FRM Exam Part I · Nonstationary Time Series

Dickey-Fuller and Augmented Dickey-Fuller Unit Root Tests

Q: What is the null hypothesis of the Dickey-Fuller test?

The null is that the series has a unit root, meaning γ = 0 or φ = 1, so it is nonstationary. The alternative is that the series is stationary, γ < 0. Rejecting the null supports stationarity.

Q: How do I compare the ADF statistic with critical values?

Compute γ̂ ÷ SE(γ̂) and compare it with the Dickey-Fuller critical value for your test form and significance level. Reject the null if the statistic is more negative than the critical value. Otherwise fail to reject.

Q: Why is the test called augmented?

The augmented version adds lagged differences of the series to the regression. These terms absorb serial correlation in the errors. The null, test statistic and decision rule are unchanged.

Q: Does failing to reject prove a unit root?

No. The test has low power, especially in small samples or when the root is close to one. Failing to reject only means you lack evidence against a unit root.

Updated 11 October 2026 · Fact-checked

The Dickey-Fuller test checks whether a series has a unit root. You regress the change in the series on its lagged level and test whether that coefficient is zero. The null is a unit root (nonstationary). If the t-statistic is more negative than the Dickey-Fuller critical value, you reject the null.

Understand Testing for Unit Roots: Dickey-Fuller Test

A unit root means a series behaves like a random walk: shocks never fade, and the mean and variance change over time. Many price and rate series look like this. Regressing one such series on another can give misleading results, so you need a formal test before modelling.

Start with an AR(1): Y(t) = φY(t−1) + ε(t). A unit root means φ = 1. Subtract Y(t−1) from both sides and you get ΔY(t) = γY(t−1) + ε(t), where γ = φ − 1. So a unit root means γ = 0. A stationary series has γ < 0 (that is, |φ| < 1).

The Dickey-Fuller (DF) test runs this regression and tests H0: γ = 0 (unit root, nonstationary) against H1: γ < 0 (stationary). It is a one-sided, lower-tail test. The key trap: under the null, the usual t-statistic does not follow a Student's t distribution. You must compare it with special Dickey-Fuller critical values, which are more negative than normal ones.

The test comes in three forms: no constant (pure random walk), with a constant (drift), and with a constant and a time trend. Choose the form that matches the data. Critical values differ across forms.

The augmented Dickey-Fuller (ADF) test adds lagged differences of Y to the regression: ΔY(t) = α + βt + γY(t−1) + Σ δ(i)ΔY(t−i) + ε(t). These extra terms soak up serial correlation in the errors, which would otherwise invalidate the test. The null and decision rule stay the same.

A failure to reject does not prove a unit root. The test has low power, especially when φ is close to 1 or the sample is short. Treat non-rejection as evidence consistent with a unit root, not proof.

Key formulas to remember

DF regression (no constant): ΔY(t) = γY(t−1) + ε(t), with γ = φ − 1; Unit root means γ = 0.
DF regression with constant and trend: ΔY(t) = α + βt + γY(t−1) + ε(t); Use when the series may have drift or a deterministic trend.
ADF regression: ΔY(t) = α + βt + γY(t−1) + Σ δ(i)ΔY(t−i) + ε(t); Lagged differences remove serial correlation in errors.
Hypotheses: H0: γ = 0 (unit root); H1: γ < 0 (stationary); One-sided test on the lower tail.
Test statistic: DF = γ̂ ÷ SE(γ̂); Compare with Dickey-Fuller critical values, not standard t values.
Decision rule: Reject H0 if DF < critical value (more negative); Rejection supports stationarity.

How to solve Testing for Unit Roots: Dickey-Fuller Test questions

Use this sequence for any question that asks you to set up, run or interpret a Dickey-Fuller or ADF test.

1Identify the form of the test: no constant, constant only, or constant plus trend. Use the data description or the regression given.
2Write the hypotheses: H0: γ = 0 (unit root), H1: γ < 0 (stationary). If the question uses φ, then H0: φ = 1.
3Find the coefficient on the lagged level Y(t−1) and its standard error. Ignore the lagged-difference terms.
4Compute the test statistic: γ̂ ÷ SE(γ̂).
5Compare it with the Dickey-Fuller critical value for that form and significance level. Remember these values are negative.
6Reject H0 only if the statistic is lower (more negative) than the critical value. Otherwise fail to reject.
7State the conclusion in words: stationary or cannot reject a unit root. If nonstationary, note that differencing is the usual remedy.

