What is the Augmented Dickey-Fuller test?

The Augmented Dickey-Fuller (ADF) test is a statistical test used to check whether a time series has a unit root, meaning it is non-stationary. A stationary series has constant mean and variance over time. The ADF test extends the basic Dickey-Fuller test by adding lagged difference terms to control for autocorrelation in the errors.

How do I interpret the ADF test p-value?

If the ADF p-value is less than 0.05, you reject the null hypothesis of a unit root — the series is stationary. If p >= 0.05, you fail to reject H0 — the series is likely non-stationary. You can also compare the ADF statistic to critical values: if the ADF statistic is more negative than the critical value, the series is stationary.

What is the difference between ADF and KPSS tests?

The ADF test has a null hypothesis of non-stationarity (unit root), while the KPSS test has a null hypothesis of stationarity. They are complementary: if both agree (ADF rejects H0 and KPSS does not reject H0), you have strong evidence of stationarity.

What are the three model specifications for the ADF test?

The ADF test can be run with: (1) No constant, no trend; (2) Constant only — most common; (3) Constant and trend — for series with a linear deterministic trend. Choose the model that matches the visual pattern of your data.

How do I report ADF test results in APA format?

Example: 'An Augmented Dickey-Fuller test indicated that the time series was stationary (ADF = -4.32, lag = 2, p = .003), leading to rejection of the null hypothesis of a unit root.' Always include the ADF statistic, lags used, and p-value.

Can I use the ADF test on ecological or biological time series?

Yes. The ADF test is applicable to any time series data, including ecological monitoring data, wildlife tracking data, and biological measurements. Non-stationarity is common in ecological series due to seasonal trends and long-term changes.

What is the ADF test formula?

The ADF regression: deltaY_t = alpha + beta*t + gamma*Y_{t-1} + sum(delta_i * deltaY_{t-i}) + epsilon_t. The null hypothesis is gamma = 0 (unit root present). The test statistic is tau = gamma_hat / SE(gamma_hat).

Augmented Dickey Fuller Test Calculator (Free ADF Unit Root Test) | Stats Unlock

Augmented Dickey-Fuller Test Calculator (Free ADF Unit Root Test) | StatsUnlock

📋 Enter Your Time Series Data

Sample Dataset:

Series Name (editable)

Time Series Values — comma-separated or one per line

Enter numeric values in time order (earliest first). Comma or newline separated.

— values loaded

Upload CSV or Excel file (.csv, .txt, .xlsx, .xls)

Supports .csv, .txt, .xlsx, .xls — click a column button to select it as your time series. Selected columns highlighted green.

Enter values manually (one per row, in time order):

t	Value

⚙ Test Configuration

Significance Level (α)

Model Specification

Lag Selection Method

Fixed Lag (if selected above)

Max Lags to Search (0 = Schwert rule)

ACF Max Lag for Plot

🔍 Interpretation of Results & How to Write Up in Research

Run the ADF test above to see your detailed interpretation and write-up templates here.

🧮 Technical Notes & Formulas

The Augmented Dickey-Fuller test uses the following regression equations, depending on the model specification:

①

ADF Regression (With Constant)

ΔY_t = α + γY_t-1 + δ₁ΔY_t-1 + … + δ_pΔY_t-p + ε_t

ΔY_tFirst difference of the series at time t: Y_t − Y_t-1

αIntercept (constant) — accounts for non-zero mean of the series

γUnit root coefficient. H₀: γ = 0 (unit root / non-stationary). H₁: γ < 0 (stationary)

δ_iCoefficients on lagged differences — absorb serial correlation from the error term

pOptimal lag order chosen by AIC, BIC, or fixed specification

②

ADF Regression (Constant + Trend)

ΔY_t = α + βt + γY_t-1 + Σδ_iΔY_t-i + ε_t

βtDeterministic linear trend — use when the series trends visually over time

H₀Unit root: γ = 0 — the series is trend non-stationary

H₁No unit root: γ < 0 — the series is trend-stationary

③

ADF Test Statistic (τ)

τ = γ̂ / SE(γ̂)

γ̂OLS estimate of the unit root coefficient from the ADF regression

SE(γ̂)Standard error of γ̂ computed from OLS residuals

NoteThe ADF τ does NOT follow a standard t-distribution — critical values come from MacKinnon (1994) tables

④

Optimal Lag via AIC

AIC(p) = n · ln(SSR/n) + 2(p + k)

nEffective sample size (observations used after lag adjustment)

SSRSum of squared residuals from the ADF regression at lag p

kNumber of deterministic regressors (1 for constant; 2 for constant + trend)

RuleSelect lag p that minimises AIC; lower AIC = better model fit with less complexity

⑤

Schwert (1989) Maximum Lag Rule

p_max = floor( 12 × (n/100)^0.25 )

nTotal number of observations in the original time series

p_maxUpper bound on the lag search grid for AIC or BIC selection

Examplesn = 50 → p_max = 10; n = 100 → p_max = 12; n = 200 → p_max = 15

⑥

MacKinnon (1994) p-value Approximation

p ≈ Φ(a₀ + a₁/T + a₂/T²)

TEffective sample size (n minus lag order)

a₀–a₂Response surface regression coefficients — differ by model specification (nc / c / ct)

ΦStandard normal CDF — transforms the polynomial into a probability

NoteFinite-sample correction is applied; results match MacKinnon (1994) Table 1

⑦

First-Order Differencing

ΔY_t = Y_t − Y_t−1

ΔY_tFirst-differenced value at time t — removes a stochastic trend

Y_tOriginal series value at time t

UseApply once if the series is I(1). If still non-stationary, apply again (I(2)). Re-run ADF after each round of differencing.

📖 How to Use This ADF Test Calculator — Step-by-Step Guide

Enter or Upload Your Time Series

Paste values into the text area (comma-separated or one per line), upload a CSV/Excel file and click a column, or use the manual grid. Values must be numeric and in time order (oldest first). Minimum 10 observations required.

Edit the Series Name

Click the editable Series Name field and type a meaningful label. This name will appear on charts and in the downloaded report. Example: "CPI_Monthly", "Rainfall_mm_1990-2020", "Population_Count".

Choose a Sample Dataset (Optional)

Select from five built-in datasets — GDP growth, temperature, wildlife population index, stock price, or rainfall — to explore how the tool works before using your own data. Each illustrates different stationarity patterns.

Set the Significance Level (α)

Choose α = 0.05 for standard research (default). Use α = 0.01 for strict hypothesis testing or financial applications. Use α = 0.10 for exploratory or pilot studies. The critical values displayed adjust automatically.

Choose the Model Specification

"With Constant" is the most common choice and handles series with non-zero means. Select "Constant + Trend" if the series plots show a clear upward or downward slope. "No Constant" is rarely used in practice — only for series that genuinely fluctuate around zero.

Select Lag Selection Method

AIC (default) balances fit and parsimony and is best for most applications. BIC is more conservative (fewer lags) and preferred for small samples or where parsimony is important. "Fixed Lag" lets you specify exactly p lags based on theory or prior studies.

Click "Run ADF Test"

The calculator runs OLS regression for each candidate lag (up to the Schwert maximum), selects the best lag by the chosen criterion, computes the ADF τ statistic, and applies MacKinnon (1994) critical values and p-value approximation.

Read the Stationarity Verdict

A green banner confirms the series is stationary (I(0)) — safe to use directly in ARIMA, regression, and other models. A red banner means non-stationary (I(1)) — apply first-order differencing (ΔY_t = Y_t − Y_t-1) and re-run.

Inspect the Four Diagnostic Charts

Chart 1 shows the raw series (look for trends and structural shifts). Chart 2 shows the first-differenced series (should look stationary). Chart 3 shows the ACF (should decay to near-zero quickly for a stationary series). Chart 4 compares your ADF τ to the 1%, 5%, and 10% critical values visually.

Copy or Download Your Results

Use the five write-up cards — APA 7th, Thesis, Plain Language, Structured Abstract, Pre-Registration — to copy formatted results directly into your manuscript. Download the full report as a .txt file or print to PDF for your records and supplementary materials.

❓ Frequently Asked Questions

What is the Augmented Dickey-Fuller (ADF) test?

The Augmented Dickey-Fuller test is a hypothesis test used to determine whether a time series has a unit root, meaning it is non-stationary. A stationary series has constant mean, variance, and autocorrelation structure over time — a prerequisite for many time series models. The ADF test extends the basic Dickey-Fuller test by including lagged difference terms (ΔY_t-1, …, ΔY_t-p) in the regression to remove autocorrelation from the error term. It is one of the most widely used pre-processing steps before fitting ARIMA, VAR, or cointegration models.

What is the null hypothesis of the ADF test?

The null hypothesis (H₀) of the ADF test is that the series has a unit root and is non-stationary. The alternative hypothesis (H₁) is stationarity — or trend-stationarity when the constant+trend model is used. If the ADF p-value is below your chosen significance level α (e.g., 0.05), you reject H₀ and conclude the series is stationary. If p ≥ α, you fail to reject H₀ — the series is likely non-stationary and should be differenced.

