Can there be parameter redundancy in an ARMA process where constant is non null?

Question

I am asked to identify an ARMA(p, q) process from an equation, and to avoid parameter redundancy. The equation is of the form :

$\varphi(L)y_t = c + \theta(L)\varepsilon_t$

with $\varphi(L) = 1 - \sum\limits_{i = 1}^p\varphi_jL^j$,
$\theta(L) = \sum\limits_{j=0}^q\theta_jL^j = 1 + \sum\limits_{j=1}^q\theta_jL^j$ as $\theta_0 = 1$,

and $y_tL^s = y_{t-s}$

to represent something like : $y_t = c + \varphi_1y_{t-1} + ... + \varphi_py_{t-p} + \varepsilon_t + \theta_1\varepsilon_{t-1} + ... + \theta_q\varepsilon_{t-q}$ (*)

I have been taught then that if $\varphi(z)$ and $\theta(z)$ have common roots, then there are parameters redundancy and we have to cancel common factors of the polynomial, so we know the real orders p and q of the process. I was not precise, but for me it seems to be true only if $c = 0$.

Then the teacher gave us series of process under the form (*), asking to identify them and "watch out for parameters redundancy". But all of these processes have a non-null $c$ constant part.

There cannot be any parameter redundancy if $c$ is non-null, am I right ?

Answer 1

Yes, you are completely right. Assume $c$ is not 0. If $\theta(z)$ is 0, then the right side of the equation is $c$, so the left side can't b 0, so neither $y_t$ nor $\phi(z)$ can be 0. So no redundancy is possible.

Answer 2

Parameter redundancy depends only on the roots of the polynomials and not on the constant. Therefore, there can also be parameter redundancy in ARMA models that include a constant. Moreover, after cancelling the common factors out, the new ARMA(p-1,q-1) model has a different constant.