Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 2 / a / Solution

Only overlapping innovations contribute to the autocovariance function of the moving-average process. Set $c_0=1,c_1={\theta }_1,c_2={\theta }_2$. The overlap formula is ${\gamma }_k={\sigma }^2{\sum }_j{c}_j{c}_{j+|k|}$, with coefficients outside $0,1,2$ set to zero. ThusThe time-series spectral density is the Fourier sum of this finite covariance sequence:It also equals ${\sigma }^2|1+{\theta }_1{e}^{-i\lambda }+{\theta }_2{e}^{-2i\lambda }{|}^2/(2\pi )$, which directly verifies nonnegativity. Integrating from $-\pi$ gives the requested time-series spectral distribution:Extend it by zero below $-\pi$ and by ${\gamma }_0$ above $\pi$. In particular its total mass is the process variance, not necessarily one.