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:

The thing that we teach our students about the DF tests is that their null distributions are non-standard, even (large T) asymptotically. Also, the null distributions of the DF tests differ according to which of the three versions of the DF regression above are estimated by least squares, prior to the "t-test" of H₀: γ = 0, against H_A: γ < 0.

Specifically, these distributions are functionals of standard Brownian motions. For that reason, appropriate critical values were obtained by simulation methods and tabulated for a limited range of situations by Dickey and Fuller (1979, 1981); and more extensive tables were constructed by MacKinnon (1990), and extended and re-published in MacKinnon (2010).

This is all well and good, but there is a really important point that most students never seem to be told about, except in relatively advanced courses on non-stationary time series analysis. This is that these non-standard distributions, and hence "special" critical values that we worry about so much when using DF tests, come about only because of a very strong assumption that is made about the underlying population data-generating process (DGP) for the series Y_t.

This assumption is that the series is generated according to a process that has no drift or trend components. That is, it's assumed that the DGP is of the form:

                                     Y_t = Y_t-1 + v_t   ;     v_t ~ N[0 , σ²]  .                          (2)

There is no intercept or trend in equation (2).

If, instead, we assume that equation (2) does include either an intercept (drift term) and / or a linear trend term, the the DF test statistics have null distributions that are asymptotically standard normal. in this case, no special tables are used in order to apply the test(s)! This is a result that applies only for very large T, and we still have non-standard distributions in the finite-sample case.

This result is very well known, and a proof is provided in Hamilton (1994), for example. For some reason, though, most students of econometrics don't seem to be aware of it.

References

Dickey, D. A. and W. A. Fuller, 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427–431.

Dickey, D. A. and W. A. Fuller, 1981.Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49, 1057-1072.

Hamilton, J. D., 1994. Time Series Analysis. Princeton University Press, Princeton, NJ.

MacKinnon, J. G., 1990. Critical values for cointegration tests. Discussion Paper 90-4, Department of Economics, University of California, San Diego.

MacKinnon, J. G., 2010. Critical values for cointegration tests. Working Paper No. 1227, Department of Economics, Queen's University.

© 2014, David E. Giles