How do I interpret the ADF statistic?

The ADF test statistic (τ) is the t-ratio for the unit root coefficient γ in the ADF regression. More negative values mean stronger evidence against the null hypothesis (non-stationarity). You compare τ to MacKinnon critical values: if τ is more negative than the critical value at your α level (e.g., −2.86 at 5% for the constant model), you reject H₀. You can also use the MacKinnon p-value directly: p < α means stationary.

How many lags should I use in the ADF test?

The most common approach is automatic selection using AIC (Akaike Information Criterion), which fits the ADF model at each candidate lag and picks the one minimising AIC. BIC is more conservative and favours fewer lags. The Schwert (1989) rule p_max = int(12·(n/100)⁵·²⁵) sets the maximum lag to search. For most research, AIC automatic selection (the default here) is recommended. Manual fixed lag is appropriate when theory or prior literature specifies a particular lag structure.

What model specification should I choose?

Three options: (1) No constant — for series that oscillate around zero (rare in practice, e.g., some financial return differentials). (2) Constant only — the default; use for most series with a non-zero mean but no visible long-run trend. (3) Constant + trend — use when the raw series plot shows a clear upward or downward slope. When uncertain, inspect Chart 1 (the time series plot) and consider testing with constant+trend if the plot suggests a trend. Choosing the wrong specification can bias the results.

What do I do if my time series is non-stationary?

Apply first-order differencing: ΔY_t = Y_t − Y_t-1. Re-run the ADF test on the differenced series. If stationary after one differencing, the original series is integrated of order 1 (written I(1)), and you should use d=1 in your ARIMA model. If still non-stationary, apply second differencing (I(2)). Most economic and ecological time series are I(1). Note: over-differencing can introduce spurious moving-average patterns — stop as soon as stationarity is confirmed.

What is the difference between the ADF test and KPSS test?

The ADF test (H₀: unit root / non-stationary) and KPSS test (H₀: stationary) are complementary. Ideally, run both: if ADF rejects H₀ and KPSS fails to reject H₀, you have strong bilateral evidence of stationarity. If they conflict — ADF rejects but KPSS also rejects — the series may be trend-stationary or contain a structural break. Running both tests is standard practice in econometrics and increasingly expected in ecology and biology time series papers.

Can I use this ADF calculator for ecological or biological time series?

Yes. The ADF test applies to any evenly-spaced numeric time series, including wildlife population counts, NDVI vegetation index, temperature anomalies, stream discharge, and animal activity indices. Non-stationarity is particularly common in ecological data due to long-term directional changes (population decline, climate trends) and seasonal patterns. Run the ADF test before fitting models like ARIMA, SARIMA, or dynamic occupancy models to ensure valid inference.

How do I report ADF test results in APA 7th edition?

Format: "An Augmented Dickey-Fuller test with AIC lag selection (p = 2) indicated that the [series name] was stationary, τ([T]) = [value], p = [value]. The null hypothesis of a unit root was rejected at α = .05 (MacKinnon 1994 critical value: [value])." Always include: τ statistic, effective sample size T (in parentheses), MacKinnon p-value, lag count, and model specification. If non-stationary, state that first-differencing was applied and the I(1) classification.

What are MacKinnon critical values and why are they different from standard t-values?

Under the null hypothesis of a unit root, the ADF test statistic does not follow a standard t-distribution — because when the null is true, the data-generating process is an I(1) random walk, not a stationary I(0) process. MacKinnon (1994) derived the correct asymptotic distributions by simulation and fitted polynomial response-surface regressions to approximate critical values and p-values for different model specifications (no constant, constant, constant+trend) and sample sizes. This calculator implements MacKinnon (1994) coefficients to give accurate results at 1%, 5%, and 10% levels.

🗺 When to Use the Augmented Dickey-Fuller Test

This free Augmented Dickey-Fuller test calculator is designed for researchers, analysts, and students who need to check stationarity as part of a time series analysis workflow. Use the ADF test when:

You plan to fit an ARIMA, SARIMA, or VAR model and need to determine the differencing order (d).
You are testing for cointegration (Engle-Granger two-step) and need to confirm each series is I(1) before testing for a long-run relationship.
You are analysing economic or financial time series (GDP, inflation, exchange rates, stock prices, interest rates).
You are pre-processing ecological, environmental, or biological time series (population indices, NDVI, rainfall, temperature) before fitting dynamic models.
You need to avoid the spurious regression problem: regressing one non-stationary series on another produces misleadingly high R² values and invalid t-statistics.
A reviewer, thesis supervisor, or journal requires you to document stationarity testing as part of your time series methods section.
You want to determine the integration order of variables before including them in a distributed lag (ARDL) or error-correction model (ECM).

