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

Question

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)?

Answer 1

This does not imply overestimation bias. We expect a negative autocorrelation in high- and ultra-high-frequency (every trade) data due to bid-ask bounce. Bounce occurs when buy and sell orders trading at the offer and bid are interspersed; that yields what seems to be returns even when the bid, ask, and midpoint do not change.

The Roll (1984) model examines this bounce in a theoretical market and determines that the negative autocovariance can be used to estimate the bid-ask spread.

What if ultra-high-frequency returns do not exhibit a negative autocovariance? That is rare and suggestive of very strong trending behavior. In that case, the best estimate of the bid-ask spread would be the minimum tick size -- since the Roll model offers you no useful information.