Dynamical approach in studying stability condition of exponential (GARCH) models

Definition 2.1

Definition 2.1 Singular point and it's stability (Ozaki 1985; Salim and Youns, 2012)

A point $(\zeta )$ is a singular point of a function $f(x)$ if satisfies: $$\zeta =f(\zeta )$$ and for the model:

$$x_t=f(x_{t-1}\text{,}{x}_{t-2}\text{,}\dots \text{,}{x}_{t-p})$$ the singular point $(\zeta )$ is defined as a point for which every trajectory of the model (2.1) beginning sufficiently near $(\zeta )$ approaches either for $t\to \infty \mathit{or}t\to -\infty$ . If $(\zeta )$ approaches for $t\to \infty$ we called $(\zeta )$ a ''stable singular point'' and if it approaches for $t\to \infty$ we called $(\zeta )$ an ''unstable singular point''.

Definition 2.2

Definition 2.2 Limit cycle and it's stability (Ozaki 1985; Salim and Youns 2012)

A limit cycle of the model (2.1) is defined as an isolated and closed trajectory of the form: $$x_{t+1}\text{,}{x}_{t+2}\text{,}\dots \text{,}{x}_{t+q}=x_t$$ such that $q$ is a positive integer. The word ''closed'' means that if the initial value $(x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_p)$ belongs to the limit cycle, then: $$(x_{1+\mathit{kq}}\text{,}{x}_{2+\mathit{kq}}\text{,}\dots x_{p+\mathit{kq}})=(x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_p)\mathit{for}\mathit{any}\mathit{integer}k$$ the word ''isolated'' means that every trajectory beginning sufficiently near the limit cycle approaches it either for $t\to \infty \mathit{or}t\to -\infty$ .

If it approaches the limit cycle for $t\to \infty$ we call it a ''stable limit cycle'', and if it approaches the limit cycle for $t\to \infty$ we call it an ''unstable limit cycle''. The smallest integer $q$ which satisfies this definition is called the period of the limit cycle of the model (2.1).

Definition 2.3

Definition 2.3 Autoregressive model of general order (Priestely, 1981)

A linear Autoregressive model of order $p$ denoted by AR( $p$ ) has the form: $$x_t+a_1{x}_{t-1}+a_2{x}_{t-2}+\cdots +a_p{x}_{t-p}=z_t\mathit{where}{z}_t\sim \mathit{iid}N(0\text{,}{\sigma }_t^2)\mathit{and}{a}_1\text{,}{a}_2\text{,}\dots \text{,}{a}_p\mathit{are}\mathit{constants}$$ by using a Backward shift operator the AR( $p$ ) model can be written in: $$\alpha (B)x_t=z_t\mathit{where}\alpha (B)=1+a_{1}z+a_2{z}^2+\cdots +a_p{z}^p$$ the general solution of the model is given by: $$x_t=f(t)+{\alpha }^{-1}(B)z_t$$ such that $f(t)$ is the complementary function which is the solution of homogenous difference equation: $$\alpha (B)x_t=0$$ and $f(t)$ in general will has the form: $$f(t)=A_1{\mu }_1^t+A_2{\mu }_2^t+\cdots +A_p{\mu }_p^t$$ where $A_1\text{,}{A}_2\text{,}\dots \text{,}{A}_p$ is an arbitrary constants and ${\mu }_1\text{,}{\mu }_2\text{,}\dots \text{,}{\mu }_p$ is the roots of the polynomial $$g(z)=z^k+a_1{z}^{k-1}+\cdots +a_k$$ the autoregressive model AR( $p$ ) is an asymptotically stationary process if: $${\displaystyle \underset{t\to \infty }{\lim }}f(t)=0$$

which implies that all the roots of $g(z)$ must lie inside the unit circle, $$\text{i.e.}|{\mu }_i|<1\mathit{for}i=1\text{,}2\text{,}\dots \text{,}p$$

Theorem 3.1

Theorem 3.1 Francq and Zakoian, 2010

Assume that $g(z_{t-j})={\gamma }_j(z_{t-j})+{\beta }_j[|z_{t-j}|-\text{E}|z_{t-j}|]$ is not almost surely equal to zero and that the polynomials $\alpha (z)={\sum }_{j=1}^P{z}^j\text{and}\beta (\text{z})=1-{\sum }_{i=1}^Q{\alpha }_i{z}^i$ have no common root with $\alpha (z)$ not identically null then the EGARCH(Q,P) model admit a strictly stationary iff the roots of $\beta (z)$ are outside the unit circle.

