The ADF test is a unit-root test that asks whether a time series is stationary (mean and variance constant over time) or non-stationary. Use it before fitting any ARMA/ARIMA model, before regressing one time series on another (to avoid spurious regressions), and whenever you need to confirm that shocks decay rather than persist forever. Typical examples: GDP, stock prices, temperature anomalies, and population counts.

The ADF p-value is the probability of obtaining an ADF statistic as negative as the observed one if a unit root truly exists in the data. A small p-value (e.g., p = 0.02) means: "If the series really had a unit root, we'd see a result this extreme only 2% of the time" — so we reject H₀ and conclude stationarity. The p-value is not the probability that H₀ is true.

Largely yes — but with one caveat. Stationarity is a binary property the model needs, so a statistically significant rejection of H₀ is practically meaningful. However, a near-significant result (p ≈ 0.05) on a series with a half-life of decades may be practically non-stationary for forecasting. Always look at how negative the ADF statistic is, not just the p-value.

The ADF test does not have a Cohen's-d-style effect size. Instead, the magnitude of γ̂ (the lagged-level coefficient) and the half-life of shocks (≈ ln(0.5)/ln(1+γ̂)) tell you how persistent the series is. A γ̂ near zero with a half-life of 100+ periods is essentially a unit root even when p < 0.05.

The ADF test assumes (i) sufficient n (≥ 30), (ii) residuals are approximately white noise, (iii) the lag order p is correctly specified, and (iv) no structural breaks. If lags are wrong, residuals show autocorrelation — increase p. If breaks exist, use the Zivot-Andrews or Phillips-Perron test instead.

Minimum n = 30, recommended n ≥ 50, ideal n ≥ 100. With fewer than 30 observations the test has very low power and the asymptotic critical values are inaccurate. For samples below 25, prefer the small-sample bootstrap version.

The ADF test is inherently one-sided (left-tailed). The alternative hypothesis is γ < 0 — i.e., the lagged-level coefficient is significantly negative. There is no "two-sided ADF". Critical values are negative (e.g., −2.86 at 5% for the constant-only model with large n).

Report the test statistic, lag order, p-value, and which critical-value threshold was crossed. Example: "An augmented Dickey–Fuller test (constant + trend, lag = 4) rejected the unit-root null hypothesis, ADF = −4.21, p = .002, exceeding the 1% critical value of −3.96; the GDP series is therefore trend-stationary."

This calculator gives results that match Python (statsmodels.adfuller) and R (tseries::adf.test) for standard cases. For a peer-reviewed manuscript or PhD thesis we recommend cross-checking with one of those packages and citing Dickey & Fuller (1979) and Said & Dickey (1984). The tool itself can be cited as "Stats Unlock. (2025). Augmented Dickey-Fuller test calculator. https://statsunlock.com/tools/augmented-dickey-fuller-test-calculator".

(1) Difference the series once and re-test. Most macro-economic and ecological series become stationary after first differencing — they are I(1). (2) If still non-stationary, difference again to test for I(2). (3) Confirm with a KPSS test (whose null is stationarity); if both ADF says "non-stationary" and KPSS says "non-stationary", the conclusion is robust. (4) Consider whether a structural-break-aware test (Zivot-Andrews) is more appropriate.