Requirements for an AR(1) model to be covariance stationary

[Bradleyz]

I have the 08 secret sauce book and it says in order for an AR(1) model to be covariance stationary:
1. b1 must be < 0
2. if b1 = 0, there is a unit root
3. The mean reverting level must be defined
Now, this confused me because I know that b1 cannot = 1, because that would mean an undefined mean reverting level and a random walk. Shouldn’t it say “if b1 = 1 there is a unit root?” Also, why does b1 have to be < 0?

 

[philip.platt]

it says “1” instead of “0” in the 2009 version:
1. b1 must be < 1
2. if b1 = 1, there is a unit root
3. The mean reverting level must be defined

 

[wyantjs]

Yes, b1 = 1 means unit root. Although, you might be seing something that refers to b1 = 0 after first-differencing has been done. In this case, b1 = 0 gives a unit root. If b1 > 1, the model explodes. Notice:
y_t = b1*y_t-1 + e
= b1*(b1*y_t-2 + e) + e
= b1*(b1*(b1*y_t-3 +e) +e) + e
This process continues such that
y_t = b1^n*y_t-n + e
If b1 > 1, the process explodes. Or, in other words, the process has infinite memory, and a shock never dies out. If b1 < 1, the shock will die out since b1^n -> 0 as n -> infinity.