= Solution
For the <moving-average process of order one>, $\gamma(0)=\sigma^2(1+\theta_1^2)$, $\gamma(1)=\sigma^2\theta_1$, and $\gamma(2)=0$. Regressing each endpoint on the single intervening observation leaves residual <covariance> $\gamma(2)-\gamma(1)^2/\gamma(0)$ and residual <variance> $\gamma(0)-\gamma(1)^2/\gamma(0)$. Therefore its lag-two <partial autocorrelation function> is
$$
\alpha(2)=\frac{\rho(2)-\rho(1)^2}{1-\rho(1)^2}
=\boxed{-\frac{\theta_1^2}{1+\theta_1^2+\theta_1^4}.}
$$
The denominator is positive. In particular, the moving-average autocorrelation cutoff at lag one does not make the lag-two partial correlation zero unless $\theta_1=0$.
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