Model Specification Decision Guide

Situation	Recommended Model	Rationale
Series with no trend; mean ≠ 0	Constant only	Most common default; accounts for non-zero mean without over-specifying
Series with clear upward or downward trend	Constant + Trend	Prevents spurious rejection due to deterministic trend component
Series fluctuating around zero	No Constant	Parsimonious; rarely appropriate in real-world data
Uncertain — inspect plot first	Try Constant, then Constant + Trend	Visual inspection + information criteria (AIC) help guide the choice

🏁 Conclusion

The Augmented Dickey-Fuller (ADF) test is one of the most fundamental tools in time series analysis. Before building any forecasting or structural model — ARIMA, VAR, cointegration, or error-correction — you must first confirm whether your series is stationary. A non-stationary series violates the assumptions of most standard models, producing spurious regression results, misleading forecasts, and invalid hypothesis tests (Granger & Newbold, 1974). The ADF test gives you an objective, peer-reviewed statistical criterion for making that determination.

This free ADF test calculator handles the complete workflow: automatic lag selection via AIC or BIC using the Schwert (1989) maximum lag rule, three model specifications (no constant, constant only, constant + trend), MacKinnon (1994) critical values and p-value approximation, and four diagnostic charts that make the results visual and easy to communicate. You do not need R, Python, Stata, or specialist econometrics software — the tool runs entirely in your browser and produces publication-ready output instantly.

When the ADF test rejects the null hypothesis (p < α), your series is stationary — you can proceed directly to model fitting without differencing. When it fails to reject (p ≥ α), apply first-order differencing and re-run the test. Most real-world series — economic indicators, wildlife population counts, climate variables — are integrated of order one (I(1)), so one round of differencing is usually sufficient to achieve stationarity.

Always complement ADF results with visual inspection of the raw time series plot and the ACF. If your series contains a structural break (policy shift, ecological disturbance, market crash), consider the Zivot-Andrews (1992) test, which allows for one unknown break point — the ADF test may incorrectly fail to reject H₀ in the presence of structural breaks. For multiple series, consider the Im-Pesaran-Shin (2003) panel unit root test.

Reporting transparently is central to reproducible time series research. Use the five write-up templates in the Interpretation panel above — APA 7th edition, Thesis, Plain Language, Structured Abstract, and Pre-Registration — to produce correctly formatted results for any journal or institutional requirement. Always report the ADF statistic, lag count, model specification, and exact MacKinnon p-value in your methods section so readers can fully evaluate and replicate your analysis.

📚 References

The following peer-reviewed references support the Augmented Dickey-Fuller test methodology, unit root testing theory, lag selection criteria, and stationarity analysis described in this augmented Dickey-Fuller test calculator and unit root test guide.

Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366), 427–431. https://doi.org/10.2307/2286348
Dickey, D. A., & Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49(4), 1057–1072. https://doi.org/10.2307/1912517
MacKinnon, J. G. (1994). Approximate asymptotic distribution functions for unit-root and cointegration tests. Journal of Business & Economic Statistics, 12(2), 167–176. https://doi.org/10.1080/07350015.1994.10510005
MacKinnon, J. G. (2010). Critical values for cointegration tests (Queen’s Economics Department Working Paper No. 1227). Queen’s University. https://ideas.repec.org/p/qed/wpaper/1227.html
Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics, 7(2), 147–159. https://doi.org/10.1080/07350015.1989.10509723
Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1–3), 159–178. https://doi.org/10.1016/0304-4076(92)90104-Y
Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71(3), 599–607. https://doi.org/10.1093/biomet/71.3.599
Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836. https://doi.org/10.2307/2171846
Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75(2), 335–346. https://doi.org/10.1093/biomet/75.2.335
Zivot, E., & Andrews, D. W. K. (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business & Economic Statistics, 10(3), 251–270. https://doi.org/10.1080/07350015.1992.10509904
Hamilton, J. D. (1994). Time series analysis. Princeton University Press. https://press.princeton.edu/books
Enders, W. (2014). Applied econometric time series (4th ed.). Wiley. https://www.wiley.com
Granger, C. W. J., & Newbold, P. (1974). Spurious regressions in econometrics. Journal of Econometrics, 2(2), 111–120. https://doi.org/10.1016/0304-4076(74)90034-7
Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723. https://doi.org/10.1109/TAC.1974.1100705
Im, K. S., Pesaran, M. H., & Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115(1), 53–74. https://doi.org/10.1016/S0304-4076(03)00092-7