Condition for Stationarity for AR(p)

[Delvesy888]

Hi all,
I am rather confused with a section of chapter 12: Time Series (1)...

For a general AR(p) model, we are told that if all of the roots of the characteristic equation of a general AR(p) model have modulus greater than 1, then the process is stationary.

For an AR(1) model, this implies that we require the constant alpha to be less than 1. This result indicates that this is the only condition we require for AR(1) to be stationary.

However, prior to this, we showed that for an AR(1) model to be stationary, we require alpha less than 1, but also require the mean and the variance of X_0 to have particular forms. OR we require the process to have infinite history.

The result that is "proved" in the notes, doesn't imply that we need any other conditions for stationarity. This implies a contradiction?

If anyone can help me understand this better, that would be greatly appreciated.

Thanks very much.

 

[Delvesy888]

Anybody able to offer their interpretation? Thank you.

 

[r_v.s]

My understanding of this is that irrespective of mean(X0) and Var(X0), in the long run an AR(1) is stationary if mod(alpha) <1. The alpha less than 1 condition is all that is required for a process with infinite history to be stationary.

 

[John Potter]

Hi Delvesy888,

I agree with what has been said by r_v.s here. Your understanding seems to be spot on when you point out that for the roots of the char poly to be greater than 1 for AR(1) process, we merely require that alpha be less than 1.

Could you please let me know which section of the notes you felt contradicted this? (It may be that further explanation is required)

John

 

[Delvesy888]

Hi John,
Sorry for the late reply.
The section of the notes that I am explicitly talking about is (the converse of) Result 12.2 in section 3.4 of chapter 12.

The section states that: if the roots of the characteristic polynomial are all greater than 1 in absolute value, then the AR process is stationary.
The result does not state anything about an infinite history.

From this result alone, we would say that AR(1) is therefore stationary if and only if the coefficient alpha is less than 1 in absolute value i.e. we would not place any restrictions on an infinite history.

However, according to page 24 of chapter 12 (the last bold statement on the page), we require alpha less than 1 AND the further requirements.

I agree with what r_v.s has said, also.

 

[John Potter]

Hi Delvesy,

Yes, I see what you mean, it is a bit confusing. This section is all about whether the mean and variance are constant in the short-term. I think the word "requirements" is a bit confused here though, it almost means "consequences".

For the exam, I would just stick with the roots of the characteristic polynomial being greater than 1 in magnitude.

John