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

An MA($q$) process iswhere $({\epsilon }_t)$ is white noise of variance ${\sigma }^2$. For MA(1), $X_t={\epsilon }_t+\theta {\epsilon }_{t-1}$ andThe transformation $(\theta ,{\sigma }^2)\mapsto (1/\theta ,{\theta }^2{\sigma }^2)$ leaves this autocovariance unchanged. A Gaussian process is determined by its mean and covariance, so the parameters are not identifiable unless one selects, for example, the invertible representative $|\theta |<1$.