPSS ----
with warnings.catch_warnings():
    warnings.simplefilter("ignore")   # p-value is table-limited
    stat, p, used_lag, crit = kpss(y, regression=regression, nlags=lag)
print(f"KPSS = {stat:.4f}, lag = {used_lag}, p = {p}")   # 0.0856, 2, 0.1
print("Critical values:", crit)

# ---- 4. ADF cross-check ----
adf_stat, adf_p, *_ = adfuller(y, autolag="AIC")
print(f"ADF = {adf_stat:.4f}, p = {adf_p:.4f}")

# ---- 5. Plain-English verdict ----
if stat > crit["5%"]:
    print("Reject stationarity: difference or detrend before modelling.")
else:
    print("Do not reject: the series is consistent with stationarity.")

# ---- 6. One 300 dpi figure ----
t = np.arange(1, T + 1)
fit = np.full(T, y.mean()) if regression == "c" else np.polyval(np.polyfit(t, y, 1), t)
S = np.cumsum(y - fit)
fig, ax = plt.subplots(2, 1, figsize=(7, 5), sharex=True)
ax[0].plot(t, y, "o-", color="#16a34a")
ax[0].plot(t, fit, "--", color="#dc2626")
ax[0].set_title(f"KPSS = {stat:.4f} (lag {used_lag})")
ax[1].plot(t, S, color="#7c3aed")
ax[1].axhline(0, ls=":", color="grey")
ax[1].set_xlabel("Time"); ax[1].set_ylabel("Running sum S(t)")
fig.tight_layout(); fig.savefig("kpss_plot.png", dpi=300)

11.2 What each line does

11.3 What the figure shows

The same two panels as the R figure: the series with its fit, and the running sums. Both scripts give KPSS = 0.0856 with lag 2.

11.4 Problems and fixes

11.5 One-liners

kpss(y, regression="ct", nlags=2)        # trend version
kpss(np.diff(y), regression="c")      # after first differencing
from pmdarima.arima import ndiffs; ndiffs(y, test="kpss")

Table 1. KPSS critical values (Kwiatkowski et al., 1992)

What to conclude: if η is above the value in your α column, reject stationarity.

Table 2. Default lag by sample size

What to conclude: the long rule is safer with persistent data but weakens the test in short series.

Table 3. Reading η (level version)

What to conclude: values in the amber band call for a second test (ADF) and a look at the chart.

Table 4. Reading η (trend version)

What to conclude: rejecting the trend version usually means a unit root, so difference the series.

Table 5. Minimum series length

What to conclude: under 20 points, treat the result as a hint, not a decision.

Table 6. Which transformation after KPSS

What to conclude: never difference a trend-stationary series; it adds a non-invertible MA term.

Eight worked KPSS examples. Each uses the default short lag and the Bartlett kernel. Example 1 is the same worked dataset used in Sections 7, 9, 10 and 11.

Design statement: one variable, measured at equal time intervals, in time order, with no gaps.

1. Fix the time step (day, week, month, year) before collecting data.
2. Record the same quantity in the same units every time.
3. Keep the method constant. A new instrument or survey design can create a false break.
4. Record a date for every value.
5. Leave missing periods blank, never as 0.
6. Write down any known events (policy change, dam, new trap layout) with dates.
7. Decide on seasonal adjustment before testing if the data are monthly or quarterly.
8. Collect at least 50 points where you can; 20 is the practical minimum.
9. Pre-register the version (level or trend), lag rule and α.
10. Run KPSS and ADF together and report both.

Minimum sample: 20 observations, 50 or more preferred. Pre-registration line: "Stationarity will be assessed with a KPSS level test, lag = trunc(4(T/100)^0.25), Bartlett kernel, α = .05." Common mistakes: mixing time steps, filling gaps with zeros, testing before removing a known seasonal cycle.

Four real examples

• Ecology: monthly sloth bear detections per camera trap before fitting a count time-series model.
• Economics: inflation rates before an ARIMA forecast.
• Hydrology: annual peak flow to check for a climate trend.
• Public health: weekly case counts before outbreak forecasting.

Decision tree

• Want to test a unit root directly? Use the ADF test or Phillips-Perron test.
• Suspect one break at an unknown date? Use Zivot-Andrews or KPSS with breaks.
• Checking residuals of a fitted model? Use the Ljung-Box test.
• Seasonal unit root? Use OCSB or Canova-Hansen tests.

Assumptions

1. Equally spaced, ordered data. Check: dates. If broken, the running sums mean nothing.
2. No structural break. Check: chart 1. If broken, KPSS over-rejects (says non-stationary when each part is stationary).
3. Seasonality removed. Check: ACF spikes at lag 12 or 4. If broken, results can go either way.
4. Linear trend (trend version). Check: chart 1. A curved trend makes KPSS over-reject.
5. Enough lags for the autocorrelation. Check: chart 4. Too few lags with strong positive autocorrelation makes KPSS over-reject; too many reduce power.
6. Large sample. Critical values are asymptotic. With T < 20 the size is distorted, usually toward over-rejection.

Limitations

• Cannot handle breaks. Alternative: KPSS with structural breaks (Carrion-i-Silvestre et al., 2005) or Zivot-Andrews.
• Sensitive to lag choice. Alternative: report both short and long lags.
• Over-rejects near-unit-root but stationary series (Caner and Kilian, 2001). Alternative: confirm with ADF or ERS DF-GLS.
• Cannot detect seasonal unit roots. Alternative: Canova-Hansen test.
• Cannot separate long memory from a unit root. Alternative: fractional integration (d) estimation.

The KPSS test asks one question: can this series be treated as stable around a mean or a straight trend? It is the natural partner to the ADF test because its null hypothesis is the opposite one.

▶ Run the analysis above and this paragraph will summarise your result: η = ___, lag = ___, decision ___.

Always report the version, the lag and the kernel, show a plot of the series, and pair the result with ADF. When the two tests disagree, look for breaks and seasonality before you difference anything.

Used this way, the KPSS test calculator gives you a defensible, reproducible answer to whether your time series is stationary.

Peer-reviewed sources behind the methods in this KPSS test calculator: stationarity and unit-root testing, long-run variance estimation, lag selection and structural breaks.

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