# Bid-Ask spread in Roll's model: Negative autocovariance of returns and informational content

Currently studying on techniques to estimate the bid-ask spread. Perhaps the most widely known model is the Roll model (1984). Let $$P_t$$ indicate log prices

$$\begin{cases} Bid_t=P_t-c, \\ Ask_t=P_t+c, \end{cases}$$

Where $$c=\sqrt{-Cov(\Delta P_t, \Delta P_{t-1})}$$.

My question comes down to the interpretation of autocovariance. My intuition is that positive autocorrelation in returns, imply presence of informed traders (Probability of having a trade in the same direction after a buy\sell is higher), and dealer sets a higher spread due to adverse selection. On the other, with negative autocorrelation, a trade is likely to be followed by a trade in the opposite direction (Sell after a Buy and vice versa) and dealer does not have adverse selection risk, so he\she should set a low spread. In the literature, some researchers take the absolute autocovariance to deal with undefined spread. Does this imply overestimation bias ("Expensive" spread when there is negative autocorrelation)?