The (Non-) Standard Asymptotics of Dickey-Fuller Tests

One of the most widely used tests in econometrics is the (augmented) Dickey-Fuller (DF) test. We use it in the context of time series data to test the null hypothesis that a series has a unit root (i.e., it is I(1)), against the alternative hypothesis that the series is I(0), and hence stationary. If we apply the test to a first-differenced time series, then the null is that the series is I(2), and the alternative hypothesis is that it is I(1), and so on.

Suppose that the time series in question is {Y_t; t = 1, 2, 3, ......, T}. The so-called "Dickey-Fuller regression" is a least squares regression of the form:

                           ΔY_t = [α + β t] + γY_t-1 + [Σ δ_j ΔY_t-j] + ε_t   .                 (1)

Here, terms in square brackets are optional; and of these the "p" ΔY_t-j terms are the "augmentation terms", whose role is to ensure that the there is no autocorrelation in the equation's residuals.

Standard econometrics packages allow for three versions of (1):

• No drift - no trend: that is, the (α + β t) terms are omitted.
• Drift - no trend: the intercept (drift term) is included, but the linear trend term is not.
• Drift - and - trend: both of the α and (β t) terms are included.

For example, here's the dialogue box that you see when you go to apply the DF test using the EViews package: