Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 37 / 3 / c / Solution

For the moving-average process of order one, $\gamma (0)={\sigma }^2(1+{\theta }_1^2)$, $\gamma (1)={\sigma }^2{\theta }_1$, and $\gamma (2)=0$. Regressing each endpoint on the single intervening observation leaves residual covariance $\gamma (2)-\gamma (1{)}^2/\gamma (0)$ and residual variance $\gamma (0)-\gamma (1{)}^2/\gamma (0)$. Therefore its lag-two partial autocorrelation function isThe denominator is positive. In particular, the moving-average autocorrelation cutoff at lag one does not make the lag-two partial correlation zero unless ${\theta }_1=0$.