Augmented Dickey-Fuller Unit Root Tests
How do we know when to difference time series data to make it stationary? You use the Augmented Dickey-Fuller t-statistic.
Here are the various cases of the test equation:
a. When the time series is flat (i.e. doesn’t have a trend) and potentially slowturning around zero, use the following test equation:
Δz t = θz t −1 + α 1 Δz t −1 + α 2 Δz t − 2 + L + α p Δz t − p + a t

where the number of augmenting lags (p) is determined by minimizing the
Schwartz Bayesian information criterion or minimizing the Akaike information criterion or lags are dropped until the last lag is statistically significant. EVIEWS allows all of these options for you to choose from.
Notice that this test equation does not have an intercept term or a time trend.
What you want to use for your test is the t-statistic associated with the
Ordinary least squares estimate of θ . This is called the Dickey-Fuller tstatistic. Unfortunately, the Dickey-Fuller t-statistic does not follow a standard t-distribution as the sampling distribution of this test statistic is skewed to the left with a long, left-hand-tail. EVIEWS will give you the correct critical values for the test, however. Notice that the test is left-tailed.
The null hypothesis of the Augmented Dickey-Fuller t-test is
H0 :θ = 0

(i.e. the data needs to be differenced to make it stationary) versus the alternative hypothesis of

H1 : θ < 0

b.

(i.e. the data is stationary and doesn’t need to be differenced) When the time series is flat and potentially slow-turning around a non-zero value, use the following test equation:
Δz t = α 0 + θz t −1 + α 1 Δz t −1 + α 2 Δz t − 2 + L + α p Δz t − p + a t .

Notice that this equation has an intercept term in it but no time trend.
Again, the number of augmenting lags (p) is determined by minimizing the
Schwartz Bayesian