Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 6 / c

Since $X_1$ is the sum of two independent centered Gaussian innovations,For any nonzero $a\in {\mathbb{R}}^n$,Expanding each $X_t={\epsilon }_t+\theta {\epsilon }_{t-1}$ expresses this as ${\sigma }^2$ times a sum of squared innovation coefficients. If all coefficients vanished, the coefficient of the latest innovation gives $a_n=0$, and backward induction gives every $a_t=0$, a contradiction. Hence $a^T\Sigma a>0$ and the covariance matrix is positive definite.

A method-of-moments estimator usesUnder $|\theta |<1$, choose the invertible rootwith the continuous value zero when ${\hat{r}}_1=0$. Alternatively maximize the exact Gaussian likelihood using the positive-definite covariance matrix from part (i), producing ${\hat{\theta }}_{\mathrm{ML}}$. Given either estimate,is the moment estimate; likelihood estimation may instead profile ${\sigma }^2$. Under a fixed interior parameter $|\theta |<1$ and standard stationary ergodic finite-moment regularity, both estimators are consistent and asymptotically normal, with Gaussian maximum likelihood asymptotically efficient.