Augmented Dickey–Fuller test

In statistics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis depends 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 procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model

Δ

y

t

=

α

+

β

t

+

γ

y

t

−

1

+

δ

1

Δ

y

t

−

1

+

⋯

+

δ

p

−

1

Δ

y

t

−

p

+

1

+

ε

t

,

{\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},}

where

α

{\displaystyle \alpha }

is a constant,

β

{\displaystyle \beta }

the coefficient on a time trend and

p

{\displaystyle p}

the lag order of the autoregressive process. Imposing the constraints

α

=

0

{\displaystyle \alpha =0}

and

β

=

0

{\displaystyle \beta =0}

corresponds to modelling a random walk and using the constraint

β

=

0

{\displaystyle \beta =0}

corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous those of the Dickey–Fuller test. (See that article 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 must be determined in order to use the test.