Adjacent products of independent Gaussian variables (${ϵ}_1={\eta }_1,{ϵ}_i={\eta }_{i-1}{\eta }_i$)

For independent standard normal random variables ${\eta }_1,\dots ,{\eta }_n$, define the displayed errors for $i\ge 2$. Each has mean zero and variance one. Nonadjacent errors depend on disjoint sets of independent variables. Adjacent covariances vanish because their product contains an independent centered factor. Thus their covariance matrix is $I_n$.

They are nevertheless dependent: the Gaussian fourth moment gives $\mathbb{E}[{ϵ}_i^2{ϵ}_{i+1}^2]=3$, and hence $\mathrm{Cov}({ϵ}_i^2,{ϵ}_{i+1}^2)=2$. Constant variance and zero pairwise covariance can conceal dependence in magnitudes. This is useful when interpreting a residual plot in linear regression.