Lemma 3.1

Lemma 3.1 Mohammad and Ghaffar, 2016

Let ${\alpha }_1\text{,}{\alpha }_2\text{,}\dots \text{,}{\alpha }_r$ be a non-negative real numbers. The following polynomial:

$$P(z)=1-{\displaystyle \sum _{i=1}^r}{\alpha }_i{z}^i$$

does not have a roots inside and on the unit circle if and only if $P(z)>1$ .

Proof

Let be $P(1)>0$ and let ${\alpha }_1\text{,}{\alpha }_2\text{,}\dots \text{,}{\alpha }_r$ are non-negative real numbers.

Thus: ${\sum }_{i=1}^r{\alpha }_i>0$ this implies: $1-{\sum }_{i=1}^r{\alpha }_i<1$ so that: $P(1)<1$ then by hypothesis we obtain: $0<P(1)<1=P(0)$ this means the polynomial (3.1) has no any roots inside and on the unit circle since all the roots $(m)$ that lie between 0 and 1 doesn't verify $P(m)=0$ .

Conversely; suppose that polynomial 3.1 doesn't have any roots inside the unit circle, so that $|h|⩾1$ is one of the roots of polynomial 3.1, and then: $$|h{|}^j⩾1\mathit{for}j=1\text{,}2\text{,}\dots \text{,}r$$ and since ${\sum }_{j=1}^r{\alpha }_j>0$ , then ${\sum }_{j=1}^r{\alpha }_j<{\sum }_{j=1}^r{\alpha }_j|h{|}^j$ $$\Rightarrow 1-{\displaystyle \sum _{j=1}^r}{\alpha }_j>1-{\displaystyle \sum _{j=1}^r}{\alpha }_j|h{|}^j$$ $$\text{this implies}P(1)>P(h)$$ and since h is one of the roots then $P(h)=0$ , so that $P(1)>0$ . □

Proposition 3.1

The non-zero singular point $\zeta$ for EGARCH(Q,P) model is stable if: $${\displaystyle \sum _{i=1}^Q}{\alpha }_i<1$$

