(Variance ratio of an MA(1))

Bid–ask bounce and stale prices are often modelled by MA(1) returns $r_t = \mu + \varepsilon_t + \theta\varepsilon_{t-1}$ with white-noise $\varepsilon_t$. (a) Compute $\rho_1$ and $\rho_j$ for $j \ge 2$. (b) Derive $\operatorname{Var}(r_t(k))$ and $VR(q)$ in closed form, and the limit as $q \to \infty$. (c) Evaluate $VR(5)$ and the limit for $\theta = -0.2$ and $\theta = 0.3$. (d) Contrast the shape of $VR(q)$ in $q$ with the AR(1) case of Proposition 2.7.