Consecutive trade price changes in a stock have a first-order autocovariance of -0.000225 in dollars squared. Under the Roll model, what bid-ask spread does that imply, and what has gone wrong if the sample autocovariance comes out positive?

Consecutive trade price changes in a stock have a first-order autocovariance of -0.000225 in dollars squared. Under the Roll model, what bid-ask spread does that imply, and what has gone wrong if the sample autocovariance comes out positive?

Approach: Write the trade price as an efficient price plus a signed half spread with independent trade direction, then take the covariance of two successive price changes and invert it.

$0.03. Roll writes the trade price as p_t = m_t + c*q_t, with q_t equal to +1 or -1 with equal probability and independent of the efficient price m_t, which follows a random walk. Then Cov(Δp_t, Δp_{t-1}) = -c^2, so c = sqrt(0.000225) = 0.015 and the full bid-ask spread is 2c = $0.03. A positive sample autocovariance leaves the estimator undefined, since it can only arise when trade directions are serially correlated through order splitting or when the efficient price itself trends, and both break the independence and random walk assumptions. In practice the Roll estimate is biased low whenever one large order is worked in a single direction across many prints.

Follow-up: How do you correct the estimate when trade directions carry an autocorrelation of 0.3 from order splitting?

Key concepts: Roll model, autocovariance, bid-ask spread, random walk.