Proof

Let $\zeta$ be the non-zero singular point of EGARCH(Q,P). near the non-zero singular or in the neighborhood of $\zeta$ with sufficiently small radius ${\zeta }_t$ such that: $${\zeta }_t^n\to 0\mathit{for}n⩾2$$ and by using the variational equation: $${\sigma }_s^2=\zeta +{\zeta }_s\mathit{where}s=t\text{,}t-1\text{,}\dots \text{,}t-Q$$ and by substitute this equation in (3.3) we get: $$\text{E}[\log (\zeta +{\zeta }_t)]=w+{\displaystyle \sum _{i=1}^Q}{\alpha }_i\text{E}[\log (\zeta +{\zeta }_{t-i})]$$ $$\text{E}\left[\log \left(\zeta \left(1+\frac{{\zeta }_t}{\zeta }\right)\right)\right]=w+{\displaystyle \sum _{i=1}^Q}{\alpha }_i\text{E}\left[\log \left(\zeta \left(1+\frac{{\zeta }_{t-i}}{\zeta }\right)\right)\right]$$ $$\text{E}[\log \zeta ]+\text{E}\left[\log \left(1+\frac{{\zeta }_t}{\zeta }\right)\right]=w+{\displaystyle \sum _{i=1}^Q}{\alpha }_i\left\{\text{E}[\log \zeta ]+\text{E}\left[\log \left(1+\frac{{\zeta }_{t-i}}{\zeta }\right)\right]\right\}$$ From (3.3) and by using expansion of $\mathit{log}(1+x)$ and since ${\zeta }_t^n\to 0\forall n⩾2$ , we get: $$\text{E}\left[\frac{{\zeta }_t}{\zeta }\right]={\displaystyle \sum _{i=1}^Q}{\alpha }_i\text{E}\left[\frac{{\zeta }_{t-i}}{\zeta }\right]$$ multiply both sides by $\zeta$ as a constant, we obtain: $$\text{E}\left[{\zeta }_t-{\displaystyle \sum _{i=1}^Q}{\alpha }_i{\zeta }_{t-i}\right]=0$$ this is a linear difference equation of order Q. If we can put the polynomial ${\zeta }_t-{\sum }_{i=1}^Q{\alpha }_i{\zeta }_{t-i}=0$ by using lag operator $(L^i)$ in the form: $${\zeta }_t\left(1-{\displaystyle \sum _{i=1}^Q}{\alpha }_i{L}^i\right)=0$$ then the stability requires that the all roots of the polynomial: $$P(L)=1-{\displaystyle \sum _{i=1}^Q}{\alpha }_i{L}^i$$ lie outside the unit circle which is by lemma (1) corresponding to the condition ${\sum }_{i=1}^Q{\alpha }_i<1$ . □

Example 3.1

By using Matlab (R2016a) software for modeling weekly volume trading S&P500 index from Dec. 2004–Oct. 2017 downloaded from yahoo finance website with 670 observation in EGARCH(1,1) model we notice the general form of the series data and the return series in the following figure (Fig. 1):We can check for the heteroskedasticity from the autocorrelation and partial autocorrelation function plot of the squared return errors of the series data also we used Ljung–Box Q-test to check the heteroskedasticity (Fig. 2).

After fitting the data with EGARCH(1,1) model we can see in Fig. 3 that the forecasted conditional variance ${\sigma }_t^2$ converge to the unconditional variance ${\sigma }^2$ which is corresponding with the non-zero singular point value as in Eq. (3.4).

Proposition 4.1

The limit cycle with period $k$ of EGARCH(1,1) is orbitally stable if the following condition satisfied: $$\left|{\displaystyle \prod _{i=1}^k}\text{E}\left[\frac{{\sigma }_{t+k-i}^2}{{\sigma }_{t+k-1-i}^2}\right]\right|<{\left|\frac{1}{{\alpha }_1}\right|}^k\text{.}$$

Proof

Suppose that the model possess a limit cycle of period $k$ namely: $${\sigma }_t^2\text{,}{\sigma }_{t-1}^2\text{,}{\sigma }_{t-2}^2\text{,}\dots \text{,}{\sigma }_{t-k}^2={\sigma }_t^2$$ near the neighborhood of each point of this limit cycle, let ${\zeta }_t$ be the radius of the neighborhood sufficiently small such that: $${\zeta }_t^n\to 0\mathit{for}n⩾2$$ by using the assignment equation $${\sigma }_s^2={\sigma }_s^2+{\zeta }_s\text{where}s=t\text{,}t-1$$ $$\text{and}{\zeta }_s\mathit{is}\text{independent with}{\sigma }_s^2$$ we substitute this equation in the model EGARCH(1,1) which is can get it by putting Q = P=1 in Eq. (3.2) and taking the mathematical expectation:

