Checking for stationarity [X-valid’ed]

Posted in Books, Statistics, University life with tags , , , , , , , , on January 16, 2012 by xi'an

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.