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P1.T2.Diebold, Ch8: AR(p) Properties-covariant-stationary

Thread starter #1
In Reference to R15.P1.T2.DIEBOLD_CH8_Topic: AR(p) Properties-COVARIANT-STATIONARY :-
Wanted to clarify if the AR(p) Property of Covariance Stationarity should include the conditions that the Mean and the variance be Stable/Constant ..?
The Inverse of the Roots of the Lag Operator is a Mandatory condition - so that has to be met for the Process to be Covar Stationary. But that does not not guarantee Covar Stationarity- so does it mean in addition, the Mean and the Variance has to be stable and Constant for the AR(p) process to be Covar Stationary..? :(:(
Very grateful for any insights on this.

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @gargi.adhikari Diebold is not here changing the definition of covariance stationary, which is fundamental: this is the 2-part condition that (1) the mean of the time series is stable [constant] over time; and 2. that the series' autocovariance is stable over time; this second condition is that covariance(t, t+h) is the same for all points in time (t) and therefore depends only on the displayment (aka, lag). The important property of covariance stationary is interesting with respect to MA() and AR()
  • For moving average MA(.), "Note that the requirements of covariance stationarity (constant unconditional mean, constant and finite unconditional variance, autocorrelation depends only on displacement) are met for any MA(1) process, regardless of the values of its parameters ... just as the MA(1) process was covariance stationary for any value of its parameters, so too is the finite-order MA(q) process."
  • For the autoregresssive AR(.), "Recall that a finite-order moving average process is always covariance stationary, but that certain conditions must be satisfied for invertibility, in which case an autoregressive representation exists. For autoregressive processes, the situation is precisely the reverse. Autoregressive processes are always invertible -- in fact invertibility isn’t even an issue, as finite-order autoregressive processes already are in autoregressive form -- but certain conditions must be satisfied for an autoregressive process to be covariance stationary ... [and then, the basis for our note above, emphasis mine, although we apparently did not pickup the "if and only if":] An AR(p) process is covariance stationary if and only if the inverses of all roots of the autoregressive lag operator polynomial M(L) are inside the unit circle."
So in my interpretation he is clearly saying, per the biconditional "if and only if" that it is true that: if the inverses of all roots of the autoregressive lag operator polynomial M(L) are inside the unit circle, then the AR(p) process is covariance stationary (in which case, per the definition, that series has constant mean and stable autocovariance); i.e., roots inside the unit circle are both necessary and sufficient. Although I admit that at first glace (I just don't have current time to dive into this now, sorry, we are slammed) the actual footnote here surprises me: "footnote 4: A necessary condition for covariance stationarity, which is often useful as a quick check, is summation of φ < 1 (in the text above). If the condition is satisfied, the process may or may not be stationary, but if the condition is violated, the process can’t be stationary."; I'm not sure why this summation, which otherwise appears to be identical to the unit root condition, is only necessary and not sufficient (?!). This footnote we pulled up into the main text. I hope that's helpful!

(@Nicole Seaman can we please ask Deepa if the summation above should contain phi, φ, rather than what we've got in there, I think it's actually the symbol for "empty set" in which case the notation is mistaken and not helping with a very hard idea...).
Thread starter #3
Thanks so much @David Harper CFA FRM for taking the time during this hour of crunch to patiently elaborate on my question and I do see now where I misinterpreted the statements- you cleared that up now. Very Thankful for that.
Hate to bother you on this but do have a follow up question to clarify...It's ok if you address my follow up question later on after the crunch time is over... :)

So you mentioned that the condition for an AR(p) Process to be Covariant Stationary is :- " Roots inside the unit circle are both necessary and sufficient "
But then, if this condition of "Roots being inside the Unit Circle ( that is the absolute value of Phi <1) is satisfied, the AR(p) Process still May Not be Covariant Stationary.. in that case, the condition("Roots being inside the Unit Circle") is not sufficient..? Am a bit foggy on this ....:(:(
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David Harper CFA FRM

David Harper CFA FRM
Staff member
@gargi.adhikari no worries, well, I do need to bookmark it for after the crunch because as I was answering your question earlier today, and looking at Diebold, I got stuck on exactly this point. I had expected the summation of, phi (φ), referenced in the footnote to also be necessarily and sufficient (i.e., an "if and only if," to match the i.i.f. in the body: "if and only if the inverses of all roots of the autoregressive lag operator polynomial M(L) are inside the unit circle."). I am also confused by this (at first glance, these conditions appear identical to me) :( ... and this is why I dislike texts like Diebolds that omit key numerical examples, so i will have to return to this later. In the meantime, Deepa is amending some of this note, so I hope to make this clearer when we re-issues this note. Thanks for your help, and your patience!