$$\text{E}[\log {\sigma }_t^2]=w+{\alpha }_1\text{E}[\log {\sigma }_{t-1}^2]$$ We get: $$\text{E}[\log ({\sigma }_t^2+{\zeta }_t)]=w+{\alpha }_1\text{E}[\log ({\sigma }_{t-1}^2+{\zeta }_{t-1})]$$ $$\text{E}\left[\log \left({\sigma }_t^2\left(1+\frac{{\zeta }_t}{{\sigma }_t^2}\right)\right)\right]=w+{\alpha }_1\text{E}\left[\log \left({\sigma }_{t-1}^2\left(1+\frac{{\zeta }_{t-1}}{{\sigma }_{t-1}^2}\right)\right)\right]$$ $$\text{E}[\log {\sigma }_t^2]+\text{E}\left[\text{log}\left(1+\frac{{\zeta }_t}{{\sigma }_t^2}\right)\right]=w+{\alpha }_1\left\{\text{E}[\log ({\sigma }_{t-1}^2)]+\text{E}\left[\mathit{log}\left(1+\frac{{\zeta }_{t-1}}{{\sigma }_{t-1}^2}\right)\right]\right\}$$ from (4.1) we get: $$\text{E}[\log {\sigma }_t^2]+\text{E}\left[\text{log}\left(1+\frac{{\zeta }_t}{{\sigma }_t^2}\right)\right]=\text{E}[\log {\sigma }_t^2]+{\alpha }_1\left\{\text{E}\left[\mathit{log}\left(1+\frac{{\zeta }_{t-1}}{{\sigma }_{t-1}^2}\right)\right]\right\}$$ and by using expansion of $\log (1+x)$ and since ${\zeta }_t^n\to 0\forall n⩾2$ , we obtain: $$E\left[\frac{{\zeta }_t}{{\sigma }_t^2}\right]={\alpha }_1\text{E}\left[\frac{{\zeta }_{t-1}}{{\sigma }_{t-1}^2}\right]$$ since ${\zeta }_s$ is independent with ${\sigma }_s^2$ , then: $$E[{\zeta }_t].\text{E}\left[\frac{1}{{\sigma }_t^2}\right]={\alpha }_1.\text{E}[{\zeta }_{t-1}].\text{E}\left[\frac{1}{{\sigma }_{t-1}^2}\right]$$ So:

(4.2)

$$E[{\zeta }_t]=\left(\frac{{\alpha }_1\text{E}\left[\frac{1}{{\sigma }_{t-1}^2}\right]}{\text{E}\left[\frac{1}{{\sigma }_t^2}\right]}\right).\text{E}[{\zeta }_{t-1}]$$ Eq. (4.2) is a difference equation with non-constant coefficients, and it is difficult to solve analytically but we need to knowing whether ${\zeta }_t$ in (4.2) converge to zero or not, and this can be verified by checking whether $\left|\frac{E[{\zeta }_{t+k}]}{E[{\zeta }_t]}\right|$ is less than 1 or not.

Now, by iterate Eq. (4.2) for $k$ times we get: $$E[{\zeta }_{t+k}]=\left(\frac{{\alpha }_1\text{E}\left[\frac{1}{{\sigma }_{t+k-1}^2}\right]}{\text{E}\left[\frac{1}{{\sigma }_{t+k}^2}\right]}\right).\text{E}[{\zeta }_{t+k-1}]$$ equivantely: $$E[{\zeta }_{t+k}]={\displaystyle \prod _{i=1}^k}\left(\frac{{\alpha }_1\text{E}\left[\frac{1}{{\sigma }_{t+k-1-i}^2}\right]}{\text{E}\left[\frac{1}{{\sigma }_{t+k-i}^2}\right]}\right).\text{E}[{\zeta }_t]$$ and hence the limit cycle is orbitally stable if: $$\left|\frac{\text{E}[{\zeta }_{t+k}]}{\text{E}[{\zeta }_t]}\right|=\left|{\alpha }_1^k{\displaystyle \prod _{i=1}^k}\text{E}\left[\frac{\frac{1}{{\sigma }_{t+k-1-i}^2}}{\frac{1}{{\sigma }_{t+k-i}^2}}\right]\right|<1$$ or in another form: $$\left|{\displaystyle \prod _{i=1}^k}\text{E}\left[\frac{{\sigma }_{t+k-i}^2}{{\sigma }_{t+k-1-i}^2}\right]\right|<{\left|\frac{1}{{\alpha }_1}\right|}^k\square$$