Stationarity

Since we want to estimate the parameters that better adjust real data to ARMA-APARCH models, the notion of stationarity is crucial. Usually, even when the data set is non-stationary in appearance, we still are able to apply transformation techniques so that the resulting time series can be reasonably modeled as a stationary process (see Brockwell (1991) ).

The stationarity property of the combined ARMA-APARCH model is achieved when both ARMA and APARCH models are individually stationary. Usually, if the conditional distribution we are dealing with has infinite variance, it is only used in the domain where its variance remains finite. For example, the standard Student's t distribution with $\nu$ degrees of freedom is defined for any $\nu > 0$, but the ARMA-APARCH model with conditional Student's t distribution described in Wurtz (2006) is defined only for $\nu > 2$ in which case the variance of the Student's t distribution is finite.

The stationarity of the ARMA model is intimately connected with the roots of the polynomials $a(z) = 1 - a_1 z - \cdots - a_m z^m$ and $b(z) = 1 - b_1 z - \cdots - b_n z^n$ representing the ARMA part of Definition 1.1. In many cases the conditional distribution of the model has a finite variance and therefore, the condition for the existence of a stationary solution is just that $a(z) = 1 - a_1 z - \cdots - a_m z^m$ has no roots for all $z \in \mathbb{C}$ such that $\vert z\vert = 1$ . On the other hand, if the distribution has an infinite variance the conditions for the existence of stationary solutions becomes (see Brockwell (1985)),

$$a(z)b(z) \not= 0, z \in \mathbb{C}, \vert z\vert \leq 1.$$ (4)

The stationarity of APARCH models has a historical background that is worth exploring. In the context of a model where the innovations have a finite variance, Bollerslev (1986) proved the second order stationarity of the GARCH(p,q) model. Then, Nelson (1990) demonstrated the strict stationarity of the empirically important case GARCH(1,1). His results were then generalized by Bougerol (1992) for the GARCH(p,q) model. Finally, Ling (2002) showed which conditions the APARCH(p,q) model would be $\delta$-order stationary. This last result states that there exists a unique $\delta$-order stationary solution of the APARCH model if and only if

$$\sum_{i=1}^p \lambda_i\alpha_i +\sum_{j=1}^q\beta_j < 1,$$ (5)

where

$$\lambda_i = E(\vert Z_t\vert - \gamma_i Z_t)^{\delta}, Z_t \sim \mathcal{D}_{\vartheta}(0,1).$$ (6)

The case of infinite variance has a somewhat different historical background. It is worth noting that the theoretical results were all based on the important work of Bougerol (1992). The first assumption made on stable distributions is that the index of stability $\alpha$ must be greater than one, because in this case the innovations have finite first moment. According to Mittnik (1995), this assumption seems plausible since most financial time series have finite mean. The second assumption is that they must have a $\delta$-moment finite, which means that we must restrict our model to $1 < \delta < \alpha$.

Regarding the stationarity of these models, Mittnik (1995) proved the strict stationarity of the stable GARCH(p,q) model, $Z_t \sim S(\alpha,\beta,1, 0)=S(\alpha,\beta;1)$ . Then, Mittnik (2002) derived conditions for the strict stationarity of the APARCH(p,q) model with all coefficients $\gamma_i = 0$ (namely the power-GARCH model). Finally, Diongue (2008) showed that the general APARCH(p,q) model has a strictly stationary solution if and only if (2.4) and (2.5) are satisfied, however, he did not get an explicit expression for $\lambda_i$.

The same stationarity conditions, (2.4) and (2.5), are valid for an APARCH(p,q) model with $Z_t \sim GEV (\xi,1, 0)$.

The stationarity conditions of the ARMA part of the model are easy to implement computationally, but if we want to impose stationarity of the APARCH equation the problem of efficiency arises. The main problem is that equations (2.4) and (2.5) depend on $\lambda_i$. However, their computation is faster when they have an explicit expression.

Figure 1.1. shows the result of the simulation of 1000 observations of the AR(1)-GARCH(1,1) model with parameters vector $\phi=(b_1, a_1, \alpha_1,\beta_1, \psi)=(0.2, 0.3, 0.2, 0.3, 0.2, 0.2)$ and GEV and normal innovations distributions.

Figure 1.2. shows the result of the simulation of 1000 observations of the AR(1)-GARCH(1,1) model with stable and normal innovations when $\phi=(b_1, a_1, \alpha_1,\beta_1, \alpha, \beta,) = (0.2, 0.3, 0.2, 0.3, 0.2, 1.8, 0)$ and stable and normal innovations distributions.

Figura: Left side: AR (1) -GARCH (1.1) with $Z_t \sim GEV$. Right side: AR (1) -GARCH (1,1) with $Z_t \sim Normal$

Figura: Left side: AR (1) -GARCH (1.1) with $Z_t \sim S_{\alpha }$. Right side: AR (1) -GARCH (1,1) with $Z_t \sim Normal$