Quickest way: Sign and size check

When to use it: When the question gives a t-statistic and a critical value and asks for the conclusion.

Check the sign. A positive statistic can never reject the null.
Compare as negative numbers: −3.1 is more negative than −2.9, so it rejects at that level.
If rejected, answer stationary. If not, answer cannot reject a unit root.
Scan options for the trap of using a standard t or normal critical value such as −1.96.

Common mistakes in Testing for Unit Roots: Dickey-Fuller Test

Using standard t critical values such as −1.96 or −1.645.
The statistic is called a t-statistic, so students assume a t distribution.
Fix: Under the unit root null the distribution is nonstandard. Always use the Dickey-Fuller critical values given in the question.
Reading failure to reject as proof of a unit root.
Students treat the null as confirmed.
Fix: Say the data are consistent with a unit root. The test has low power, especially in small samples or when φ is near 1.
Rejecting when the statistic is less negative than the critical value.
Comparing magnitudes and ignoring the sign.
Fix: It is a lower-tail test. Reject only if the statistic is below the critical value, e.g. −3.5 < −2.9.
Confusing H0 and H1: thinking the null is stationarity.
Other tests, like KPSS, use a stationary null.
Fix: For DF and ADF, the null is always a unit root. Rejecting means stationary.
Testing the lagged difference coefficients instead of the lagged level.
The ADF regression has many coefficients.
Fix: Only the coefficient γ on Y(t−1) is tested. The δ terms are there to clean up the errors.
Applying the wrong form (no constant) to a trending series.
Students pick the simplest regression.
Fix: Match the form to the data. A trending series needs a constant and trend, with its own critical values.

Worked examples

Example 1

An ADF regression with a constant on a bond yield series gives γ̂ = −0.040 with standard error 0.020. The 5% Dickey-Fuller critical value (constant, no trend) is −2.86. What do you conclude?

Show the solution

H0: γ = 0 (unit root); H1: γ < 0.
Test statistic = −0.040 ÷ 0.020 = −2.00.
Compare: −2.00 is less negative than −2.86, so it is not below the critical value.
Do not reject H0 at 5%.

Answer: Test statistic is −2.00. Fail to reject: the series is consistent with a unit root (nonstationary).

Example 2

For an equity index, the ADF test with constant and trend gives γ̂ = −0.135 and SE = 0.030. The 1% critical value is −3.96. The analyst uses a normal 5% critical value of −1.645 and concludes the series is stationary. Evaluate the conclusion.

Show the solution

Statistic = −0.135 ÷ 0.030 = −4.50.
Correct comparison uses the Dickey-Fuller value: −4.50 < −3.96, so reject H0 even at 1%.
The analyst's conclusion of stationarity happens to be right here, but the method is wrong: −1.645 is not valid under the unit root null.
If the statistic had been −2.0, the analyst would have wrongly rejected, since −2.0 is above −3.96.

Answer: Statistic is −4.50, below −3.96, so reject the unit root null at 1% and conclude the series is stationary (trend-stationary). The method of using −1.645 is invalid.

Exam tips

Questions usually give the statistic and critical value. Do the sign-and-size comparison first, then read the options.
Memorise the direction: null equals unit root, reject equals stationary.
Expect conceptual items on why ADF adds lagged differences (to remove serial correlation) and why standard t tables fail.
If a series is nonstationary, the standard remedy is first differencing, so link this topic to random walks and spurious regression.

Practice questions from Nonstationary Time Series

Testing for Unit Roots: Dickey-Fuller Test: frequently asked questions

What is the null hypothesis of the Dickey-Fuller test?

The null is that the series has a unit root, meaning γ = 0 or φ = 1, so it is nonstationary. The alternative is that the series is stationary, γ < 0. Rejecting the null supports stationarity.

How do I compare the ADF statistic with critical values?

Compute γ̂ ÷ SE(γ̂) and compare it with the Dickey-Fuller critical value for your test form and significance level. Reject the null if the statistic is more negative than the critical value. Otherwise fail to reject.

Why is the test called augmented?

The augmented version adds lagged differences of the series to the regression. These terms absorb serial correlation in the errors. The null, test statistic and decision rule are unchanged.

Does failing to reject prove a unit root?

No. The test has low power, especially in small samples or when the root is close to one. Failing to reject only means you lack evidence against a unit root.