ADF Stationarity Test

Suppose a graph of monthly sales keeps climbing. If the graph shows autocorrelation, is that because sales are influenced by last month? Or is it because the level itself shifts over time?

In a time series, a simple upward trend and persistent random change often appear mixed together. Before applying ordinary regression or an ARIMA model to such data, you often need to check the properties of the series.

The ADF test (Augmented Dickey–Fuller test) tests whether a time series has a unit root. Having a unit root means the effect of a shock lasts a long time, so the series is likely to behave differently from an ordinary stationary series.

In this guide we use a hypothetical monthly sales index to compare the original series with its first-differenced series, and look at how to read the ADF statistic, the p-value and the critical values, as well as the test's limitations.

The ADF test is a unit root test that extends the Dickey–Fuller test so that it can include additional lagged difference terms of the series. It examines whether the current series has a unit root while accounting for autocorrelation across several time points.

The null hypothesis (H₀) of the ADF test is that a unit root exists. The alternative hypothesis (H₁) is that, given the specified deterministic terms (constant, trend, etc.), the process has no unit root. Rejecting the null hypothesis does not automatically prove stationarity in every sense.

In data whose mean and variability structure change over time, simple correlations can be exaggerated, or the model's residual structure may not be properly explained. For example, regressing two unrelated rising series on each other can produce a spurious regression that looks like a significant relationship.

A unit root test is not the only way to judge stationarity. You should look at it together with the graph, seasonality, structural changes and the results of other tests. Still, it is a useful starting point for considering whether differencing is needed.

The test statistic is a t-type statistic for γ, but it follows the Dickey–Fuller distribution rather than the usual t distribution. So you must not judge it using ordinary t-test critical values.

With too few lagged difference terms, residual autocorrelation remains; with too many, power can drop. In this example we include a constant and choose the number of lags by AIC.

We look at a monthly sales index recorded by a retail company over 10 years. This hypothetical data was generated so that a small upward pressure and irregular shocks accumulate each month. It is not any real company's sales data.

In the original series the values shift over time, and changes caused by shocks remain in later levels too. This pattern fits a series with a unit root well, but you should not conclude there is a unit root from the graph alone.

For the original series, the ADF statistic is -0.148 and the p-value is 0.944. At the 5% significance level we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that there is no unit root. This is different from proving that a unit root definitely exists.

When the test statistic is more negative than the corresponding critical value, the result points toward rejecting the null hypothesis. In an actual decision, you need to check the chosen model specification together with the p-value and the critical values.

The first difference is calculated by subtracting last month's value from this month's value. It removes the accumulated changes in level, letting you look at the data in terms of each month's change. However, not every nonstationary series is fixed by first differencing alone.

For the differenced series, the ADF statistic is -4.494 and the p-value is 0.0002018. At the 5% level we reject the unit root null hypothesis. In this hypothetical case, we can interpret the differenced series as behaving more like a stationary series than the original series.

You should not automatically add or drop a trend term just because a trend looks clear. Decide based on the context in which the data were generated, the graph and the model assumptions. When you change the specification, the interpretation of the test changes as well.

ADF and KPSS have null hypotheses pointing in opposite directions, so they can be used to complement each other. Passing the ADF test does not mean the autocorrelation problem checked by Ljung–Box is solved as well.

Time interval and missing values

For monthly data, consecutive months must all be present. Filling missing values with 0 without any justification, or analyzing irregular dates just by their order, can distort the interpretation of differences and lags.

Seasonality and structural changes

Seasonal unit roots, economic shocks and structural breaks due to policy changes can shake the results of an ordinary ADF test. If needed, consider seasonal differencing or a separate test that is robust to structural change.

Lag length and sample size

The number of lags can be chosen with AIC, BIC and the like, but with a small sample the power can be low. You should also check whether residual autocorrelation remains.

It is hard to conclude “the series is nonstationary” just because the p-value is .05 or higher. It means there is not enough evidence to reject a unit root, and it may have come out that way because of low power or the model specification.

Conversely, even if the p-value is smaller than .05, that does not guarantee constant variance, seasonal stability or the absence of structural change. Over-differencing can lose useful long-run relationships, so consider visual diagnostics and context together.

In the report, state the unit and observation period of the data, whether a constant and trend were included, how the lags were selected, the ADF statistic and p-value, and whether differencing was applied.

The ADF test does not itself make forecasts, but it provides the basis for judgments needed to set up many time series models, such as ARIMA and VAR, appropriately. To analyze long-run equilibrium relationships, you also need to separately examine each series' order of integration and whether the series are cointegrated.

The ADF test examines whether a time series has a unit root, helping you decide on differencing and model specification. Even if you fail to reject the null hypothesis for the original series, the existence of a unit root has not been proven; conversely, even if you reject it, not every form of stationarity is guaranteed.

Only by looking together at the deterministic terms used in the test, the lag selection method, the graph and the possibility of structural change can you correctly convey what the results mean.