Every time-series model rests on an implicit assumption of stationarity, and regressions estimated without checking it are among the most common corrections demanded in theses and journal reviews. A unit root test provides formal evidence on whether a series is stationary and is the first step before any ARDL, VAR or cointegration analysis. This guide sets out the hypotheses of the ADF, PP and KPSS trio, the choice of lags and deterministic terms, and a concrete decision strategy for when the tests disagree.
What stationarity means: three conditions
A series is (weakly) stationary when three conditions hold together: its mean is constant over time, its variance is constant over time, and the autocovariance between any two observations depends only on the lag separating them, not on the calendar date. A unit root is the most common violation of this structure: in the AR(1) model yt = ρyt−1 + εt, the series is a random walk when ρ = 1 — shocks never die out, they accumulate permanently, and the variance grows without bound. When ρ < 1, the series reverts to its mean. Every unit root test is, at heart, a test of that ρ = 1 boundary.
Spurious regression: a high R² and a meaningless relationship
Regress two entirely independent random walks on each other and, despite the absence of any genuine relationship, you will usually obtain significant t statistics and a high R² — the phenomenon known in the literature as spurious regression, and the central danger of running regressions in levels on non-stationary data. The classic warning sign is an R² that exceeds the Durbin-Watson statistic (for instance R² = 0.85 against DW = 0.30). The chart below shows a typical random-walk pattern: long swings with no tendency to return to a fixed mean.
The unit root test trio: ADF, PP and KPSS
ADF (Augmented Dickey-Fuller): the null hypothesis is that the series contains a unit root; the alternative is stationarity. Lagged differences of the dependent variable are added to the test equation to mop up serial correlation in the errors. The lag length is typically chosen by AIC or SIC: too few lags distort the size of the test, too many drain its power. The unit root null is rejected when the test statistic is more negative than the ADF critical value.
PP (Phillips-Perron): tests the same null as the ADF but handles serial correlation through a non-parametric (Newey-West) correction to the test statistic rather than by adding lags to the equation. There is no lag-length decision to defend and the test is robust to heteroskedasticity; the trade-off is possible size distortion in small samples and in series with strong negative MA components. KPSS turns the game around: its null hypothesis is that the series is stationary (around a level or a trend), with a unit root as the alternative. This LM-type test mirrors the ADF/PP and supplies confirmatory evidence.
| Test | Null hypothesis (H₀) | Alternative (H₁) | Approach |
|---|---|---|---|
| ADF | Unit root present | Series is stationary | Parametric; lagged differences (AIC/SIC selection) |
| PP | Unit root present | Series is stationary | Non-parametric Newey-West correction |
| KPSS | Series is stationary | Unit root present | LM test; hypotheses reversed |
| Zivot-Andrews | Unit root, no break | Stationary with one endogenous break | Break date estimated from the data |
Deterministic terms: none, constant, or constant plus trend
Whether the test equation includes a constant and/or a linear trend directly changes the critical values and the power of the test. The practical rule: include a constant if the series fluctuates around a non-zero level; use constant plus trend if the plot shows a clear drift; the no-constant, no-trend form is appropriate only for series oscillating around zero (returns, for example). Omitting a needed component biases the test towards finding a unit root, while including redundant ones reduces power. When in doubt, start from the most general specification (with trend) and simplify sequentially according to the significance of the deterministic terms.
A joint decision strategy and the order of integration
- Run ADF (or PP) together with KPSS; because their hypotheses point in opposite directions, they corroborate one another.
- If ADF rejects the unit root and KPSS does not reject stationarity, treat the series as I(0); if ADF fails to reject and KPSS rejects, treat it as I(1).
- If the two tests conflict, the verdict is inconclusive: revisit the lag choice, question the deterministic specification, and test for a structural break.
- Difference an I(1) series once and re-run the tests on the differences; if the first difference is still non-stationary the series may be I(2) (price-level series are the usual suspects). Unnecessary differencing over-differences the data and destroys long-run information.
When a series contains a structural break (a crisis, an exchange-rate regime change, a pandemic), the power of the standard ADF test falls sharply and a genuinely stationary series can be misclassified as having a unit root. The standard remedy is the Zivot-Andrews test, which determines the break date endogenously from the data. On the software side, EViews offers all three tests and their break-augmented variants in a single unit-root menu; Stata provides dfuller, pperron and kpss, while in R the urca and tseries packages cover the same ground. The stationarity verdict is a precondition for the next steps — the ARDL bounds test or Johansen cointegration — and for studies with a panel dimension, see our panel data guide on model choice.
A unit root test is not a formality; it is the structural survey of the ground every later model will stand on.
Frequently Asked Questions
What should I do when the ADF and KPSS tests disagree?
First revisit the lag length and the deterministic terms, since most conflicts stem from misspecification. If the disagreement persists, test for a structural break with Zivot-Andrews, report both results transparently, and justify your final classification in the text.
How do I choose the lag length in the ADF test?
The standard approach is automatic selection by the AIC or SIC information criteria, with SIC favouring more parsimonious models. Good practice is to verify that no serial correlation remains in the residuals at the chosen lag and to show that the conclusion survives a few alternative lag choices.
What does it mean if my series is still non-stationary after first differencing?
The series may be I(2), in which case second differencing is needed, but I(2) behaviour is relatively rare in economic data; rule out seasonality, structural breaks and outliers first. Differencing twice without cause complicates interpretation and erases long-run relationship information.
What does Celsus offer for stationarity and unit root analysis?
Celsus delivers visual and formal stationarity analysis of your series, the ADF-PP-KPSS battery, break-augmented tests, and a reasoned verdict on the order of integration. EViews, Stata or R output is supplied with thesis- and journal-ready tables and fully reproducible scripts.