How do you test for a unit root and what does the Dickey-Fuller test actually tell you?

The Dickey-Fuller (DF) test is the standard tool for detecting unit roots (non-stationarity) in time series data. Understanding its mechanics is critical for CFA Level II.

The Setup:

Start with an AR(1) model: x_t = b0 + b1 * x_(t-1) + e_t

Subtract x_(t-1) from both sides:

Delta(x_t) = b0 + (b1 - 1) * x_(t-1) + e_t

Delta(x_t) = b0 + g * x_(t-1) + e_t (where g = b1 - 1)

Hypotheses:

• H0: g = 0 (equivalently, b1 = 1) — the series has a unit root (non-stationary)
• Ha: g < 0 (equivalently, b1 < 1) — the series is stationary

Critical Difference from Regular t-Tests:

Under the null hypothesis of a unit root, the test statistic does NOT follow a standard t-distribution. It follows the Dickey-Fuller distribution, which has critical values that are MORE negative than standard t-values. This means:

• A regular t-test would reject the null too often (find false stationarity)
• You must use DF critical values, not standard t-tables
• For example, at 5% significance, the DF critical value might be -2.86, while the standard t-critical is only -1.96

Decision Rule:

• If test statistic < DF critical value (more negative): reject H0, series IS stationary
• If test statistic > DF critical value (less negative): fail to reject H0, series has a unit root

Worked Example:

You run a DF test on quarterly inflation data and get a test statistic of -3.42. The 5% DF critical value is -2.86.

Since -3.42 < -2.86, you reject the null hypothesis. The inflation series is stationary — you can proceed with AR modeling directly.

Now suppose you test a stock price level and get a test statistic of -1.85. Since -1.85 > -2.86, you fail to reject the null. The price series likely has a unit root. You should first-difference the data (use returns instead of prices) before modeling.

The Augmented Dickey-Fuller (ADF) Test:

The basic DF test assumes errors are not autocorrelated. If they are, use the ADF test, which adds lagged differences as additional regressors:

Delta(x_t) = b0 + g x_(t-1) + Sum[c_i Delta(x_(t-i))] + e_t

The additional lags absorb the autocorrelation, making the test valid. The hypotheses and critical values remain the same.

Exam Tip: The CFA exam will likely give you the test statistic and critical value and ask you to interpret the result. Know the decision rule cold: more negative test stat than critical value = stationary.

Practice unit root interpretation in our CFA Level II Quantitative Methods course.