Why the Dickey Fuller test in time series analysis tests for B=1 and not for B>1?

[Santiago Valdivieso]

I read that a serie is not stationary in an AR(p) when B's in absolute values are greater than 1. Then, why the Dickey fuller just test's for the null hypothesis that B's are equal to 1 (d=B-1=0) and not for if B's>1?

[Andrew Musau]

The (simplest) AR(1) equation is

$$y_{t}= \gamma y_{t-1}+ v_{t}\;\;\;\;\;(Eq. 1)$$

where \(\gamma\) is the coefficient on lagged \(y\) and there is no constant and no trend. So there are 3 possibilities.

1. Eq. 1 is a stationary AR(1) process if \(|\gamma|<1\).
2. Eq. 1 follows random walk if \(\gamma=1\).
3. Eq. 1 is an explosive series if \(|\gamma|>1\).

Therefore, as you note, both 2 and 3 represent nonstationary series, but we reformulate Eq. 1 as

$$y_{t}- y_{t-1}= (\gamma-1)y_{t-1}+ v_{t}\;\;\;\;\;(Eq. 2)$$

and test \(\beta=0\) in the Eq. 3 (derived from Eq.2)

$$\Delta y_{t}= \beta y_{t-1}+ v_{t}\;\;\;\;\;(Eq. 3)$$

where \(\Delta y_{t}= y_{t}- y_{t-1}\) and \(\beta= \gamma-1\). That is, we test for a unit root and only determine #2 (Eq. 1 follows random walk). The reason we do not consider #3 (Eq. 1 is an explosive series) is simple: we do not expect that economic series are explosive and therefore do not consider this. Normally, you will usually graph the series in advance and have idea of what you are dealing with.

[Santiago Valdivieso]

Andrew Musau Thank you!