Autocovariance of an MA(1) process (${\gamma }_0={\sigma }^2(1+{\theta }^2),{\gamma }_1={\sigma }^2\theta$)

Only observations at lag one share a noise term. Thus ${\gamma }_0={\sigma }^2(1+{\theta }^2)$, ${\gamma }_{\pm 1}={\sigma }^2\theta$, and every other lag has zero covariance. This tridiagonal finite covariance structure makes the finite-sample innovations of an MA(1) process particularly simple.