= Uncorrelated returns with nonlinear predictability
{title2=$X_t=a_{t-2}a_{t-1}+a_t$}
For independent Rademacher signs $a_t$, these increments have mean zero, variance two and zero pairwise nonzero-lag covariance. Nevertheless $\mathbb E[X_tX_{t-1}X_{t-2}]=1$, so they are dependent. The observed past feature $Z_t=X_{t-1}X_{t-2}$ has $\mathbb E[X_tZ_t]=1$ and $\mathbb EZ_t^2=4$; its least-squares coefficient for forecasting $X_t$ is $1/4$. Higher-order dependence can therefore carry predictive information when a return autocorrelation plot is flat.
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