Augmented Dickey Fuller Test

Augmented Dickey Fuller test is the augmented version of dickey fuller test. The Augmented Dickey Fuller Test (ADF) is a unit root test for checking stationarity.

 

The hypotheses for the test:

• The null hypothesis for this test is that there is a unit root.

• The alternate hypothesis differs slightly depending on which equation you’re using, but is usually stationarity or trend-stationarity.

 

 

Consider an AR(3):

yt  = θ1yt-1 + θ2yt-2+ θ3yt-3 + εt

A unit root in θ(L)=1 − θ1L − θ2L2 − θ3L3 corresponds to θ(1) = 0.

 

The test is most easily performed by rewriting the model:

yt − yt−1 = (θ1 − 1)yt−1 + θ2yt−2 + θ3yt−3 + εt

yt − yt−1 = (θ1 − 1)yt−1 + (θ2 + θ3)yt−2 + θ3(yt−3 − yt−2) + εt

yt − yt−1 = (θ1 + θ2 + θ3 − 1)yt−1 + (θ2 + θ3)(yt−2 − yt−1) + θ3(yt−3 − yt−2) + εt

?yt = πyt−1 + c1?yt−1 + c2?yt−2 + εt

 

where,  π = θ1 + θ2 + θ3 − 1 = − θ(1)

c1 = − (θ2 + θ3)

c2 = −θ3

The hypothesis θ(1) = 0 again corresponds to

H0 : π = 0    HA : π < 0

The t−test for H0 is denoted as the augmented Dickey-Fuller (ADF) test.