Augmented Dickey–Fuller Test

An Augmented Dickey–Fuller Test is a statistical test for the null hypothesis of whether a unit root is present in a given time series.

• AKA: ADF.
• See: Unit Root Test, Dickey–Fuller Test, Phillips–Perron Test, ADF-GLS Test, KPSS Test, Breusch–Godfrey Test, Ljung–Box Test, Durbin–Watson Test, Akaike Information Criterion, Bayesian Information Criterion, Hannan–Quinn Information Criterion.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Augmented_Dickey–Fuller_test Retrieved 2016-08-07
  • In statistics and econometrics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis of whether a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models.

The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence.

Testing procedure

The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model

[math]\displaystyle{ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t, }[/math]

where [math]\displaystyle{ \alpha }[/math] is a constant, [math]\displaystyle{ \beta }[/math] the coefficient on a time trend and [math]\displaystyle{ p }[/math] the lag order of the autoregressive process. Imposing the constraints [math]\displaystyle{ \alpha = 0 }[/math] and [math]\displaystyle{ \beta = 0 }[/math] corresponds to modelling a random walk and using the constraint [math]\displaystyle{ \beta = 0 }[/math] corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous to the ones discussed on Dickey–Fuller test (see that page for a discussion on dealing with uncertainty about including the intercept and deterministic time trend terms in the test equation.)

By including lags of the order p the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p has to be determined when applying the test. One possible approach is to test down from high orders and examine the t-values on coefficients. An alternative approach is to examine information criteria such as the Akaike information criterion, Bayesian information criterion or the Hannan–Quinn information criterion.

The unit root test is then carried out under the null hypothesis [math]\displaystyle{ \gamma = 0 }[/math] against the alternative hypothesis of [math]\displaystyle{ \gamma \lt 0. }[/math] Once a value for the test statistic

[math]\displaystyle{ DF_\tau = \frac{\hat{\gamma}}{SE(\hat{\gamma})} }[/math]

is computed it can be compared to the relevant critical value for the Dickey–Fuller Test. If the test statistic is less (this test is non symmetrical so we do not consider an absolute value) than the (larger negative) critical value, then the null hypothesis of [math]\displaystyle{ \gamma = 0 }[/math] is rejected and no unit root is present.

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