*2.2. Stationarity of PHARCH(m,p) Models*

We first give a necessary condition for Model (3) to be stationary. We know that *E rt*−*irt*−*<sup>j</sup>* = 0, ∀*i* = *j*, and if *rt* is stationary, we must have

$$E\left(r\_t^2\right) = E\left(\sigma\_t^2\right) = \mathbb{C}\_0 + E\left(r\_t^2\right) \sum\_{i=1}^m a\_i \mathbb{C}\_i + E\left(\sigma\_t^2\right) \sum\_{i=1}^p b\_{i,\*}$$

so

$$E\left[r\_t^2\right] = \frac{C\_0}{1 - \left(\sum\_{i=1}^m a\_i \mathbf{C}\_i + \sum\_{i=1}^p b\_i\right)}.$$

Therefore,

$$
\Sigma\_{i=1}^{m} a\_i \mathbb{C}\_i + \sum\_{i=1}^{p} b\_i < 1. \tag{4}
$$

To prove a sufficient condition it will be necessary to represent the PHARCH(*m,p*) as a Markov process. We use the definitions given in the previous section, so the process

$$X\_t = (r\_{t-1}, \dots, r\_{t-a\_m+1}, \sigma\_t, \dots, \sigma\_{t-p+1}),\tag{5}$$

whose elements follow Equation (3), is also a *T*-chain.

The proofs of the following results are based on Dacorogna et al. (1996), and they are given in the Appendix A.

**Proposition 1.** *The Markov Chain Xt that represents a PHARCH(m,p) process is a T-chain.*

**Proposition 2.** *The Markov Chain that represents a PHARCH(m,p) process is recurrent with an invariant probability measure (stationary distribution), and its second moments are finite if the condition given in (4) is satisfied.*

Note that if *εt* ∼ *<sup>t</sup>*(*v*) (a Student's *t* distribution with *ν* degrees of freedom) in (3), then the necessary and sufficient condition becomes

$$\sum\_{i=1}^{m} \frac{\upsilon}{\upsilon - 2} a\_i \mathbb{C}\_i + \sum\_{i=1}^{p} b\_i < 1, \quad \text{for } \upsilon > 2.$$
