Dickey-Fuller Test

The Dickey-Fuller test is a statistical procedure for deciding whether a time series contains a unit root. A series with a unit root has no fixed level to which it returns. Each shock shifts it permanently, as in a random walk. A series without one is pulled back toward a mean or trend. The test was introduced by David Dickey and Wayne Fuller in a paper published in 1979.

The question matters because most statistical methods assume stationarity, and applying them to unit-root series produces misleading results. In finance the test, and more often its extension the augmented Dickey-Fuller test, is used to judge whether prices, spreads, interest rates or valuation ratios revert to a mean.

The starting point is a first-order autoregression, $y_t = \rho \, y_{t-1} + \varepsilon_t$. If $\rho = 1$, the series is a random walk. Subtracting $y_{t-1}$ from both sides gives the form used in the test:

• $\Delta y_t$ is the change in the series, $y_t - y_{t-1}$.
• $\gamma$ equals $\rho - 1$.
• $\varepsilon_t$ is a random error.

The null hypothesis is $\gamma = 0$, a unit root. The alternative is $\gamma$ below zero, which means the series is stationary. The test statistic is the estimated coefficient divided by its standard error:

The statistic is computed exactly like a t-statistic from a linear regression. The difference lies in its distribution. When the null hypothesis is true, the regressor $y_{t-1}$ is itself a random walk, and the usual theory behind the t-test breaks down. The statistic does not follow Student's t-distribution, even in very large samples. Its distribution is shifted toward negative values.

Dickey and Fuller worked out this distribution by simulation and published tables. The critical value depends on what else is in the regression. For large samples at the 5 percent level it is about −1.95 with no constant, −2.86 with a constant, and −3.41 with a constant and a time trend. An ordinary one-sided t-test would use −1.65. Using the ordinary value would reject the unit root far too often and find mean reversion where there is none.

• No constant. The regression above, suitable for a series that would fluctuate around zero if stationary, such as a regression residual.
• With a constant. A constant term is added, for a series that would fluctuate around a non-zero mean, such as an interest rate spread.
• With a constant and trend. A time trend is also added, for a series that may be stationary around a rising or falling line, such as the logarithm of output.

Choosing the wrong version reduces the test's ability to detect stationarity or biases it, so the choice should follow from a plot of the data and from what is known about the series.

An analyst fits the version with a constant to 250 daily observations of the spread between two bond yields. The estimated autoregressive coefficient is $\hat{\rho} = 0.95$, so $\hat{\gamma} = -0.05$, with a standard error of 0.02. The statistic is −0.05 ÷ 0.02 = −2.5.

This is not more negative than −2.86. The unit root cannot be rejected at the 5 percent significance level, even though the estimate of $\rho$ is below one and an ordinary t-test would have called the result significant. Had the standard error been 0.015, the statistic would have been −3.33, below −2.86, and the spread would be judged stationary.

The test supports mean reversion strategies. A trader who intends to buy a spread when it is low and sell when it is high first needs evidence that the spread does return to its mean. In pairs trading, the test is applied to the residual from regressing one price on another; a stationary residual indicates cointegration. Economists use the test before modeling series such as inflation, exchange rates and price-to-earnings ratios, to decide whether to work with levels or with changes.

Asset prices themselves nearly always fail to reject the unit root, which is consistent with the view that price changes are hard to predict. Returns, being changes in log prices, nearly always reject it.

The basic test assumes the errors $\varepsilon_t$ are uncorrelated over time. Real series often have richer dynamics, and when the errors are serially correlated the test gives wrong answers. The augmented version adds lagged changes to the regression to deal with this and has largely replaced the original in applied work.

The test also has low power. A series with $\rho$ of 0.98 is stationary but hard to distinguish from a random walk without a very long sample, so a failure to reject is weak evidence. Structural breaks, such as a one-time shift in the mean, can make a stationary series appear to have a unit root. And when the test is run across many candidate series, some will pass by chance, so results from a broad search need confirmation on fresh data as part of sound hypothesis testing.