= Solution
An MA($q$) process is
$$
X_t=\varepsilon_t+\theta_1\varepsilon_{t-1}+\cdots+\theta_q\varepsilon_{t-q},
$$
where $(\varepsilon_t)$ is white noise of variance $\sigma^2$. For MA(1), $X_t=\varepsilon_t+\theta\varepsilon_{t-1}$ and
$$
\gamma(0)=\sigma^2(1+\theta^2),\qquad
\gamma(1)=\gamma(-1)=\sigma^2\theta,
\qquad\gamma(h)=0\ (|h|>1).
$$
The 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 $|\theta|<1$.
Back to article page
