(MA(1) and the bid-ask bounce)

Let $X_t = e_t + \theta e_{t-1}$. (a) Show $\rho_1 = \theta/(1+\theta^2)$ and $\rho_k = 0$ for $k\ge2$, and prove $|\rho_1|\le 1/2$. (b) Show that $\theta$ and $1/\theta$ give the same autocorrelations, and explain why one restricts to $|\theta|<1$. (c) Price changes of a stock that trades alternately at the bid and the ask have negative first-order autocorrelation (the Roll model, M05). If the sample $\hat\rho_1$ of a stock's tick-by-tick price changes is $-0.3$, what is the invertible $\hat\theta$?