Roll with adverse selection

Let $m_t = m_{t-1} + \lambda q_t + \varepsilon_t$ and $p_t = m_t + c\,q_t$, with $q_t$ i.i.d. $\pm1$ equally likely, independent of the i.i.d. $\varepsilon_t$, and $c, \lambda \ge 0$. The dealer's quotes before trade $t$ are $m_{t-1} + \varepsilon_t \pm (c + \lambda)$. (a) Compute $\operatorname{Cov}(\Delta p_t, \Delta p_{t-1})$ and the Roll estimate of the spread. (b) Decompose the effective half-spread into realised half-spread and price impact as in (5.1), with the post-trade mid $m_t$. (c) Which component does the Roll estimator measure? Interpret.