= Solution
A method-of-moments estimator uses
$$
\widehat r_1=\frac{\widehat\gamma(1)}{\widehat\gamma(0)}
\approx\frac{\theta}{1+\theta^2}.
$$
Under $|\theta|<1$, choose the invertible root
$$
\widehat\theta_{\mathrm{MM}}
=\frac{1-\sqrt{1-4\widehat r_1^2}}{2\widehat r_1},
$$
with the continuous value zero when $\widehat r_1=0$. Alternatively maximize the exact Gaussian likelihood using the positive-definite covariance matrix from part (i), producing $\widehat\theta_{\mathrm{ML}}$. Given either estimate,
$$
\widehat\sigma^2=\frac{\widehat\gamma(0)}{1+\widehat\theta^2}
$$
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.
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