Autoregressive conditional heteroscedasticity (${\sigma }_t^2={\alpha }_0+{\sum }_j{\alpha }_j{X}_{t-j}^2$)

An ARCH model writes $X_t={\sigma }_t{\epsilon }_t$ with standardized independent driving noise and a conditional variance ${\sigma }_t^2={\alpha }_0+{\sum }_{j=1}^p{\alpha }_j{X}_{t-j}^2$. Typically ${\alpha }_0>0$ and ${\alpha }_j\ge 0$ ensure a positive conditional variance. It allows a changing conditional scale even when $X$ is a stationary process.

With symmetric driving noise whose sign is independent of its magnitude, changing the sign of one driving variable changes the corresponding $X_t$ but leaves future conditional scales unchanged. Consequently $\mathrm{Cov}(X_t,f(X_{t+h}))=0$ for $h>0$ and square-integrable $f(X_{t+h})$. This conclusion uses symmetry, beyond the martingale difference sequence property.

For ${\alpha }_0>0$ and $0<{\alpha }_2<1$, the model $X_t=\sqrt{{\alpha }_0+{\alpha }_2{X}_{t-2}^2}{\epsilon }_t$ splits into independent even-time and odd-time chains in its stationary causal solution. With standard normal noise, $\mathbb{E}{X}_t^2={\alpha }_0/(1-{\alpha }_2)$, and its fourth moment is finite exactly when $3{\alpha }_2^2<1$.

The stationary squared process has the positive series $X_t^2={\alpha }_0{\sum }_{k\ge 0}{\alpha }_2^k{\prod }_{j=0}^k{\epsilon }_{t-2j}^2$. Even and odd observations therefore use disjoint families of independent random variables. This explains zero lag-one covariance of the squares despite dependence at lag two.

Volatility clustering means that large absolute observations tend to be followed by further large absolute observations, and small ones by small ones. In an ARCH process, persistence is in the conditional variance rather than necessarily in the signed observations.