Could anyone clarify the difference between a time series that has a unit root and a time series that is not covariance stationary , if any there is? Seems like these terms are used interchangeably.
Merci !
Not covariance stationary is when the root is bigger than one, unit root means a root of one.
There . . . that’s better.
so am I correct if I say that:
- b1 < 1 : covariance stationary
- b1 > 1 or b1=1 : not covariance stationary
- and if b1 = 1 : unit root (i.e. random walk)
When using the Dickey Fuller test, if we fail to reject H0 (b1=1), we can conclude that the time series has a unit root. Fine.
If we reject H0, the time series does not have a unit root, but how can we also conclude that the time series is covariance stationary? Couldn’t b1 be >1 (no unit root but no covariance stationarity either) ? This doesn’t make sense to me… !
Remember that the slope in the Dickey-Fuller test is (b1 - 1), let’s name it “g1”.
The Ho in the D-F test is that g1 = 0, which means that an unit root exists.
You must reject the Ho of the D-F test in order to say that the original time serie was covariance stationary.
If you fail to reject the Ho, then there is unit root (the serie is not covariance stationary)
b1 - 1 = 0
b1 = 1 >>> unit root
Hope this helps.
I agree with you on the unit root (that if we fail to reject H0, there is no unit root), but my question is, how do we know that no unit root also means that the series is covariance stationary?
Can’t we have a series that has NO unit root and that is NOT covariance stationary? (when b1 > 1 ?)
I’m still confused but thanks for your help!
Highlights:
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Unit root
-
Covariance stationary
-
Mean reversion
_______________________________
The fact here is that:
If a time serie has unit root, then it is not covariance stationary and has no mean reversion.
The other side is that a time serie doesn’t have unit root, therefore it is covariance stationary and has mean reversion.
The scenario you display is incompatible with both scenarios above. My is answer is No.
Let me know your comments please.
Regards.
Almost.
- _ −1 < _ b1 < 1: covariance stationary
- _ b1 ≤ −1 _ or b1 ≥ 1: not covariance stationary
- b1 = 1, unit walk
Thanks a lot.
Agreed. Unit root => Not covariance stationary
I don’t agree on that one though. I understand that if a time series doesn’t have a unit root, it can be either when b1<1 (covariance stationary) or when b1>1 (no unit root… but not covariance stationary either). That’s why I don’t understand how we can come to the conclusion that No unit root => covariance stationarity.
Ok, I get your worry.
I think that is it better to talk about explosive roots and stable roots.
Explosive ones are roots that are equal or more than 1, so all of them are Not covariance stationary (also have no mean reversion). Therefore if the series has a unit root of 1, or 1.1 or 2 or 3 (-1, -1.1, -2 and -3 as well), then the time serie is not covariance stationary.
Stable roots are less than 1 or higher than -1 (approaching to 0), so the time serie is covariance stationary.
Remember that D-F Test Ho is that g1 = 0, which implies that the original slope (b1) could be or not at least equal to 1. Assume for a moment that the real slope b1 is equal to 2, DF test would classify the original serie as non-stationary? Surely yes, because despite we test it assuming a slope equal to 1, being the real slope 2, the test also fails to reject that g1=0 indicating, therefore, that the original serie was Not covariance stationary.
In conlussion, when someone tells you that a serie has unit root, it means the root is 1 or even higher than 1. Maybe it is a slopy way to refer to a “explosive root” time series.
Hope this helps.
It is definitely _ not _ true that the lack of a unit root means that the series is covariance stationary. If b1 = 2, for example, or if b1 = -1.
lol… complications…
Unit Root & Stationarity are the opposite to each other.
Unit Root = Non-Stationary Series
Covariance Stationary = No Unit Root