 Uncorrelated returns with nonlinear predictability

For independent Rademacher signs $a_t$, these increments have mean zero, variance two and zero pairwise nonzero-lag covariance. Nevertheless $\mathbb{E}[X_t{X}_{t-1}{X}_{t-2}]=1$, so they are dependent. The observed past feature $Z_t=X_{t-1}{X}_{t-2}$ has $\mathbb{E}[X_t{Z}_t]=1$ and $\mathbb{E}{Z}_t^2=4$; 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.