Checking for stationarity [X-valid’ed]

While working with Jean-Michel Marin on the revision of Bayesian Core, and more specifically on the time series chapter, I was wondering about the following problem:

It is well-known [at least to readers of  Bayesian Core] that an AR(p) process

x_t=\sum_{i=1}^p \varrho_i x_{t-i} + \epsilon_t

is causal and stationary if and only if the roots of the polynomial

\mathcal{P}(u) = 1 - \sum_{i=1}^p \varrho_i u^i

are all outside the unit circle in the complex plane. This defines an implicit (and unfriendly!) parameter space for the original parameters of the AR(p) model. In particular, when considering a candidate parameter, to determine whether or not the constraint is satisfied implies checking for the root of the associated polynomial. The question  I asked on Cross Validated a few days ago was whether or not there existed a faster algorithm than the naïve one that consists in (a) finding the roots of P and (b) checking none one them is inside the unit circle. Two hours later I got a reply from J. Bowman about the Schur-Cohn procedure that answers the question about the roots in O() steps without going through the determination of the roots. (This is presumably the same Issai Schur as in Schur’s lemma.) However,  J. Bowman also pointed out that the corresponding order for polynomial root solvers is O()! Nonetheless, I think the Schur-Cohn procedure is way faster.

3 Responses to “Checking for stationarity [X-valid’ed]”

  1. […] am producing utilization of wordpress for crafting my blog and, even although I am reasonably happy using the method in add-on to the actuality that just one […]

  2. didn’t know you were into representation theory.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: