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...).

## Stay connected