 Parity decomposition of a lag-two ARCH process

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.