hard · Market Microstructure
A stock has a true (efficient) price following a random walk with per-trade innovation variance σ_u^2. The observed transaction price is the efficient price plus a bid-ask bounce component, where each trade is buyer- or seller-initiated with equal probability and the half-spread is s. Under the Roll model, the first-order autocovariance of observed price changes is -s^2. An analyst estimates the implied spread from this autocovariance but the stock's order flow is actually positively autocorrelated (trade-direction indicator has serial correlation ρ > 0), violating Roll's independence assumption.
Relative to the true effective spread, how does the standard Roll estimator behave?
- It is biased downward, because positive order-flow autocorrelation makes the price-change autocovariance less negative (closer to zero), shrinking the spread implied by 2√(-Cov).
- It is biased upward, because persistent runs of same-side trades amplify the magnitude of the negative autocovariance and inflate 2√(-Cov).
- It remains unbiased, because the random-walk efficient price component is orthogonal to order flow and absorbs the autocorrelation, leaving the bounce covariance intact.
- It is biased downward only when the autocovariance turns positive and the estimator is undefined, but otherwise it correctly recovers the spread.
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