In statistics, the Dickey–Fuller test tests the null hypothesis of whether a unit root is present in an autoregressive model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is named after the statisticians David Dickey and Wayne Fuller, who developed the test in 1979.
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Explanation
A simple AR(1) model is
where
The regression model can be written as
where
There are three main versions of the test:
1. Test for a unit root:
2. Test for a unit root with drift:
3. Test for a unit root with drift and deterministic time trend:
Each version of the test has its own critical value which depends on the size of the sample. In each case, the null hypothesis is that there is a unit root,
The intuition behind the test is as follows. If the series
It is notable that
may be rewritten as
with a deterministic trend coming from
There is also an extension of the Dickey–Fuller (DF) test called the augmented Dickey–Fuller test (ADF), which removes all the structural effects (autocorrelation) in the time series and then tests using the same procedure.
Dealing with uncertainty Fuller Test about including the intercept and deterministic time trend terms
Which of the three main versions of the test should be used is not a minor issue. The decision is important for the size of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is one) and the power of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is not one). Inappropriate exclusion of the intercept or deterministic time trend term leads to bias in the coefficient estimate for δ, leading to the actual size for the unit root test not matching the reported one. If the time trend term is inappropriately excluded with the
Use of prior knowledge about whether the intercept and deterministic time trend should be included is of course ideal but not always possible. When such prior knowledge is unavailable, various testing strategies (series of ordered tests) have been suggested, e.g. by Dolado, Jenkinson, and Sosvilla-Rivero (1990) and by Enders (2004), often with the ADF extension to remove autocorrelation. Elder and Kennedy (2001) present a simple testing strategy that avoids double and triple testing for the unit root that can occur with other testing strategies, and discusses how to use prior knowledge about the existence or not of long-run growth (or shrinkage) in y. Hacker and Hatemi-J (2010) provide simulation results on these matters, including simulations covering the Enders (2004) and Elder and Kennedy (2001) unit-root testing strategies. Simulation results are presented in Hacker (2010) which indicate that using an information criterion such as the Schwarz information criterion may be useful in determining unit root and trend status within a Dickey–Fuller framework.