Unit Root Tests: Augmented Dickey-Fuller and KPSS
Check whether time series are stationary before modelling them. The ADF test has a unit root as its null; the KPSS test has stationarity as its null. Running both, in levels and first differences, gives the order of integration of each series.
Updated · By Bibin XavierData 160 × 6
| Quartertext · 160 levels | Log_GDPnumeric | Log_Consumptionnumeric | Inflationnumeric | Interest_Ratenumeric | Money_Growthnumeric |
|---|---|---|---|---|---|
| 2000Q1 | 9.2061 | 8.6572 | 1.17 | 1.32 | 4.05 |
| 2000Q2 | 9.2005 | 8.6695 | 1.97 | 3.2 | 3.9 |
| 2000Q3 | 9.216 | 8.6727 | 2.61 | 3.54 | 4.84 |
| 2000Q4 | 9.2115 | 8.6923 | 2.15 | 2.53 | 3.4 |
| 2001Q1 | 9.224 | 8.6957 | 1.98 | 3.14 | 1.25 |
| 2001Q2 | 9.2286 | 8.6884 | 1.47 | 3.11 | 2.17 |
| 2001Q3 | 9.232 | 8.7052 | 1.55 | 2.65 | 1.8 |
| 2001Q4 | 9.2207 | 8.6907 | 2.18 | 2.43 | 3.5 |
Variables and options
Results
160 rows · example dataUnit root and stationarity tests
| Series | Form | ADF τ | Lags | ADF p | KPSS η | KPSS p | Order |
|---|---|---|---|---|---|---|---|
| Log_GDP | Level | 0.907 | 1 | .993 | 1.849 | < .010 | I(1) |
| First difference | -13.862 | 0 | < .001 | 0.220 | > .100 | ||
| Log_Consumption | Level | 0.512 | 2 | .985 | 1.847 | < .010 | I(1) |
| First difference | -12.292 | 1 | < .001 | 0.142 | > .100 | ||
| Inflation | Level | -5.193 | 2 | < .001 | 0.049 | > .100 | I(0) |
| First difference | -7.052 | 7 | < .001 | 0.098 | > .100 |
Note. ADF: H₀ unit root, MacKinnon (1994) p-values; 5% critical value ≈ -2.88 (MacKinnon, 2010). KPSS: H₀ stationarity, p-values interpolated between .01 and .10 (Kwiatkowski et al., 1992), Newey-West bandwidth chosen automatically (Hobijn et al., 2004). A series is I(0) when ADF rejects and KPSS does not.
Log_GDP over time
ACF and PACF of Log_GDP
Ljung-Box test for Log_GDP
| Lag | Q | p |
|---|---|---|
| 1 | 157.26 | < .001 |
| 4 | 601.44 | < .001 |
| 8 | 1127.86 | < .001 |
| 12 | 1580.31 | < .001 |
| 20 | 2263.11 | < .001 |
Note. H₀: no autocorrelation up to the given lag.
Interpretation
Log_GDP is I(1); Log_Consumption is I(1); Inflation is I(0). Because some series are I(1), test for cointegration before estimating a model in levels, or model the first differences.
How to cite this tool
Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.
APA 7
Xavier, B. (2026). Unit Root Tests: Augmented Dickey-Fuller and KPSS [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/unit-root-test-adf-kpss/
BibTeX
@software{xavier_unitroottest_2026,
author = {Xavier, Bibin},
title = {Unit Root Tests: Augmented Dickey-Fuller and KPSS},
publisher = {FreeStatisticalTools.com},
year = {2026},
url = {https://freestatisticaltools.com/unit-root-test-adf-kpss/},
note = {Web application}
}Suggested sentence for your methods section: “The analysis was carried out with the free Unit Root Tests: Augmented Dickey-Fuller and KPSS tool on FreeStatisticalTools.com (Xavier, 2026).”
What is a unit root test?
Most time-series methods assume stationary data. The augmented Dickey-Fuller test checks for a unit root; the KPSS test checks the opposite null of stationarity. Together, in levels and first differences, they reveal whether a series is I(0), I(1) or higher.
When to use a unit root test
Run unit root tests on every series before time-series regression, VAR, Granger causality or cointegration analysis, to avoid spurious regressions.
Assumptions
- Rows are in time order with equal spacing and no gaps.
- Include a trend in the ADF regression if the series trends in levels.
How to use this tool
- Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
- Choose the variables. Select the time series (rows in time order).
- Set the options. Fill in adf deterministic terms, adf lag selection, maximum lag (0 = automatic) and kpss null.
- Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.
How to report a unit root test in APA style
Report a table of ADF (and KPSS) statistics for each series in levels and first differences, with lag length and deterministic terms.
Frequently asked questions
How do I interpret the ADF test?
The null hypothesis is a unit root (non-stationary). A p-value below .05 rejects it, so the series is stationary.
Why use both ADF and KPSS?
They have opposite null hypotheses. Agreement (ADF rejects, KPSS does not) gives strong evidence of stationarity; disagreement suggests the evidence is weak.
Should I include a trend?
Include a constant and trend if the series trends upward or downward in levels; a constant only if it fluctuates around a non-zero mean.
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