Common mistake 1 with Unit root tests

Skipping this step. It is how student dissertations end up reporting an R² of 0.97 that means nothing.

Common mistake 2 with Unit root tests

Differencing everything reflexively - if the series are cointegrated, differencing throws away the long-run relationship you were looking for.

Common mistake 3 with Unit root tests

These tests have low power in short samples. Failing to reject is weak evidence.

Common mistake 4 with Unit root tests

Treating the ADF and KPSS verdicts as interchangeable. Their nulls are opposites, so 'ADF did not reject' and 'KPSS rejected' are the same conclusion stated twice, while one rejecting and the other not is genuinely inconclusive.

EconStatsCraft › Unit root tests

Unit root tests

Augmented Dickey–Fuller and KPSS. Run this on every series before any time-series regression - non-stationary data produce spurious results.

What it needs

Series to test

The intuition

A stationary series has a gravitational pull towards its mean: push it away and it comes back. A unit-root series has no such anchor - every shock is permanently absorbed into the level, so the series wanders. Regressing one wanderer on another finds a relationship that is really just two independent random walks happening to drift the same way, and the t-statistic diverges rather than converging, so it "confirms" the relationship more strongly the more data you collect.

When to use it

On EVERY series, before any time-series regression. This is the single most important pre-check in applied time-series work.

Where the estimator comes from

Write y_t=\rho y_{t-1}+\varepsilon_t. Subtract y_{t-1}: \Delta y_t=(\rho-1)y_{t-1}+\varepsilon_t=\gamma y_{t-1}+\varepsilon_t. A unit root means \rho=1, i.e. \gamma=0. The awkward part is that under H_0 the regressor y_{t-1} is non-stationary, so the statistic does NOT have a t distribution - it follows the Dickey–Fuller distribution, which is why the critical values are much more negative than -1.96. The 'augmented' part adds lags of \Delta y to soak up serial correlation so the test statistic is valid.

The model

\Delta y_t=\alpha+\beta t+\gamma y_{t-1}+\sum_{i=1}^{p}\delta_i\,\Delta y_{t-i}+\varepsilon_t,\qquad H_0:\ \gamma=0\ \text{(unit root)}

What it assumes

You have chosen the right deterministic terms - constant, or constant plus trend.
Enough lags to leave the residuals white - too few invalidates the test.
No structural break. If there is one, use Zivot–Andrews instead.
The series is a single regime. A series that is stationary around one mean and then around another will look like a unit root to these tests, which is why a break test is the right next step after a puzzling non-rejection.

Statistical properties

ADF has notoriously low power: it fails to reject far too often when ρ is close to but below 1 (say 0.95).
That low power is why KPSS is run alongside - it reverses the null, so agreement between the two is much stronger evidence than either alone.
Including an unnecessary trend costs power; omitting a necessary one biases towards non-rejection.
Too few lags leaves serial correlation and invalidates the test; too many costs power. AIC selection is the usual compromise.
A structural break makes ADF fail to reject almost regardless of the truth - use Zivot–Andrews.

How to read the output

ADF H₀ = "there IS a unit root". A small p-value means stationary.
KPSS H₀ = "the series IS stationary". They are deliberately opposite; read them together.
Both agree stationary → use levels. Both agree non-stationary → difference it.
They disagree → the series is borderline. Say so and check both specifications.
If the level is I(1) but the first difference is I(0), the series is "integrated of order one".
ADF and KPSS have OPPOSITE nulls, so agreement between them is the informative case. ADF failing to reject while KPSS also fails to reject means the sample is simply not decisive, and reporting only one of the two hides that.
The lag order matters and is reported. Too few lags leaves autocorrelation in the residuals and over-rejects; too many costs power. If the verdict flips across sensible lag choices, that is the finding.
The critical values are not from a normal or t distribution and cannot be compared with ordinary ones. A statistic of -2.5 is unremarkable here and would be decisive in a regression.

A worked reading

For log GDP: ADF = 0.4707 (p = 0.984, 2 lags) and KPSS = 2.0582 (p ≤ 0.01, against a 5% critical value of 0.463). The two tests have opposite null hypotheses and both point the same way - ADF cannot reject a unit root, KPSS rejects stationarity - which is the clean case. The growth rate, by contrast, gives ADF = -5.9245 (p < 0.0001) and KPSS = 0.1359 (p ≥ 0.1): again both agree, this time on stationarity. So log GDP is I(1). The practical consequence is that a regression of log GDP on another I(1) series is spurious unless the two are cointegrated, and that is a separate test rather than something to assume.

Symptom → diagnosis → what to do

If you see this	It means	Do this
ADF fails to reject, KPSS rejects	Both point to non-stationary	Difference the series and re-test
ADF rejects, KPSS fails to reject	Both point to stationary	Use levels
Both reject	Possibly fractionally integrated, or a break	Check for breaks; consider ARFIMA
Neither rejects	Not enough data to tell	Say so honestly; report both specifications
Still non-stationary after two differences	Almost certainly a break, not I(2)	Run Zivot–Andrews before differencing again

Common mistakes

Skipping this step. It is how student dissertations end up reporting an R² of 0.97 that means nothing.
Differencing everything reflexively - if the series are cointegrated, differencing throws away the long-run relationship you were looking for.
These tests have low power in short samples. Failing to reject is weak evidence.
Treating the ADF and KPSS verdicts as interchangeable. Their nulls are opposites, so 'ADF did not reject' and 'KPSS rejected' are the same conclusion stated twice, while one rejecting and the other not is genuinely inconclusive.

Where to read this properly

Dickey & Fuller (1979)
Kwiatkowski, Phillips, Schmidt & Shin (1992)
Granger & Newbold (1974) - spurious regressions in econometrics
Enders (2014), Applied Econometric Time Series, 4th ed., Wiley, chapter 4

Related Time series - one series

Time-varying coefficients (Kalman) · Correlogram (ACF / PACF) · ARIMA / SARIMA · Rolling-origin cross-validation · Theta method · Unobserved components · STL forecasting · Forecast combination · Markov regime switching · Change frequency (resample) · ARCH / GARCH volatility · Decompose & filter · Spectral analysis · Exponential smoothing · Forecast accuracy

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Unit root tests

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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