Auto-Regressive & Conditionally Heteroskedatic

• def. Skedaticity = Variance
• def. Heteroskedatic = Different Variance Motivation. There are in practice many time series where there are clusters of high volatility, and clusters of low volatility. See for example Stylized Facts of Financial Return Series. Thus we have a model to address that. def. Auto-regressive Conditionally Heteroscedatic (ARCH) Process. Let:

1. “Random Source” $\{Z_t\}$ be $\text{SWN}(0,1)$
   1. We may/may not model a distribution for this. If we do it’s often Normal or Student’s T.
2. “Volatility Setting” $\{{\sigma }_t\}$ be a always-positive process defined as: ${\sigma }_t^2={\alpha }_0+{\sum }_{i=1}^p{\alpha }_i{X}_{t-1}^2$
   1. By this definition, at time $t$ with information ${\mathcal{F}}_t$, ${\sigma }_{t+1}$ is already known, i.e. pre-visible.
3. $\{X_t\}$ is a process $X_t={\sigma }_t{Z}_t$ Properties.

• Assuming $Z_t\perp {\mathcal{F}}_{t-1}$ → $\mathbb{E}[X_t|{\mathcal{F}}_{t-1}]=0$
  • Thus Martingale Difference process
• Assuming $\{X_t\}$ is white noise → $\text{Var}[X_t|{\mathcal{F}}_{t-1}]={\sigma }_t^2$
  • …and since ${\sigma }_t^2$ is weighted sum of past $X_t$’s…
  • Thus the process makes volatility clusters

Stochastic Difference Equation §

Consider the squared of the $\text{ARCH}(1)$ process:

$$X_t={\alpha }_0{Z}_t^2+{\alpha }_1{Z}_t^2{X}_{t-1}^2$$

This is analyzed as a Stochastic Recurrence Relation, and solution given by:

$$X_t^2={\alpha }_0\sum _{i=1}^{\infty }\left({\alpha }_1^i\prod _{j=0}^i{Z}_{t-j}^2\right)$$

For this to be stationary: thm. $\text{ARCH}(1)$ is a white noise (=cov-stationary & uncorrelated) iff ${\alpha }_1<1$