Dickey–Fuller Test

A Dickey–Fuller Test is a statistical test for the null hypothesis of whether a unit root is present in a autoregressive model.

• See: Augmented Dickey–Fuller Test, Unit Root Test, Phillips–Perron Test, ADF-GLS Test, KPSS Test, Breusch–Godfrey Test, Ljung–Box Test, Durbin–Watson Test, Time Series Analysis.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Dickey–Fuller_test Retrieved 2016-08-07
  • 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.

Explanation: A simple AR(1) model is

[math]\displaystyle{ y_{t}=\rho y_{t-1}+u_{t}\, }[/math]

where [math]\displaystyle{ y_{t} }[/math] is the variable of interest, [math]\displaystyle{ t }[/math] is the time index, [math]\displaystyle{ \rho }[/math] is a coefficient, and [math]\displaystyle{ u_{t} }[/math] is the error term. A unit root is present if [math]\displaystyle{ \rho = 1 }[/math]. The model would be non-stationary in this case.

The regression model can be written as

[math]\displaystyle{ \nabla y_{t}=(\rho-1)y_{t-1}+u_{t}=\delta y_{t-1}+ u_{t}\, }[/math]

where [math]\displaystyle{ \nabla }[/math] is the first difference operator. This model can be estimated and testing for a unit root is equivalent to testing [math]\displaystyle{ \delta = 0 }[/math] (where [math]\displaystyle{ \delta \equiv \rho - 1 }[/math]). Since the test is done over the residual term rather than raw data, it is not possible to use standard t-distribution to provide critical values. Therefore this statistic [math]\displaystyle{ t }[/math] has a specific distribution simply known as the Dickey–Fuller table.

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